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name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #20 to #21: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='topology'>Topology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a></strong> (<a class='existingWikiWord' href='/nlab/show/diff/general+topology'>point-set topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/point-free+topology'>point-free topology</a>)</p> <p>see also <em><a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/functional+analysis'>functional analysis</a></em> and <em><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></em></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology'>Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhood</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base for the topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood+base'>neighbourhood base</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>finer/coarser topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a>, <a class='existingWikiWord' href='/nlab/show/diff/interior'>interior</a>, <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation</a>, <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sobriety</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/uniformly+continuous+map'>uniformly continuous function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>embedding</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a>, <a class='existingWikiWord' href='/nlab/show/diff/subnet'>sub-net</a>, <a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href='Top#UniversalConstructions'>Universal constructions</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>initial topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a>, <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a>,</p> </li> <li> <p>fiber space, <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>space attachment</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a>, <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+telescope'>mapping telescope</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/colimits+of+normal+spaces'>colimits of normal spaces</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric space</a>, <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/metrisable+topological+space'>metrisable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a>, <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular space</a>, <a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact space</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+topological+space'>sequentially compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+compact+topological+space'>countably compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/sigma-compact+topological+space'>sigma-compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+paracompact+topological+space'>countably paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/strongly+compact+topological+space'>strongly compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second-countable space</a>, <a class='existingWikiWord' href='/nlab/show/diff/first-countable+space'>first-countable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+contractible+space'>locally contractible space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+connected+topological+space'>locally connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simply+connected+space'>simply-connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/semi-locally+simply-connected+topological+space'>locally simply-connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+space'>topological vector space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Banach+space'>Banach space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+space'>Hilbert space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a>, <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+space'>codiscrete space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Sierpinski+space'>Sierpinski space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/order+topology'>order topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/specialization+topology'>specialization topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Scott+topology'>Scott topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a>, <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a>, <a class='existingWikiWord' href='/nlab/show/diff/ball'>ball</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/torus'>torus</a>, <a class='existingWikiWord' href='/nlab/show/diff/annulus'>annulus</a>, <a class='existingWikiWord' href='/nlab/show/diff/M%C3%B6bius+strip'>Moebius strip</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/polytope'>polytope</a>, <a class='existingWikiWord' href='/nlab/show/diff/polyhedron'>polyhedron</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/projective+space'>projective space</a> (<a class='existingWikiWord' href='/nlab/show/diff/real+projective+space'>real</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+projective+space'>complex</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path'>path</a>, <a class='existingWikiWord' href='/nlab/show/diff/loop'>loop</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping spaces</a>: <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/topology+of+uniform+convergence'>topology of uniform convergence</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/diff/path+space'>path space</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Zariski+topology'>Zariski topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cantor+space'>Cantor space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mandelbrot+set'>Mandelbrot space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peano+curve'>Peano curve</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/line+with+two+origins'>line with two origins</a>, <a class='existingWikiWord' href='/nlab/show/diff/long+line'>long line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Sorgenfrey+line'>Sorgenfrey line</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/K-topology'>K-topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dowker+space'>Dowker space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Warsaw+circle'>Warsaw circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hawaiian+earring+space'>Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+implies+sober'>Hausdorff spaces are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/schemes+are+sober'>schemes are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+images+of+compact+spaces+are+compact'>continuous images of compact spaces are compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces'>closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact'>open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff'>quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+number+lemma'>Lebesgue number lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces'>sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+totally+bounded'>sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous'>continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+injections+are+embeddings'>closed injections are embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+maps+to+locally+compact+spaces+are+closed'>proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings'>injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+sigma-compact+spaces+are+paracompact'>locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+second-countable+spaces+are+sigma-compact'>locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+regular+spaces+are+paracompact'>second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/CW-complexes+are+paracompact+Hausdorff+spaces'>CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Urysohn%27s+lemma'>Urysohn's lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tietze+extension+theorem'>Tietze extension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tube+lemma'>tube lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael%27s+theorem'>Michael's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brouwer%27s+fixed+point+theorem'>Brouwer's fixed point theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+invariance+of+dimension'>topological invariance of dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jordan+curve+theorem'>Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/intermediate+value+theorem'>intermediate value theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/extreme+value+theorem'>extreme value theorem</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological homotopy theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>, <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+extension+property'>homotopy extension property</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+cofiber+sequence'>cofiber sequence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a></p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a><ul><li><a href='#the_category_of_pointed_topological_spaces'>The category of pointed topological spaces</a></li><li><a href='#ForgettingAndAdjoiningBasepoints'>Forgetting and adjoining basepoints</a></li><li><a href='#wedge_sum_and_smash_product'>Wedge sum and Smash product</a></li><li><a href='#mapping_cocones'>Mapping (co-)cones</a></li></ul></li><li><a href='#properties'>Properties</a><ul><li><a href='#general'>General</a></li><li><a href='#relation_to_onepoint_compactification'>Relation to one-point compactification</a></li><li><a href='#SmashMonoidalDiagonals'>Smash-monoidal diagonals</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>A <em>pointed topological space</em> (often <em><a class='existingWikiWord' href='/nlab/show/diff/pointed+space'>pointed space</a></em>, for short) is a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> equipped with a choice of one of its <a class='existingWikiWord' href='/nlab/show/diff/point'>points</a> (<a class='existingWikiWord' href='/nlab/show/diff/element'>elements</a>). If the inclusion of that point is a <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a> then one speaks of a <em><a class='existingWikiWord' href='/nlab/show/diff/well-pointed+topological+space'>well-pointed topological space</a></em>.</p> <p>Although this concept may seem simple, pointed topological spaces play a central role for instance in <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a> as domains for <a class='existingWikiWord' href='/nlab/show/diff/reduced+cohomology'>reduced</a> <a class='existingWikiWord' href='/nlab/show/diff/generalized+%28Eilenberg-Steenrod%29+cohomology'>generalized (Eilenberg-Steenrod) cohomology theories</a> and as an ingredient for the definition of <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectra</a>.</p> <p>One reason why pointed topological spaces are important is that the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> which they form is an intermediate stage in the <a class='existingWikiWord' href='/nlab/show/diff/stabilization'>stabilization</a> of <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a> (the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical homotopy theory of topological spaces</a>) to <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable homotopy theory</a>:</p> <p>The category of pointed topological spaces has a <a class='existingWikiWord' href='/nlab/show/diff/zero+object'>zero object</a> (the <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a> itself) and the canonical <a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a> on pointed spaces is the <em><a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash product</a></em>, which is non-<a class='existingWikiWord' href='/nlab/show/diff/cartesian+monoidal+category'>cartesian monoidal category</a>, in contrast to the plain <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product of topological space</a>.</p> <h2 id='definition'>Definition</h2> <p>A <em>pointed topological space</em> is a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,\tau)</annotation></semantics></math> equipped with a choice of point <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math>. A <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphism</a> between pointed topological space <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,x)</annotation></semantics></math> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(Y,y)</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f \colon X \to Y</annotation></semantics></math> which preserves the chosen basepoints in that <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>f(x) = y</annotation></semantics></math>.</p> <h3 id='the_category_of_pointed_topological_spaces'>The category of pointed topological spaces</h3> <p>Stated in the language of <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a>, this means that pointed topological spaces are the <em><a class='existingWikiWord' href='/nlab/show/diff/pointed+object'>pointed objects</a></em> in the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> of topological spaces. This is the <a class='existingWikiWord' href='/nlab/show/diff/under+category'>coslice category</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>Top^{\ast/}</annotation></semantics></math> of topological spaces “under” the <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>\ast</annotation></semantics></math>:</p> <p>an <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>Top^{\ast/}</annotation></semantics></math> is equivalently a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo lspace='verythinmathspace'>:</mo><mo>*</mo><mo>→</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x \colon \ast \to (X,\tau)</annotation></semantics></math>, which is equivalently just a choice of point in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, and a <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>Top^{\ast/}</annotation></semantics></math> is a morphism <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f \colon X \to Y</annotation></semantics></math> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> (hence a continuous function), such that this triangle <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>commutes</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mi>x</mi></mpadded></msup><mo>↙</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mpadded width='0'><mi>y</mi></mpadded></msup></mtd> <mtd /> <mtd /></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd /> <mtd><munder><mo>⟶</mo><mi>f</mi></munder></mtd> <mtd /> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && \ast \\ & {}^{\mathllap{x}}\swarrow && \searrow^{\mathrlap{y}} && \\ X && \underset{f}{\longrightarrow} && Y } </annotation></semantics></math></div> <p>which equivalently means that <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>f(x) = y</annotation></semantics></math>.</p> <h3 id='ForgettingAndAdjoiningBasepoints'>Forgetting and adjoining basepoints</h3> <div class='num_defn' id='BasePointAdjoined'> <h6 id='definition_2'>Definition</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>forgetful functor</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mo>→</mo><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top^{\ast/} \to Top</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> given by forming the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a> (<a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>) with a <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a> (“adjoining a base point”), this is denoted by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msub><mo stretchy='false'>)</mo> <mo>+</mo></msub><mo>≔</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⊔</mo><mo>*</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>Top</mi><mo>⟶</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (-)_+ \coloneqq (-) \sqcup \ast \;\colon \; Top \longrightarrow Top^{\ast/} \,. </annotation></semantics></math></div></div> <h3 id='wedge_sum_and_smash_product'>Wedge sum and Smash product</h3> <div class='num_example' id='WedgeSumAsCoproduct'> <h6 id='example'>Example</h6> <p>Given two pointed topological spaces <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,x)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(Y,y)</annotation></semantics></math>, then:</p> <ol> <li> <p>their <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian product</a> in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>Top^{\ast/}</annotation></semantics></math> is simply their <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \times Y</annotation></semantics></math> equipped with the <a class='existingWikiWord' href='/nlab/show/diff/pair'>pair</a> of basepoints <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X\times Y, (x,y))</annotation></semantics></math>;</p> </li> <li> <p>their <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>Top^{\ast/}</annotation></semantics></math> has to be computed using the second clause in <a href='pointed+object#LimitsAndColimitsOfPointedObjects'>this prop.</a>: since the point <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>\ast</annotation></semantics></math> has to be adjoined to the diagram, it is given not by the coproduct in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> (which is the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a>), but by the <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a> in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> of the form:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mi>x</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mi>y</mi></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd><mpadded lspace='-50%width' width='0'><mrow><msup><mrow /> <mrow><msub><mrow /> <mrow><mo stretchy='false'>(</mo><mi>po</mi><mo stretchy='false'>)</mo></mrow></msub></mrow></msup></mrow></mpadded></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><mo>∨</mo><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ \ast &\overset{x}{\longrightarrow}& X \\ {}^{\mathllap{y}}\downarrow &\mathclap{{}^{{}_{(po)}}}& \downarrow \\ Y &\longrightarrow& X \vee Y } \,. </annotation></semantics></math></div> <p>This is called the <em><a class='existingWikiWord' href='/nlab/show/diff/wedge+sum'>wedge sum</a></em> operation on pointed objects.</p> <p>This is the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a> of the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a> under the <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a> which identifies the two basepoints:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∨</mo><mi>Y</mi><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><mi>X</mi><mo>⊔</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>∼</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> X \vee Y \;\simeq\; (X \sqcup Y)/(x \sim y) </annotation></semantics></math></div></li> </ol> <p>Generally for a set <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mo stretchy='false'>(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{(X_i,x_i)\}_{i \in I}</annotation></semantics></math> of pointed topological spaces</p> <ol> <li> <p>their <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> is formed in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>, as the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> with the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>Tychonoff topology</a>, with the <a class='existingWikiWord' href='/nlab/show/diff/tuple'>tuple</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>i</mi></msub><msub><mo stretchy='false'>)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo>∈</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>(x_i)_{i \in I} \in \underset{i \in I}{\prod} X_i</annotation></semantics></math> of basepoints being the new basepoint;</p> </li> <li> <p>their <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> is formed by the <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a> in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> over the <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> with a basepoint adjoined, and is called the <a class='existingWikiWord' href='/nlab/show/diff/wedge+sum'>wedge sum</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∨</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\vee_{i \in I} X_i</annotation></semantics></math>, which is the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a> of the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a> with all the basepoints identified:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∨</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>X</mi> <mi>i</mi></msub><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mrow><mo>(</mo><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>X</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo stretchy='false'>/</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>∼</mo><msub><mi>x</mi> <mi>j</mi></msub><msub><mo stretchy='false'>)</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \underset{i \in I}{\vee} X_i \;\simeq\; \left(\underset{i \in I}{\sqcup} X_i\right)/(x_i \sim x_j)_{i,j \in I} \,. </annotation></semantics></math></div></li> </ol> </div> <div class='num_example'> <h6 id='example_2'>Example</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a>, then for every <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/quotient+object'>quotient</a> of its <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-skeleton by its <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n-1)</annotation></semantics></math>-skeleton is the <a class='existingWikiWord' href='/nlab/show/diff/wedge+sum'>wedge sum</a>, def. <a class='maruku-ref' href='#WedgeSumAsCoproduct'>1</a>, of <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-spheres, one for each <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-cell of <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mi>n</mi></msup><mo stretchy='false'>/</mo><msup><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>≃</mo><munder><mo>∨</mo><mrow><mi>i</mi><mo>∈</mo><msub><mi>I</mi> <mi>n</mi></msub></mrow></munder><msup><mi>S</mi> <mi>n</mi></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> X^n / X^{n-1} \simeq \underset{i \in I_n}{\vee} S^n \,. </annotation></semantics></math></div></div> <div class='num_defn' id='SmashProductOfPointedObjects'> <h6 id='definition_3'>Definition</h6> <p>The <em><a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash product</a></em> of pointed topological spaces is the <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>∧</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mo>×</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mo>⟶</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'> (-)\wedge(-) \;\colon\; Top^{\ast/} \times Top^{\ast/} \longrightarrow Top^{\ast/} </annotation></semantics></math></div> <p>given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∧</mo><mi>Y</mi><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><mo>*</mo><munder><mo>⊔</mo><mrow><mi>X</mi><mo>⊔</mo><mi>Y</mi></mrow></munder><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> X \wedge Y \;\coloneqq\; \ast \underset{X\sqcup Y}{\sqcup} (X \times Y) \,, </annotation></semantics></math></div> <p>hence by the <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a> in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math><span> of<del class='diffmod'> he</del><ins class='diffmod'> the</ins><del class='diffmod'> frm</del><ins class='diffmod'> form</ins></span></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo>⊔</mo><mi>Y</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>,</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><msub><mi>id</mi> <mi>Y</mi></msub><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>po</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><mo>∧</mo><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ X \sqcup Y &\overset{(id_X,y),(x,id_Y) }{\longrightarrow}& X \times Y \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& X \wedge Y } \,. </annotation></semantics></math></div> <p>In terms of the <a class='existingWikiWord' href='/nlab/show/diff/wedge+sum'>wedge sum</a> from def. <a class='maruku-ref' href='#WedgeSumAsCoproduct'>1</a>, this may be written concisely as the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a> (<a href='quotient+space#QuotientBySubspace'>this def</a>) of the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> by the <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a> constituted by the <a class='existingWikiWord' href='/nlab/show/diff/wedge+sum'>wedge sum</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∧</mo><mi>Y</mi><mo>≃</mo><mfrac><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><mrow><mi>X</mi><mo>∨</mo><mi>Y</mi></mrow></mfrac><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> X \wedge Y \simeq \frac{X\times Y}{X \vee Y} \,. </annotation></semantics></math></div></div> <p>t <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></p> <table><thead><tr><th>symbol</th><th>name</th><th>category theory</th></tr></thead><tbody><tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \times Y</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>Top^{\ast/}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∨</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \vee Y</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/wedge+sum'>wedge sum</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>Top^{\ast/}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∧</mo><mi>Y</mi><mo>=</mo><mfrac><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><mrow><mi>X</mi><mo>∨</mo><mi>Y</mi></mrow></mfrac></mrow><annotation encoding='application/x-tex'>X \wedge Y = \frac{X \times Y}{X \vee Y}</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash product</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a> in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>Top^{\ast/}</annotation></semantics></math></td></tr> </tbody></table> <div class='num_example' id='WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces'> <h6 id='example_3'>Example</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>Top</mi></mrow><annotation encoding='application/x-tex'>X, Y \in Top</annotation></semantics></math>, with <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>Y</mi> <mo>+</mo></msub><mo>∈</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>X_+,Y_+ \in Top^{\ast/}</annotation></semantics></math>, def. <a class='maruku-ref' href='#BasePointAdjoined'>1</a>, then</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mo>+</mo></msub><mo>∨</mo><msub><mi>Y</mi> <mo>+</mo></msub><mo>≃</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>⊔</mo><mi>Y</mi><msub><mo stretchy='false'>)</mo> <mo>+</mo></msub></mrow><annotation encoding='application/x-tex'>X_+ \vee Y_+ \simeq (X \sqcup Y)_+</annotation></semantics></math>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mo>+</mo></msub><mo>∧</mo><msub><mi>Y</mi> <mo>+</mo></msub><mo>≃</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><msub><mo stretchy='false'>)</mo> <mo>+</mo></msub></mrow><annotation encoding='application/x-tex'>X_+ \wedge Y_+ \simeq (X \times Y)_+</annotation></semantics></math>.</p> </li> </ul> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>By example <a class='maruku-ref' href='#WedgeSumAsCoproduct'>1</a>, <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mo>+</mo></msub><mo>∨</mo><msub><mi>Y</mi> <mo>+</mo></msub></mrow><annotation encoding='application/x-tex'>X_+ \vee Y_+</annotation></semantics></math> is given by the colimit in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> over the diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd /> <mtd /> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd /> <mtd><mo>↙</mo></mtd> <mtd /> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mspace width='thinmathspace' /><mspace width='thinmathspace' /></mtd> <mtd><mo>*</mo></mtd> <mtd /> <mtd /> <mtd /> <mtd><mo>*</mo></mtd> <mtd><mspace width='thinmathspace' /><mspace width='thinmathspace' /></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ && && \ast \\ && & \swarrow && \searrow \\ X &\,\,& \ast && && \ast &\,\,& Y } \,. </annotation></semantics></math></div> <p>This is clearly <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊔</mo><mo>*</mo><mo>⊔</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>A \sqcup \ast \sqcup B</annotation></semantics></math>. Then, by definition <a class='maruku-ref' href='#SmashProductOfPointedObjects'>2</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><msub><mi>X</mi> <mo>+</mo></msub><mo>∧</mo><msub><mi>Y</mi> <mo>+</mo></msub></mtd> <mtd><mo>≃</mo><mfrac><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>⊔</mo><mo>*</mo><mo stretchy='false'>)</mo><mo>×</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>⊔</mo><mo>*</mo><mo stretchy='false'>)</mo></mrow><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>⊔</mo><mo>*</mo><mo stretchy='false'>)</mo><mo>∨</mo><mo stretchy='false'>(</mo><mi>Y</mi><mo>⊔</mo><mo>*</mo><mo stretchy='false'>)</mo></mrow></mfrac></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><mfrac><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>⊔</mo><mi>X</mi><mo>⊔</mo><mi>Y</mi><mo>⊔</mo><mo>*</mo></mrow><mrow><mi>X</mi><mo>⊔</mo><mi>Y</mi><mo>⊔</mo><mo>*</mo></mrow></mfrac></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>⊔</mo><mo>*</mo><mspace width='thinmathspace' /><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} X_+ \wedge Y_+ & \simeq \frac{(X \sqcup \ast) \times (X \sqcup \ast)}{(X\sqcup \ast) \vee (Y \sqcup \ast)} \\ & \simeq \frac{X \times Y \sqcup X \sqcup Y \sqcup \ast}{X \sqcup Y \sqcup \ast} \\ & \simeq X \times Y \sqcup \ast \,. \end{aligned} </annotation></semantics></math></div></div> <div class='num_example' id='StandardReducedCyclinderInTop'> <h6 id='example_4'>Example</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>≔</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>⊂</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>I \coloneqq [0,1] \subset \mathbb{R}</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/diff/interval'>closed interval</a> with its <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean</a> <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>.</p> <p>Hence</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>I</mi> <mo>+</mo></msub><mo>∈</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'> I_+ \in Top^{\ast/} </annotation></semantics></math></div> <p>is the interval with a disjoint basepoint adjoined, def. <a class='maruku-ref' href='#BasePointAdjoined'>1</a>.</p> <p>Now for <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> any <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed topological space</a>, then the <a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash product</a> (def. <a class='maruku-ref' href='#SmashProductOfPointedObjects'>2</a>)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∧</mo><mo stretchy='false'>(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>x</mi> <mn>0</mn></msub><mo stretchy='false'>}</mo><mo>×</mo><mi>I</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> X \wedge (I_+) = (X \times I)/(\{x_0\} \times I) </annotation></semantics></math></div> <p>is the <strong><a class='existingWikiWord' href='/nlab/show/diff/reduced+cylinder'>reduced cylinder</a></strong> over <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>: the result of forming the ordinary <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a> over <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, and then identifying the interval over the basepoint of <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with the point.</p> <p>(Generally, any construction in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> properly adapted to pointed spaces is called the “reduced” version of the unpointed construction. Notably so for “<a class='existingWikiWord' href='/nlab/show/diff/reduced+suspension'>reduced suspension</a>” which we come to <a href='#MappingCones'>below</a>.)</p> <p>Just like the ordinary cylinder <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>X\times I</annotation></semantics></math> receives a canonical injection from the <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>⊔</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X \sqcup X</annotation></semantics></math> formed in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math>, so the reduced cyclinder receives a canonical injection from the coproduct <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>⊔</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X \sqcup X</annotation></semantics></math> formed in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>Top^{\ast/}</annotation></semantics></math>, which is the <a class='existingWikiWord' href='/nlab/show/diff/wedge+sum'>wedge sum</a> from example <a class='maruku-ref' href='#WedgeSumAsCoproduct'>1</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∨</mo><mi>X</mi><mo>⟶</mo><mi>X</mi><mo>∧</mo><mo stretchy='false'>(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> X \vee X \longrightarrow X \wedge (I_+) \,. </annotation></semantics></math></div></div> <h3 id='mapping_cocones'>Mapping (co-)cones</h3> <p>Recall that the <em><a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a></em> on a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a> of the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a> with the closed interval</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cone</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>0</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Cone(X) = (X \times [0,1])/( X \times \{0\} ) \,. </annotation></semantics></math></div> <p>If <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is pointed with basepoint <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math>, then the <em>reduced cone</em> is the further quotient by the copy of the interval over the basepoint</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cone</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>Cone</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Cone(X,x) = Cone(X) / ( \{x\} \times [0,1] ) \,. </annotation></semantics></math></div> <p>For <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f \colon X \to Y</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, then</p> <ol> <li> <p>the <em><a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a></em> of <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>attachment space</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>≔</mo><mi>Y</mi><msub><mo>∪</mo> <mi>f</mi></msub><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> Cyl(f) \coloneqq Y \cup_f Cyl(X) </annotation></semantics></math></div></li> <li> <p>the <em><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></em> of <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>attachment space</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cone</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>≔</mo><mi>Y</mi><msub><mo>∪</mo> <mi>f</mi></msub><mi>Cone</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> Cone(f) \coloneqq Y \cup_f Cone(X) </annotation></semantics></math></div></li> </ol> <p>accordingly if <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f \colon X \to Y</annotation></semantics></math> is a continuous function between pointed spaces which preserves the basepoint, then the analogous construction with the <a class='existingWikiWord' href='/nlab/show/diff/reduced+cylinder'>reduced cylinder</a> and the reduce cone, respectively, yield the <em>reduced mapping cyclinder</em> and the <em>reduced mapping cone</em>.</p> <p>We now say this again in terms of <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushouts</a>:</p> <div class='num_defn' id='MappingConeAndMappingCocone'> <h6 id='definition_4'>Definition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f \colon X \longrightarrow Y</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> between pointed spces, its <strong>reduced <a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></strong> is the space</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cone</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>≔</mo><mo>*</mo><munder><mo>⊔</mo><mi>X</mi></munder><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><munder><mo>⊔</mo><mi>X</mi></munder><mi>Y</mi></mrow><annotation encoding='application/x-tex'> Cone(f) \coloneqq \ast \underset{X}{\sqcup} Cyl(X) \underset{X}{\sqcup} Y </annotation></semantics></math></div> <p>in the <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimiting</a> <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>i</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd /> <mtd><msup><mo>↘</mo> <mpadded width='0'><mi>η</mi></mpadded></msup></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd /> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \array{ && X &\stackrel{f}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{i_1}} && \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) \\ \downarrow && & \searrow^{\mathrlap{\eta}} & \downarrow \\ {*} &\longrightarrow& &\longrightarrow& Cone(f) } \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cyl(X)</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/reduced+cylinder'>reduced cylinder</a> from def. <a class='maruku-ref' href='#StandardReducedCyclinderInTop'>4</a>.</p> </div> <div class='num_prop' id='ConeAndMappingCylinder'> <h6 id='proposition'>Proposition</h6> <p>The colimit appearing in the definition of the reduced <a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a> in def. <a class='maruku-ref' href='#MappingConeAndMappingCocone'>3</a> is equivalent to three consecutive <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushouts</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd> <mtd><mo stretchy='false'>(</mo><mi>po</mi><mo stretchy='false'>)</mo></mtd> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>i</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>po</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>po</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ && X &\stackrel{f}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cyl(f) \\ \downarrow &(po)& \downarrow & (po) & \downarrow \\ {*} &\longrightarrow& Cone(X) &\longrightarrow& Cone(f) } \,. </annotation></semantics></math></div> <p>The two intermediate objects appearing here are called</p> <ul> <li> <p>the plain <strong>reduced <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a></strong> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cone</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>≔</mo><mo>*</mo><munder><mo>⊔</mo><mi>X</mi></munder><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cone(X) \coloneqq \ast \underset{X}{\sqcup} Cyl(X)</annotation></semantics></math>;</p> </li> <li> <p>the <strong>reduced <a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a></strong> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>≔</mo><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><munder><mo>⊔</mo><mi>X</mi></munder><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Cyl(f) \coloneqq Cyl(X) \underset{X}{\sqcup} Y</annotation></semantics></math>.</p> </li> </ul> </div> <div class='num_defn' id='SuspensionAndLoopSpaceObject'> <h6 id='definition_5'>Definition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>X \in Top^{\ast/}</annotation></semantics></math> be any pointed topological space.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, def. <a class='maruku-ref' href='#ConeAndMappingCylinder'>1</a>, of <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding='application/x-tex'>X \to \ast</annotation></semantics></math> is called the <strong><a class='existingWikiWord' href='/nlab/show/diff/reduced+suspension'>reduced</a> <a class='existingWikiWord' href='/nlab/show/diff/suspension'>suspension</a></strong> of <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, denoted</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi><mi>X</mi><mo>=</mo><mi>Cone</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>→</mo><mo>*</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \Sigma X = Cone(X\to\ast)\,. </annotation></semantics></math></div> <p>Via prop. <a class='maruku-ref' href='#ConeAndMappingCylinder'>1</a> this is equivalently the coproduct of two copies of the cone on <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> over their base:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow /></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd> <mtd><mo stretchy='false'>(</mo><mi>po</mi><mo stretchy='false'>)</mo></mtd> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow /></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>po</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>po</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Σ</mi><mi>X</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ && X &\stackrel{}{\longrightarrow}& \ast \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cone(X) \\ \downarrow &(po)& \downarrow & (po) & \downarrow \\ {*} &\longrightarrow& Cone(X) &\longrightarrow& \Sigma X } \,. </annotation></semantics></math></div> <p>This is also equivalently the <a class='existingWikiWord' href='/nlab/show/diff/cofiber'>cofiber</a>f of <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(i_0,i_1)</annotation></semantics></math>, hence (example <a class='maruku-ref' href='#WedgeSumAsCoproduct'>1</a>) of the <a class='existingWikiWord' href='/nlab/show/diff/wedge+sum'>wedge sum</a> inclusion:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∨</mo><mi>X</mi><mo>=</mo><mi>X</mi><mo>⊔</mo><mi>X</mi><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow></mover><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mover><mo>⟶</mo><mrow><mi>cofib</mi><mo stretchy='false'>(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow></mover><mi>Σ</mi><mi>X</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> X \vee X = X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \overset{cofib(i_0,i_1)}{\longrightarrow} \Sigma X \,. </annotation></semantics></math></div></div> <div class='num_prop' id='ReducedSuspensionBySmashProductWithCircle'> <h6 id='proposition_2'>Proposition</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/reduced+suspension'>reduced suspension</a> objects (def. <a class='maruku-ref' href='#SuspensionAndLoopSpaceObject'>4</a>) induced from the standard <a class='existingWikiWord' href='/nlab/show/diff/reduced+cylinder'>reduced cylinder</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>∧</mo><mo stretchy='false'>(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(-)\wedge (I_+)</annotation></semantics></math> of example <a class='maruku-ref' href='#StandardReducedCyclinderInTop'>4</a> are isomorphic to the <a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash product</a> (def. <a class='maruku-ref' href='#SmashProductOfPointedObjects'>2</a>) with the [[circle] (the <a class='existingWikiWord' href='/nlab/show/diff/sphere'>1-sphere</a>)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>cofib</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>∨</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>∧</mo><mo stretchy='false'>(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>≃</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>∧</mo><mi>X</mi><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> cofib(X \vee X \to X \wedge (I_+)) \simeq S^1 \wedge X \,, </annotation></semantics></math></div></div> <div class='num_prop' id='UnreducedMappingConeAsReducedConeOfBasedPointAdjoined'> <h6 id='proposition_3'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f \colon X \longrightarrow Y</annotation></semantics></math> a morphism in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>, then its unreduced mapping cone with respect to the standard cylinder object <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>X \times I</annotation></semantics></math> def. \ref{TopologicalInterval}, is isomorphic to the reduced mapping cone, of the morphism <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mo>+</mo></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>X</mi> <mo>+</mo></msub><mo>→</mo><msub><mi>Y</mi> <mo>+</mo></msub></mrow><annotation encoding='application/x-tex'>f_+ \colon X_+ \to Y_+</annotation></semantics></math> (with a basepoint adjoined) with respect to the standard <a class='existingWikiWord' href='/nlab/show/diff/reduced+cylinder'>reduced cylinder</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cone</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>Cone</mi><mo stretchy='false'>(</mo><msub><mi>f</mi> <mo>+</mo></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Cone'(f) \simeq Cone(f_+) \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>By example <a class='maruku-ref' href='#WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces'>3</a>, <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cone</mi><mo stretchy='false'>(</mo><msub><mi>f</mi> <mo>+</mo></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cone(f_+)</annotation></semantics></math> is given by the colimit in <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> over the following diagram:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><mo>⊔</mo><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>id</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><mi>Y</mi><mo>⊔</mo><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi><mo>⊔</mo><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo stretchy='false'>)</mo><mo>⊔</mo><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd /> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd /> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy='false'>(</mo><msub><mi>f</mi> <mo>+</mo></msub><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast \\ \downarrow && \downarrow && \downarrow \\ X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast \\ \downarrow && && \downarrow \\ \ast &\longrightarrow& &\longrightarrow& Cone(f_+) } \,. </annotation></semantics></math></div> <p>We may factor the vertical maps to give</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><mo>⊔</mo><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>id</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><mi>Y</mi><mo>⊔</mo><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi><mo>⊔</mo><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo stretchy='false'>)</mo><mo>⊔</mo><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd /> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo><mo>⊔</mo><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd /> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>f</mi><msub><mo stretchy='false'>)</mo> <mo>+</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd /> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd /> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast \\ \downarrow && \downarrow && \downarrow \\ X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast \\ \downarrow && && \downarrow \\ \ast \sqcup \ast &\longrightarrow& &\longrightarrow& Cone'(f)_+ \\ \downarrow && && \downarrow \\ \ast &\longrightarrow& &\longrightarrow& Cone'(f) } \,. </annotation></semantics></math></div> <p>This way the top part of the diagram (using the <a class='existingWikiWord' href='/nlab/show/diff/pasting+law+for+pullbacks'>pasting law</a> to compute the colimit in two stages) is manifestly a cocone under the result of applying <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msub><mo stretchy='false'>)</mo> <mo>+</mo></msub></mrow><annotation encoding='application/x-tex'>(-)_+</annotation></semantics></math> to the diagram for the unreduced cone. Since <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msub><mo stretchy='false'>)</mo> <mo>+</mo></msub></mrow><annotation encoding='application/x-tex'>(-)_+</annotation></semantics></math> is itself given by a colimit, it preserves colimits, and hence gives the partial colimit <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cone</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>f</mi><msub><mo stretchy='false'>)</mo> <mo>+</mo></msub></mrow><annotation encoding='application/x-tex'>Cone'(f)_+</annotation></semantics></math> as shown. The remaining pushout then contracts the remaining copy of the point away.</p> </div> <h2 id='properties'>Properties</h2> <h3 id='general'>General</h3> <p>Most of the relevant constructions on pointed topological spaces are immediate specializations of the general construction discussed at <em><a class='existingWikiWord' href='/nlab/show/diff/pointed+object'>pointed object</a></em>.</p> <h3 id='relation_to_onepoint_compactification'>Relation to one-point compactification</h3> <div class='num_prop' id='OnePointCompactificationAndSmashProduct'> <h6 id='proposition_4'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/one-point+compactification+intertwines+Cartesian+product+with+smash+product'>one-point compactification intertwines Cartesian product with smash product</a>)</strong></p> <p>On the <a class='existingWikiWord' href='/nlab/show/diff/subcategory'>subcategory</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mi>LCHaus</mi></msub></mrow><annotation encoding='application/x-tex'>Top_{LCHaus}</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> on the <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact Hausdorff spaces</a> with <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper maps</a> between them, the <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> of <a class='existingWikiWord' href='/nlab/show/diff/one-point+compactification'>one-point compactification</a> (Prop. \ref{OnePointCompactificationFunctor})</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msup><mo stretchy='false'>)</mo> <mi>cpt</mi></msup><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>Top</mi> <mi>LCHaus</mi></msub><mo>⟶</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'> (-)^{cpt} \;\colon\; Top_{LCHaus} \longrightarrow Top^{\ast/} </annotation></semantics></math></div> <p>sends <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian products</a> (<a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological spaces</a>) to <a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash products</a> of <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed topological spaces</a>, hence constitutes a <a class='existingWikiWord' href='/nlab/show/diff/monoidal+functor'>strong monoidal functor</a>, in that there is a <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural</a> <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><msup><mo maxsize='1.2em' minsize='1.2em'>)</mo> <mi>cpt</mi></msup><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msup><mi>X</mi> <mi>cpt</mi></msup><mo>∧</mo><msup><mi>Y</mi> <mi>cpt</mi></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \big( X \times Y \big)^{cpt} \;\simeq\; X^{cpt} \wedge Y^{cpt} \,. </annotation></semantics></math></div></div> <p>This is briefly mentioned in, for instance, <a href='#Bredon93'>Bredon 93, p. 199</a>. The argument may be found spelled out in: <a href='https://math.stackexchange.com/a/1645794/58526'>MO:a/1645794/</a>, <a href='#Cutler20'>Cutler 20, Prop. 1.6</a>.</p> <h3 id='SmashMonoidalDiagonals'>Smash-monoidal diagonals</h3> <p>Write</p> <div class='maruku-equation' id='eq:SmashMonoidalCategoryOfPointedTopologicalSpaces'><span class='maruku-eq-number'>(1)</span><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>PointedTopologicalSpaces</mi><mo>,</mo><msup><mi>S</mi> <mn>0</mn></msup><mo>,</mo><mo>∧</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>∈</mo><mspace width='thickmathspace' /><mi>SymmetricMonoidalCategories</mi></mrow><annotation encoding='application/x-tex'> \big( PointedTopologicalSpaces, S^0, \wedge \big) \;\;\in\; SymmetricMonoidalCategories </annotation></semantics></math></div> <ul> <li> <p>for the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> of <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed topological spaces</a> (with respect to some <a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of topological spaces</a> such as <a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated topological spaces</a> or <a class='existingWikiWord' href='/nlab/show/diff/Delta-generated+topological+space'>D-topological spaces</a>)</p> </li> <li> <p>regarded as a <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric monoidal category</a> with tensor product the <a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash product</a> and unit the <a class='existingWikiWord' href='/nlab/show/diff/0-sphere'>0-sphere</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup><mspace width='thinmathspace' /><mo>=</mo><mspace width='thinmathspace' /><msub><mo>*</mo> <mo>+</mo></msub></mrow><annotation encoding='application/x-tex'>S^0 \,=\, \ast_+</annotation></semantics></math>.</p> </li> </ul> <p>This category also has a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian product</a>, given on pointed spaces <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub><mo>=</mo><mo stretchy='false'>(</mo><msub><mi>𝒳</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X_i = (\mathcal{X}_i, x_i)</annotation></semantics></math> with underlying <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒳</mi> <mi>i</mi></msub><mo>∈</mo><mi>TopologicalSpaces</mi></mrow><annotation encoding='application/x-tex'>\mathcal{X}_i \in TopologicalSpaces</annotation></semantics></math> by</p> <div class='maruku-equation' id='eq:CartesianProductOfPointedTopologicalSpaces'><span class='maruku-eq-number'>(2)</span><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub><mspace width='thickmathspace' /><mo>=</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><msub><mi>𝒳</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>×</mo><mo stretchy='false'>(</mo><msub><mi>𝒳</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mi>𝒳</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>𝒳</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> X_1 \times X_2 \;=\; (\mathcal{X}_1, x_1) \times (\mathcal{X}_2, x_2) \;\coloneqq\; \big( \mathcal{X}_1 \times \mathcal{X}_2 , (x_1, x_2) \big) \,. </annotation></semantics></math></div> <p>But since this <a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash product</a> is a non-trivial <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient</a> of the Cartesian product</p> <div class='maruku-equation' id='eq:SmashProductOfPointedSpacesQuotientDefinition'><span class='maruku-eq-number'>(3)</span><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>X</mi> <mn>1</mn></msub><mspace width='thinmathspace' /><mo>≔</mo><mspace width='thinmathspace' /><mfrac><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>∨</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow></mfrac></mrow><annotation encoding='application/x-tex'> X_1 \wedge X_1 \,\coloneqq\, \frac{X_1 \times X_2}{ X_1 \vee X_2 } </annotation></semantics></math></div> <p>it is not itself <a class='existingWikiWord' href='/nlab/show/diff/cartesian+monoidal+category'>cartesian</a>, but just <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric monoidal</a>.</p> <p>However, via the quotienting <a class='maruku-eqref' href='#eq:SmashProductOfPointedSpacesQuotientDefinition'>(3)</a>, it still inherits, from the <a class='existingWikiWord' href='/nlab/show/diff/diagonal+morphism'>diagonal morphisms</a> on underlying topological spaces</p> <div class='maruku-equation' id='eq:CartesianDiagonalOnTopologicalSpaces'><span class='maruku-eq-number'>(4)</span><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒳</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>Δ</mi> <mi>𝒳</mi></msub></mrow></mover></mtd> <mtd><mi>𝒳</mi><mo>×</mo><mi>𝒳</mi></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ \mathcal{X} &\overset{ \Delta_{\mathcal{X}} }{\longrightarrow}& \mathcal{X} \times \mathcal{X} \\ x &\mapsto& (x,x) } </annotation></semantics></math></div> <p>a suitable notion of <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category+with+diagonals'>monoidal diagonals</a>:</p> <p>\begin{definition}\label{SmashMonoidalDiagonal} [Smash monoidal diagonals]</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mspace width='thinmathspace' /><mo>∈</mo><mspace width='thinmathspace' /><mi>PointedTopologicalSpaces</mi></mrow><annotation encoding='application/x-tex'>X \,\in\, PointedTopologicalSpaces</annotation></semantics></math>, let <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>⟶</mo><mi>X</mi><mo>∧</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>D_X \;\colon\; X \longrightarrow X \wedge X</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/diff/composition'>composite</a></p> <p>\begin{xymatrix@C=7pt} X \ar@/_1.6pc/[rrrrr]|-{ \;D_X\; } \ar[rr]^-{\Delta_X} && X \times X \ar[rr] && \frac{X \times X}{ X \vee X} \ar@{}[r]|-{ =: } & X \wedge X \end{xymatrix}</p> <p>of the Cartesian <a class='existingWikiWord' href='/nlab/show/diff/diagonal+morphism'>diagonal morphism</a> <a class='maruku-eqref' href='#eq:CartesianProductOfPointedTopologicalSpaces'>(2)</a> with the <a class='existingWikiWord' href='/nlab/show/diff/coprojection'>coprojection</a> onto the defining <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a> <a class='maruku-eqref' href='#eq:SmashProductOfPointedSpacesQuotientDefinition'>(3)</a>.</p> <p>\end{definition}</p> <p>It is immediate that:</p> <p>\begin{proposition} \label{SmashMonoidalDiaginalsAreMonoidalDiagonals} The smash monoidal diagonal <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> (Def. \ref{SmashMonoidalDiagonal}) makes the <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric monoidal category</a> <a class='maruku-eqref' href='#eq:SmashMonoidalCategoryOfPointedTopologicalSpaces'>(1)</a> of <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed topological spaces</a> with <a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash product</a> a <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category+with+diagonals'>monoidal category with diagonals</a>, in that</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup><mover><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msub><mi>D</mi> <mrow><msup><mi>S</mi> <mn>0</mn></msup></mrow></msub><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><msup><mi>S</mi> <mn>0</mn></msup><mo>∧</mo><msup><mi>S</mi> <mn>0</mn></msup></mrow><annotation encoding='application/x-tex'>S^0 \overset{\;\;D_{S^0}\;\;}{\longrightarrow} S^0 \wedge S^0</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>.</p> </li> </ol> <p>\end{proposition}</p> <p>While elementary in itself, this has the following profound consequence:</p> <p>\begin{remark}[Suspension spectra have diagonals]</p> <p>Since the <a class='existingWikiWord' href='/nlab/show/diff/suspension+spectrum'>suspension spectrum</a>-<a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>PointedTopologicalSpaces</mi><mo>⟶</mo><mi>HighlyStructuredSpectra</mi></mrow><annotation encoding='application/x-tex'> \Sigma^\infty \;\colon\; PointedTopologicalSpaces \longrightarrow HighlyStructuredSpectra </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/monoidal+functor'>strong monoidal functor</a> from <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed topological spaces</a> <a class='maruku-eqref' href='#eq:SmashMonoidalCategoryOfPointedTopologicalSpaces'>(1)</a> to any standard category of <a class='existingWikiWord' href='/nlab/show/diff/highly+structured+spectrum'>highly structured spectra</a> (by <a href='Introduction+to+Stable+homotopy+theory+--+1-2#SmashProductOfFreeSpectra'>this Prop.</a>) it follows that <em><a class='existingWikiWord' href='/nlab/show/diff/suspension+spectrum'>suspension spectra</a> have <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category+with+diagonals'>monoidal diagonals</a></em>, in the form of <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformations</a></p> <div class='maruku-equation' id='eq:SmashMonoidalDiagonalOnSuspensionSpectra'><span class='maruku-eq-number'>(5)</span><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mover><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msup><mi>Σ</mi> <mn>∞</mn></msup><mo stretchy='false'>(</mo><msub><mi>D</mi> <mi>X</mi></msub><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>∧</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'> \Sigma^\infty X \overset{ \;\; \Sigma^\infty(D_X) \;\; }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big) </annotation></semantics></math></div> <p>to their respective <a class='existingWikiWord' href='/nlab/show/diff/symmetric+smash+product+of+spectra'>symmetric smash product of spectra</a>, which hence makes them into <em><a class='existingWikiWord' href='/nlab/show/diff/comonoid'>comonoid objects</a></em>, namely <em><a class='existingWikiWord' href='/nlab/show/diff/coring+spectrum'>coring spectra</a></em>.</p> <p>For example, given a <a class='existingWikiWord' href='/nlab/show/diff/generalized+%28Eilenberg-Steenrod%29+cohomology'>Whitehead-generalized cohomology theory</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>E</mi><mo>˜</mo></mover></mrow><annotation encoding='application/x-tex'>\widetilde E</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Brown+representability+theorem'>represented</a> by a <a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>ring spectrum</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>E</mi><mo>,</mo><msup><mn>1</mn> <mi>E</mi></msup><mo>,</mo><msup><mi>m</mi> <mi>E</mi></msup><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>∈</mo><mspace width='thickmathspace' /><mi>SymmetricMonoids</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Spectra</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>𝕊</mi><mo>,</mo><mo>∧</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'> \big(E, 1^E, m^E \big) \;\; \in \; SymmetricMonoids \big( Ho(Spectra), \mathbb{S}, \wedge \big) </annotation></semantics></math></div> <p>the smash-monoidal diagonal structure <a class='maruku-eqref' href='#eq:SmashMonoidalDiagonalOnSuspensionSpectra'>(5)</a> on suspension spectra serves to define the <a class='existingWikiWord' href='/nlab/show/diff/cup+product'>cup product</a> <math class='maruku-mathml' display='inline' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(-)\cup (-)</annotation></semantics></math> in the corresponding <a class='existingWikiWord' href='/nlab/show/diff/multiplicative+cohomology+theory'>multiplicative cohomology theory structure</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0b6164ae267e8c6ba1301ac25d715039573f3598_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd /> <mtd><mo maxsize='1.2em' minsize='1.2em'>[</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>c</mi> <mi>i</mi></msub></mrow></mover><msup><mi>Σ</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mi>E</mi><mo maxsize='1.2em' minsize='1.2em'>]</mo><mspace width='thinmathspace' /><mo>∈</mo><mspace width='thinmathspace' /><mover><mi>E</mi><mo>˜</mo></mover><msup><mrow /> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>⇒</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo stretchy='false'>[</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>]</mo><mo>∪</mo><mo stretchy='false'>[</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo>≔</mo><mspace width='thinmathspace' /><mo maxsize='1.8em' minsize='1.8em'>[</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mover><mo>⟶</mo><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mo stretchy='false'>(</mo><msub><mi>D</mi> <mi>X</mi></msub><mo stretchy='false'>)</mo></mrow></mover><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>∧</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mrow></mover><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>Σ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mi>E</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>∧</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>Σ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mi>E</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mover><mo>⟶</mo><mrow><msup><mi>m</mi> <mi>E</mi></msup></mrow></mover><msup><mi>Σ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mi>E</mi><mo maxsize='1.8em' minsize='1.8em'>]</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>∈</mo><mspace width='thinmathspace' /><mover><mi>E</mi><mo>˜</mo></mover><msup><mrow /> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} & \big[ \Sigma^\infty X \overset{c_i}{\longrightarrow} \Sigma^{n_i} E \big] \,\in\, {\widetilde E}{}^{n_i}(X) \\ & \Rightarrow \;\; [c_1] \cup [c_2] \, \coloneqq \, \Big[ \Sigma^\infty X \overset{ \Sigma^\infty(D_X) }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big) \overset{ ( c_1 \wedge c_2 ) }{\longrightarrow} \big( \Sigma^{n_1} E \big) \wedge \big( \Sigma^{n_2} E \big) \overset{ m^E }{\longrightarrow} \Sigma^{n_1 + n_2}E \Big] \;\; \in \, {\widetilde E}{}^{n_1+n_2}(X) \,. \end{aligned} </annotation></semantics></math></div> <p>\end{remark}</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pointed+simplicial+set'>pointed simplicial set</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/retractive+space'>retractive space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/diff/reduced+suspension'>reduced suspension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash product</a>, <a class='existingWikiWord' href='/nlab/show/diff/wedge+sum'>wedge sum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pointed+homotopy+type'>pointed homotopy type</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+pointed+topological+spaces'>classical model structure on pointed topological spaces</a></p> </li> </ul> <h2 id='references'>References</h2> <p>Textbook accounts:</p> <ul> <li id='GabrielZisman67'> <p><a class='existingWikiWord' href='/nlab/show/diff/Pierre+Gabriel'>Pierre Gabriel</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michel+Zisman'>Michel Zisman</a>, Chapters IV.4 and V.7 of <em><a class='existingWikiWord' href='/nlab/show/diff/Calculus+of+fractions+and+homotopy+theory'>Calculus of fractions and homotopy theory</a></em>, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) (<a href='https://www.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf'>pdf</a>)</p> </li> <li id='Bredon93'> <p><a class='existingWikiWord' href='/nlab/show/diff/Glen+Bredon'>Glen Bredon</a>, <em>Topology and Geometry</em>, Graduate Texts in Mathematics 139, Springer 1993 (<a href='https://link.springer.com/book/10.1007/978-1-4757-6848-0'>doi:10.1007/978-1-4757-6848-0</a>, <a href='http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf'>pdf</a>)</p> </li> </ul> <p>Review:</p> <ul> <li id='Cutler20'><a class='existingWikiWord' href='/nlab/show/diff/Tyrone+Cutler'>Tyrone Cutler</a>, <em>The category of pointed topological spaces</em> (2020) [[pdf](https://www.math.uni-bielefeld.de/~tcutler/pdf/Elementary%20Homotopy%20Theory%20II%20-%20The%20Pointed%20Category.pdf), <a class='existingWikiWord' href='/nlab/files/CutlerPointedTopologicalSpaces.pdf' title='pdf'>pdf</a>]</li> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on March 7, 2024 at 16:58:33. 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