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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"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> <h4 id="stable_homotopy_theory">Stable homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <p><em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+Homotopy+Theory">Introduction</a></em></p> <h1 id="ingredients">Ingredients</h1> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h1 id="contents">Contents</h1> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/stable+homotopy+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#history'>History</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#LongExactSequences'>Long fiber-cofiber sequences</a></li> <li><a href='#FromStableModelCategories'>From stable model categories and stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories</a></li> </ul> <li><a href='#examples'>Examples</a></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>Any <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> can be flattened, by ignoring higher morphisms, into a <a class="existingWikiWord" href="/nlab/show/1-category">1-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ho(C)</annotation></semantics></math> called its <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category</a>. The notion of a <em>triangulated structure</em> is designed to capture the additional structure canonically existing on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ho(C)</annotation></semantics></math> when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> has the property of being <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stable</a>. This structure can be described roughly as the data of an invertible <a class="existingWikiWord" href="/nlab/show/suspension+functor">suspension functor</a>, together with a collection of sequences called <em>distinguished triangles</em>, which behave like <a class="existingWikiWord" href="/nlab/show/decategorification">shadows</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+cofibre+sequence">homotopy (co)fibre sequences</a> in <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-categories">stable (infinity,1)-categories</a>, subject to various axioms.</p> <p>A central class of examples of triangulated categories are the <a class="existingWikiWord" href="/nlab/show/derived+categories">derived categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(\mathcal{A})</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>. These are the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy categories</a> of the stable <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories+of+chain+complexes">(∞,1)-categories of chain complexes</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>. However the notion also encompasses important examples coming from nonabelian contexts, like the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a>, which is the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of the <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stable</a> <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+of+spectra">(infinity,1)-category of spectra</a>. Generally, it seems that all triangulated categories appearing in nature arise as <a class="existingWikiWord" href="/nlab/show/homotopy+categories">homotopy categories</a> of <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-categories">stable (infinity,1)-categories</a> (though examples of “exotic” triangulated categories probably exist).</p> <p>By construction, passing from a <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stable (infinity,1)-category</a> to its <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> represents a serious loss of information. In practice, endowing the homotopy category with a triangulated structure is often sufficient for many purposes. However, as soon as one needs to remember the <a class="existingWikiWord" href="/nlab/show/homotopy+colimits">homotopy colimits</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+limits">homotopy limits</a> that existed in the <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stable (infinity,1)-category</a>, a triangulated structure is not enough. For example, even the <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> in a triangulated category is not functorial. Hence it is often necessary to work with some <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced</a> notion of triangulated category, like <a class="existingWikiWord" href="/nlab/show/stable+derivators">stable derivators</a>, <a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-categories">pretriangulated dg-categories</a>, <a class="existingWikiWord" href="/nlab/show/stable+model+categories">stable model categories</a> or <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-categories">stable (infinity,1)-categories</a>. See <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a> for more details.</p> <h2 id="history">History</h2> <p>The notion of triangulated category was developed by <a class="existingWikiWord" href="/nlab/show/Jean-Louis+Verdier">Jean-Louis Verdier</a> in his 1963 thesis under <a class="existingWikiWord" href="/nlab/show/Alexandre+Grothendieck">Alexandre Grothendieck</a>. His motivation was to axiomatize the structure existing on the <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> of an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>. Axioms similar to Verdier’s were given by <a class="existingWikiWord" href="/nlab/show/Albrecht+Dold">Albrecht Dold</a> and <a class="existingWikiWord" href="/nlab/show/Dieter+Puppe">Dieter Puppe</a> in a 1961 paper. A notable difference is that Dold-Puppe did not impose the octahedral axiom (TR4).</p> <h2 id="definition">Definition</h2> <p>The traditional definition of <em>triangulated category</em> is the following. But see remark <a class="maruku-ref" href="#RedundancyInDefinition"></a> below.</p> <div class="num_defn" id="TriangulatedCategory"> <h6 id="definition_2">Definition</h6> <p>A <strong>triangulated category</strong> is</p> <ul> <li> <p>an <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+translation">with (additive) translation</a>;</p> </li> <li> <p>equipped with a collection of <a class="existingWikiWord" href="/nlab/show/category+with+translation">triangles</a> called <strong><a class="existingWikiWord" href="/nlab/show/distinguished+triangles">distinguished triangles</a></strong> (dts)</p> </li> <li> <p>such that the following axioms hold</p> </li> </ul> <p><strong>TR0:</strong> every triangle isomorphic to a distinguished triangle is itself a distinguished triangle;</p> <p><strong>TR1:</strong> the triangle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>Id</mi> <mi>X</mi></msub></mrow></mover><mi>X</mi><mo>→</mo><mn>0</mn><mo>→</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \stackrel{Id_X}{\to} X \to 0 \to T X </annotation></semantics></math></div> <p>is a distinguished triangle;</p> <p><strong>TR2:</strong> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math>, there exists a distinguished triangle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi><mo>→</mo><mi>Z</mi><mo>→</mo><mi>T</mi><mi>X</mi><mspace width="thinmathspace"></mspace><mo>;</mo></mrow><annotation encoding="application/x-tex"> X \stackrel{f}{\to} Y \to Z \to T X \,; </annotation></semantics></math></div> <p><strong>TR3:</strong> a triangle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi><mover><mo>→</mo><mi>g</mi></mover><mi>Z</mi><mover><mo>→</mo><mi>h</mi></mover><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \stackrel{f}{\to} Y \stackrel{g}{\to} Z \stackrel{h}{\to} T X </annotation></semantics></math></div> <p>is a distinguished triangle precisely if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mover><mo>→</mo><mi>g</mi></mover><mi>Z</mi><mover><mo>→</mo><mi>h</mi></mover><mi>T</mi><mi>X</mi><mover><mo>→</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover><mi>T</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex"> Y \stackrel{g}{\to} Z \stackrel{h}{\to} T X \stackrel{-T(f)}{\to} T Y </annotation></semantics></math></div> <p>is a distinguished triangle;</p> <p><strong>TR4:</strong> given two distinguished triangles</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi><mover><mo>→</mo><mi>g</mi></mover><mi>Z</mi><mover><mo>→</mo><mi>h</mi></mover><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \stackrel{f}{\to} Y \stackrel{g}{\to} Z \stackrel{h}{\to} T X </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo><mover><mo>→</mo><mrow><mi>f</mi><mo>′</mo></mrow></mover><mi>Y</mi><mo>′</mo><mover><mo>→</mo><mrow><mi>g</mi><mo>′</mo></mrow></mover><mi>Z</mi><mo>′</mo><mover><mo>→</mo><mrow><mi>h</mi><mo>′</mo></mrow></mover><mi>T</mi><mi>X</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">X' \stackrel{f'}{\to} Y' \stackrel{g'}{\to} Z' \stackrel{h'}{\to} T X'</annotation></semantics></math></div> <p>and given morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mi>α</mi></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>β</mi></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>′</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>f</mi><mo>′</mo></mrow></mover></mtd> <mtd><mi>Y</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ X &\stackrel{f}{\to}& Y \\ \downarrow^\alpha && \downarrow^\beta \\ X' &\stackrel{f'}{\to}& Y' } </annotation></semantics></math></div> <p>there exists a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><mi>Z</mi><mo>→</mo><mi>Z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\gamma : Z \to Z'</annotation></semantics></math> extending this to a morphism of distinguished triangles in that the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>Z</mi></mtd> <mtd><mover><mo>→</mo><mi>h</mi></mover></mtd> <mtd><mi>T</mi><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mi>α</mi></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>β</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∃</mo><mi>γ</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>′</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>f</mi><mo>′</mo></mrow></mover></mtd> <mtd><mi>Y</mi><mo>′</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>g</mi><mo>′</mo></mrow></mover></mtd> <mtd><mi>Z</mi><mo>′</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>h</mi><mo>′</mo></mrow></mover></mtd> <mtd><mi>T</mi><mi>X</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{f}{\to}& Y &\stackrel{g}{\to}& Z &\stackrel{h}{\to}& T X \\ \downarrow^\alpha && \downarrow^{\mathrlap{\beta}} && \downarrow^{\mathrlap{\exists \gamma}} && \downarrow^{\mathrlap{T(\alpha)}} \\ X' &\stackrel{f'}{\to}& Y' &\stackrel{g'}{\to}& Z' &\stackrel{h'}{\to}& T X' } </annotation></semantics></math></div> <p>commutes;</p> <p><strong>TR5 (octahedral axiom):</strong> given three distinguished triangles of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi><mover><mo>→</mo><mi>h</mi></mover><mi>Y</mi><mo stretchy="false">/</mo><mi>X</mi><mover><mo>→</mo><mrow></mrow></mover><mi>T</mi><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>Y</mi><mover><mo>→</mo><mi>g</mi></mover><mi>Z</mi><mover><mo>→</mo><mi>k</mi></mover><mi>Z</mi><mo stretchy="false">/</mo><mi>Y</mi><mover><mo>→</mo><mrow></mrow></mover><mi>T</mi><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>X</mi><mover><mo>→</mo><mrow><mi>g</mi><mo>∘</mo><mi>f</mi></mrow></mover><mi>Z</mi><mover><mo>→</mo><mi>l</mi></mover><mi>Z</mi><mo stretchy="false">/</mo><mi>X</mi><mover><mo>→</mo><mrow></mrow></mover><mi>T</mi><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & X \stackrel{f}{\to} Y \stackrel{h}{\to} Y/X \stackrel{}{\to} T X \\ & Y \stackrel{g}{\to} Z \stackrel{k}{\to} Z/Y \stackrel{}{\to} T Y \\ & X \stackrel{g \circ f}{\to} Z \stackrel{l}{\to} Z/X \stackrel{}{\to} T X \end{aligned} </annotation></semantics></math></div> <p>there exists a distinguished triangle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo stretchy="false">/</mo><mi>X</mi><mover><mo>→</mo><mi>u</mi></mover><mi>Z</mi><mo stretchy="false">/</mo><mi>X</mi><mover><mo>→</mo><mi>v</mi></mover><mi>Z</mi><mo stretchy="false">/</mo><mi>Y</mi><mover><mo>→</mo><mi>w</mi></mover><mi>T</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Y/X \stackrel{u}{\to} Z/X \stackrel{v}{\to} Z/Y \stackrel{w}{\to} T (Y/X) </annotation></semantics></math></div> <p>such that the following big diagram commutes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mi>g</mi><mo>∘</mo><mi>f</mi></mrow></mover></mtd> <mtd></mtd> <mtd><mi>Z</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>k</mi></mover></mtd> <mtd></mtd> <mtd><mi>Z</mi><mo stretchy="false">/</mo><mi>Y</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>w</mi></mover></mtd> <mtd></mtd> <mtd><mi>T</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mi>f</mi></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mi>g</mi></msub></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mi>l</mi></msup></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mi>v</mi></msub></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mrow></mrow></msup></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mrow><mi>T</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Z</mi><mo stretchy="false">/</mo><mi>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>T</mi><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mi>h</mi></msup></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mi>u</mi></msub></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mrow></mrow></msup></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mrow><mi>T</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi><mo stretchy="false">/</mo><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd></mtd> <mtd><mi>T</mi><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{g \circ f}{\to}&& Z &&\stackrel{k}{\to}&& Z/Y &&\stackrel{w}{\to}&& T (Y/X) \\ & {}_{f}\searrow && \nearrow_{g} && \searrow^{l} && \nearrow_{v} && \searrow^{} && \nearrow_{T(h)} \\ && Y &&&& Z/X &&&& T Y \\ &&& \searrow^{h} && \nearrow_{u} && \searrow^{} && \nearrow_{T(f)} \\ &&&& Y/X &&\stackrel{}{\to}&& T X } </annotation></semantics></math></div> <p>If TR5 is not required, one speaks of a <strong>pretriangulated category</strong>.</p> </div> <div class="num_remark" id="RedundancyInDefinition"> <h6 id="remark">Remark</h6> <p>This classical definition is redundant; TR4 and one direction of TR3 follow from the remaining axioms. See (<a href="#May">May</a>).</p> <p>The octahedral axiom has many equivalent formulations, a concise account is in (<a href="#Hubery">Hubery</a>). Notice that one of the equivalent axioms, called axiom B, there, is essentially just an axiomatization of the existence of <a class="existingWikiWord" href="/nlab/show/homotopy+pushouts">homotopy pushouts</a>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>In the context of triangulated categories the translation functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">T : C \to C</annotation></semantics></math> is often called the <strong>suspension functor</strong> and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>:</mo><mi>X</mi><mo>↦</mo><mi>X</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(-)[1] : X \mapsto X[1]</annotation></semantics></math> (in an algebraic context) or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> (in a topological context). However unlike “translation functor”, suspension functor is also the term used when the invertibility is not assumed, cf. <a class="existingWikiWord" href="/nlab/show/suspended+category">suspended category</a>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,g,h)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/distinguished+triangle">distinguished triangle</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,g,-h)</annotation></semantics></math> is not generally distinguished, although it is “exact” (induces <a class="existingWikiWord" href="/nlab/show/long+exact+sequences">long exact sequences</a> in <a class="existingWikiWord" href="/nlab/show/homology">homology</a> and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>). However, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>g</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,-g,-h)</annotation></semantics></math> <em>is</em> always distinguished, since it is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,g,h)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>Z</mi></mtd> <mtd><mover><mo>→</mo><mi>h</mi></mover></mtd> <mtd><mi>T</mi><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo></mo><mi>id</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo></mo><mi>id</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo></mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>id</mi></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>g</mi></mrow></mover></mtd> <mtd><mi>Z</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>h</mi></mrow></mover></mtd> <mtd><mi>T</mi><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X & \xrightarrow{f} & Y & \xrightarrow{g} & Z & \xrightarrow{h} & T X\\ ^{id}\downarrow && ^{id} \downarrow && ^{-1} \downarrow && \downarrow^{id}\\ X & \xrightarrow{f} & Y & \xrightarrow{-g} & Z & \xrightarrow{-h} & T X} </annotation></semantics></math></div></div> <h2 id="properties">Properties</h2> <h3 id="LongExactSequences">Long fiber-cofiber sequences</h3> <p>The following prop. <a class="maruku-ref" href="#LongExactSequencesFromDistinguishedTriangle"></a> is the incarnation in the axiomatics of triangulated categories of the <a class="existingWikiWord" href="/nlab/show/long+exact+sequences+of+homotopy+groups">long exact sequences of homotopy groups</a> induced by <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequences">homotopy fiber sequences</a> in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <div class="num_lemma" id="CompositesInADistinguishedTriangleAreZero"> <h6 id="lemma">Lemma</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, def. <a class="maruku-ref" href="#TriangulatedCategory"></a>, and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>B</mi><mover><mo>⟶</mo><mi>g</mi></mover><mi>B</mi><mo stretchy="false">/</mo><mi>A</mi><mover><mo>⟶</mo><mi>h</mi></mover><mi>Σ</mi><mi>A</mi></mrow><annotation encoding="application/x-tex"> A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} B/A \overset{h}{\longrightarrow} \Sigma A </annotation></semantics></math></div> <p>a distinguished triangle, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∘</mo><mi>f</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> g\circ f = 0 </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero morphism</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Consider the <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>id</mi></mover></mtd> <mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mn>0</mn></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>Σ</mi><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi><mo stretchy="false">/</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>h</mi></mover></mtd> <mtd><mi>Σ</mi><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{id}{\longrightarrow}& A &\overset{}{\longrightarrow}& 0 &\overset{}{\longrightarrow}& \Sigma A \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}} \\ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A } \,. </annotation></semantics></math></div> <p>Observe that the top part is a distinguished triangle by axioms T1 and T2 in def. <a class="maruku-ref" href="#TriangulatedCategory"></a>. Hence by T3 there is an extension to a commuting diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>id</mi></mover></mtd> <mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mn>0</mn></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>Σ</mi><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>Σ</mi><mpadded width="0"><mi>id</mi></mpadded></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi><mo stretchy="false">/</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>h</mi></mover></mtd> <mtd><mi>Σ</mi><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{id}{\longrightarrow}& A &\overset{}{\longrightarrow}& 0 &\overset{}{\longrightarrow}& \Sigma A \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}} && \downarrow && \downarrow^{\mathrlap{\Sigma \mathrlap{id}}} \\ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A } \,. </annotation></semantics></math></div> <p>Now the commutativity of the middle square proves the claim.</p> </div> <div class="num_prop" id="LongExactSequencesFromDistinguishedTriangle"> <h6 id="proposition">Proposition</h6> <p>Consider a <a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, def. <a class="maruku-ref" href="#TriangulatedCategory"></a>, with shift functor denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> and with <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo lspace="verythinmathspace">:</mo><msup><mi>Ho</mi> <mi>op</mi></msup><mo>×</mo><mi>Ho</mi><mo>→</mo><mi>Ab</mi></mrow><annotation encoding="application/x-tex">[-,-]_\ast \colon Ho^{op}\times Ho \to Ab</annotation></semantics></math>. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any object, and for any distinguished triangle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>B</mi><mover><mo>⟶</mo><mi>g</mi></mover><mi>B</mi><mo stretchy="false">/</mo><mi>A</mi><mover><mo>⟶</mo><mi>h</mi></mover><mi>Σ</mi><mi>A</mi></mrow><annotation encoding="application/x-tex"> A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} B/A \overset{h}{\longrightarrow} \Sigma A </annotation></semantics></math></div> <p>the sequences of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a></p> <ol> <li> <p>(long cofiber sequence)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mi>A</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mover><mo>⟶</mo><mrow><mo stretchy="false">[</mo><mi>h</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow></mover><mo stretchy="false">[</mo><mi>B</mi><mo stretchy="false">/</mo><mi>A</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mover><mo>⟶</mo><mrow><mo stretchy="false">[</mo><mi>g</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow></mover><mo stretchy="false">[</mo><mi>B</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mover><mo>⟶</mo><mrow><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow></mover><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex"> [\Sigma A, X]_\ast \overset{[h,X]_\ast}{\longrightarrow} [B/A,X]_\ast \overset{[g,X]_\ast}{\longrightarrow} [B,X]_\ast \overset{[f,X]_\ast}{\longrightarrow} [A,X]_\ast </annotation></semantics></math></div></li> <li> <p>(long fiber sequence)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mover><mo>⟶</mo><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>f</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow></mover><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>B</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mover><mo>⟶</mo><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>g</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow></mover><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">/</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mover><mo>⟶</mo><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>h</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow></mover><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Σ</mi><mi>A</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex"> [X,A]_\ast \overset{[X,f]_\ast}{\longrightarrow} [X,B]_\ast \overset{[X,g]_\ast}{\longrightarrow} [X,B/A]_\ast \overset{[X,h]_\ast}{\longrightarrow} [X,\Sigma A]_\ast </annotation></semantics></math></div></li> </ol> <p>are <a class="existingWikiWord" href="/nlab/show/long+exact+sequences">long exact sequences</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Regarding the first case:</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∘</mo><mi>f</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g \circ f = 0</annotation></semantics></math> by lemma <a class="maruku-ref" href="#CompositesInADistinguishedTriangleAreZero"></a>, we have an inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>g</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>⊂</mo><mi>ker</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im([g,X]_\ast) \subset ker([f,X]_\ast)</annotation></semantics></math>. Hence it is sufficient to show that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\psi \colon B \to X</annotation></semantics></math> is in the kernel of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">[f,X]_\ast</annotation></semantics></math> in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>∘</mo><mi>f</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\psi \circ f = 0</annotation></semantics></math>, then there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \colon C \to X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>∘</mo><mi>g</mi><mo>=</mo><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\phi \circ g = \psi</annotation></semantics></math>. To that end, consider the commuting diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi><mo stretchy="false">/</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>h</mi></mover></mtd> <mtd><mi>Σ</mi><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ψ</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>id</mi></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A \\ \downarrow && {}^{\mathllap{\psi}}\downarrow \\ 0 &\overset{}{\longrightarrow}& X &\overset{id}{\longrightarrow}& X &\overset{}{\longrightarrow}& 0 } \,, </annotation></semantics></math></div> <p>where the commutativity of the left square exhibits our assumption.</p> <p>The top part of this diagram is a distinguished triangle by assumption, and the bottom part is by condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">T1</annotation></semantics></math> in def. <a class="maruku-ref" href="#TriangulatedCategory"></a>. Hence by condition T3 there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> fitting into a commuting diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi><mo stretchy="false">/</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>h</mi></mover></mtd> <mtd><mi>Σ</mi><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ψ</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ϕ</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>id</mi></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A \\ \downarrow && {}^{\mathllap{\psi}}\downarrow && \downarrow^{\mathrlap{\phi}} && \downarrow \\ 0 &\overset{}{\longrightarrow}& X &\overset{id}{\longrightarrow}& X &\overset{}{\longrightarrow}& 0 } \,. </annotation></semantics></math></div> <p>Here the commutativity of the middle square exhibits the desired conclusion.</p> <p>This shows that the first sequence in question is exact at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>B</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">[B,X]_\ast</annotation></semantics></math>. Applying the same reasoning to the distinguished traingle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>Σ</mi><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g,h,-\Sigma f)</annotation></semantics></math> provided by T2 yields exactness at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">[C,X]_\ast</annotation></semantics></math>.</p> <p>Regarding the second case:</p> <p>Again, from lemma <a class="maruku-ref" href="#CompositesInADistinguishedTriangleAreZero"></a> it is immediate that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>f</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>⊂</mo><mi>ker</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>g</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> im([X,f]_\ast) \subset ker([X,g]_\ast) </annotation></semantics></math></div> <p>so that we need to show that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\psi \colon X \to B</annotation></semantics></math> in the kernel of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>g</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">[X,g]_\ast</annotation></semantics></math>, hence such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∘</mo><mi>ψ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g\circ \psi = 0</annotation></semantics></math>, then there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\phi \colon X \to A</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>ϕ</mi><mo>=</mo><mi>ψ</mi></mrow><annotation encoding="application/x-tex">f \circ \phi = \psi</annotation></semantics></math>.</p> <p>To that end, consider the commuting diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Σ</mi><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>Σ</mi><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ψ</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi><mo stretchy="false">/</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>h</mi></mover></mtd> <mtd><mi>Σ</mi><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>Σ</mi><mi>f</mi></mrow></mover></mtd> <mtd><mi>Σ</mi><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\longrightarrow& 0 &\longrightarrow& \Sigma X &\overset{- id}{\longrightarrow}& \Sigma X \\ \downarrow^{\mathrlap{\psi}} && \downarrow \\ B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A &\overset{-\Sigma f}{\longrightarrow}& \Sigma B } \,, </annotation></semantics></math></div> <p>where the commutativity of the left square exhibits our assumption.</p> <p>Now the top part of this diagram is a distinguished triangle by conditions T1 and T2 in def. <a class="maruku-ref" href="#TriangulatedCategory"></a>, while the bottom part is a distinguished triangle by applying T2 to the given distinguished triangle. Hence by T3 there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ϕ</mi><mo stretchy="false">˜</mo></mover><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mi>X</mi><mo>→</mo><mi>Σ</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\tilde \phi \colon \Sigma X \to \Sigma A</annotation></semantics></math> such as to extend to a commuting diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Σ</mi><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>Σ</mi><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ψ</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>ϕ</mi><mo stretchy="false">˜</mo></mover></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>Σ</mi><mi>ψ</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi><mo stretchy="false">/</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>h</mi></mover></mtd> <mtd><mi>Σ</mi><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>Σ</mi><mi>f</mi></mrow></mover></mtd> <mtd><mi>Σ</mi><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\longrightarrow& 0 &\longrightarrow& \Sigma X &\overset{- id}{\longrightarrow}& \Sigma X \\ \downarrow^{\mathrlap{\psi}} && \downarrow && \downarrow^{\mathrlap{\tilde \phi}} && \downarrow^{\mathrlap{\Sigma \psi}} \\ B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A &\overset{-\Sigma f}{\longrightarrow}& \Sigma B } \,, </annotation></semantics></math></div> <p>At this point we appeal to the condition in def. <a class="maruku-ref" href="#TriangulatedCategory"></a> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo lspace="verythinmathspace">:</mo><mi>Ho</mi><mo>→</mo><mi>Ho</mi></mrow><annotation encoding="application/x-tex">\Sigma \colon Ho \to Ho</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>, so that in particular it is a <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful functor</a>. It being a <a class="existingWikiWord" href="/nlab/show/full+functor">full functor</a> implies that there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\phi \colon X \to A</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ϕ</mi><mo stretchy="false">˜</mo></mover><mo>=</mo><mi>Σ</mi><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\tilde \phi = \Sigma \phi</annotation></semantics></math>. It being faithful then implies that the whole commuting square on the right is the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> of a commuting square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ϕ</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ψ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>f</mi></mrow></munder></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{-id}{\longrightarrow}& X \\ {}^{\mathllap{\phi}}\downarrow && \downarrow^{\mathrlap{\psi}} \\ A &\underset{-f}{\longrightarrow}& B } \,. </annotation></semantics></math></div> <p>This exhibits the claim to be shown.</p> </div> <h3 id="FromStableModelCategories">From stable model categories and stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories</h3> <p>A <a class="existingWikiWord" href="/nlab/show/pointed+category">pointed</a> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is called a <em><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></em> if the canonically induced <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a>-functor on its <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Sigma \;\colon\; Ho(\mathcal{C}) \longrightarrow Ho(\mathcal{C}) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>.</p> <p>In this case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Ho(\mathcal{C}),\Sigma)</annotation></semantics></math> is a triangulated category. (<a href="#Hovey99">Hovey 99, section 7</a>, for review see also <a href="#Schwede">Schwede, section 2</a>).</p> <p>Similarly, the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy category</a> of a <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> is a triangulated category, see <a href="stable+infinity-category#TheTriangulatedHomotopyCategory">there</a>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+chain+complexes">homotopy category of chain complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(\mathcal{A})</annotation></semantics></math> in an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> (the category of chain complexes modulo <a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a>) is a triangulated category: the translation functor is the <a class="existingWikiWord" href="/nlab/show/suspension+of+chain+complexes">suspension of chain complexes</a> and the distinguished triangles are those coming from the <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> construction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi><mo>→</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">X \stackrel{f}{\to}Y \to Cone(f) \to T X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(\mathcal{A})</annotation></semantics></math>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a> (the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category</a> of the <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a>) is a triangulated category. This is also true for <a class="existingWikiWord" href="/nlab/show/parameterized+spectra">parametrized</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+spectra">equivariant</a>, etc. spectra. Also the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> called the <a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+category">Spanier-Whitehead category</a> is triangulated.</p> </li> <li> <p>The stable category of a <a class="existingWikiWord" href="/nlab/show/Quillen+exact+category">Quillen exact category</a> is <a class="existingWikiWord" href="/nlab/show/suspended+category">suspended category</a> as exhibited by <a class="existingWikiWord" href="/nlab/show/Bernhard+Keller">Bernhard Keller</a>. If the exact category is Frobenius, i.e. has enough injectives and enough projective and these two classes coincide, then the suspension of the stable category is in fact invertible, hence the stable category is triangulated. A triangulated category equivalent to a stable category of a Frobenius exact category is said to be an <a class="existingWikiWord" href="/nlab/show/algebraic+triangulated+category">algebraic triangulated category</a>.</p> </li> <li> <p>As mentioned before, the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy category</a> of a <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> is a triangulated category. Slightly more generally, this applies also to a <a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a>, and slightly less generally, it applies to a <a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a>. This includes both the preceding examples.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/localization">localization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">C/N</annotation></semantics></math> of any triangulated category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> at a <a class="existingWikiWord" href="/nlab/show/null+system">null system</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>↪</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">N \hookrightarrow C</annotation></semantics></math>, i.e. the localization using the <a class="existingWikiWord" href="/nlab/show/calculus+of+fractions">calculus of fractions</a> given by the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> such that there exists distinguished triangles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi><mo>→</mo><mi>Z</mi><mo>→</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">X \to Y \to Z \to T X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> an object of a <a class="existingWikiWord" href="/nlab/show/null+system">null system</a>, is still naturally a triangulated category, with the distinguished triangles being the triangles isomorphic to an image of a distinguished triangle under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo stretchy="false">/</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">Q : C \to C/N</annotation></semantics></math>.</p> <ul> <li>In particular, therefore, the <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> of any <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> is a triangulated category, since it is the localization of the homotopy category at the null system of acyclic complexes. This example is also the homotopy category of a stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category, and usually of a stable model category.</li> </ul> </li> <li> <p>In <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>, important examples are given by the various <a class="existingWikiWord" href="/nlab/show/triangulated+categories+of+sheaves">triangulated categories of sheaves</a> associated to a <a class="existingWikiWord" href="/nlab/show/variety">variety</a> (e.g. bounded derived category of <a class="existingWikiWord" href="/nlab/show/coherent+sheaves">coherent sheaves</a>, triangulated category of <a class="existingWikiWord" href="/nlab/show/perfect+complexes">perfect complexes</a>).</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+triangle">exact triangle</a>, <a class="existingWikiWord" href="/nlab/show/distinguished+triangle">distinguished triangle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a>, <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+triangulated+category">tensor triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/t-structure">t-structure</a>, <a class="existingWikiWord" href="/nlab/show/Bridgeland+stability+condition">Bridgeland stability condition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/suspended+category">suspended category</a>, <a class="existingWikiWord" href="/nlab/show/n-angulated+category">n-angulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+lattice">Bousfield lattice</a>, <a class="existingWikiWord" href="/nlab/show/thick+subcategory">thick subcategory</a>, <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+collection">exceptional collection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/well-generated+triangulated+category">well-generated triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/compactly+generated+triangulated+category">compactly generated triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum+of+a+triangulated+category">spectrum of a triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+categories+of+sheaves">triangulated categories of sheaves</a>, <a class="existingWikiWord" href="/nlab/show/Bondal-Orlov+reconstruction+theorem">Bondal-Orlov reconstruction theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+subcategory">triangulated subcategory</a>, <a class="existingWikiWord" href="/nlab/show/full+triangulated+subcategory">full triangulated subcategory</a>, <a class="existingWikiWord" href="/nlab/show/periodic+triangulated+category">periodic triangulated category</a></p> </li> </ul> <h2 id="references">References</h2> <p>The concept orignates in the thesis</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Verdier%2C+Jean-Louis">Verdier, Jean-Louis</a>, <em>Des Catégories Dérivées des Catégories Abéliennes</em>, Astérisque <strong>239</strong> (1996) [<a href="https://smf.emath.fr/publications/des-categories-derivees-des-categories-abeliennes">doi:10.24033/ast.364</a>, <a href="http://www.numdam.org/issues/AST_1996__239__R1_0">numdam:AST_1996__239__R1_0</a>, <a href="http://www.numdam.org/item/AST_1996__239__R1_0.pdf">pdf</a>]</li> </ul> <p>Similar axioms were already given in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Albrecht+Dold">Albrecht Dold</a>, <a class="existingWikiWord" href="/nlab/show/Dieter+Puppe">Dieter Puppe</a>, <em>Homologie nicht-additiver Funktoren</em>, Annales de l’Institut Fourier (Université de Grenoble) 11: 201–312, 1961, <a href="https://eudml.org/doc/73776">eudml</a>.</li> </ul> <p>Monographs:</p> <ul> <li id="GelfandManin96"> <p><a class="existingWikiWord" href="/nlab/show/Sergei+Gelfand">Sergei Gelfand</a>, <a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, Section IV of: <em><a class="existingWikiWord" href="/nlab/show/Methods+of+homological+algebra">Methods of homological algebra</a></em>, transl. from the 1988 Russian (Nauka Publ.) original, Springer (1996, 2002) [<a href="https://doi.org/10.1007/978-3-662-12492-5">doi:10.1007/978-3-662-12492-5</a>]</p> </li> <li id="Neeman01"> <p><a class="existingWikiWord" href="/nlab/show/Amnon+Neeman">Amnon Neeman</a>, <em>Triangulated Categories</em>, Annals of Mathematics Studies <strong>213</strong>, Princeton University Press (2001) [<a href="https://press.princeton.edu/titles/7102.html">ISBN:9780691086866</a>, <a href="http://www.mi-ras.ru/~akuznet/homalg/Neeman%20Triangulated%20categories.pdf">pdf</a>]</p> </li> </ul> <p>Discussion of the relation to <a class="existingWikiWord" href="/nlab/show/stable+model+categories">stable model categories</a> originates in</p> <ul> <li id="Hovey99"><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, section 7 of <em>Model Categories</em> Mathematical Surveys and Monographs, Volume 63, AMS (1999) (<a href="https://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf">pdf</a>, <a href="http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover">Google books</a>)</li> </ul> <p>Other surveys:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Masaki+Kashiwara">Masaki Kashiwara</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, Section 10 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></em>, Grundlehren der Mathematischen Wissenschaften <strong>332</strong>, Springer (2006) [<a href="https://link.springer.com/book/10.1007/3-540-27950-4">doi:10.1007/3-540-27950-4</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf">pdf</a>]</p> </li> <li id="Hubery"> <p><a class="existingWikiWord" href="/nlab/show/Andrew+Hubery">Andrew Hubery</a>, <em>Notes on the octahedral axiom</em>, (<a href="http://math-www.uni-paderborn.de/user/hubery/static/Octahedral.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Masaki+Kashiwara">Masaki Kashiwara</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, section 10 <em><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></em>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, section 3 of <em><a class="existingWikiWord" href="/nlab/show/Stable+Infinity-Categories">Stable Infinity-Categories</a></em>,</p> </li> <li id="Noohi08"> <p><a class="existingWikiWord" href="/nlab/show/Behrang+Noohi">Behrang Noohi</a>, <em>Lectures on derived and triangulated categories</em>, pp. 383-418 in <a class="existingWikiWord" href="/nlab/show/Masoud+Khalkhali">Masoud Khalkhali</a>, <a class="existingWikiWord" href="/nlab/show/Matilde+Marcolli">Matilde Marcolli</a> (eds.), <em>An Invitation to Noncommutative Geometry</em>, World Scientific (2008) (<a href="https://doi.org/10.1142/9789812814333_0006">doi:10.1142/9789812814333_0006</a> <a href="http://arxiv.org/abs/0704.1009">arXiv:0704.1009</a>).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fernando+Muro">Fernando Muro</a> (<a href="http://personal.us.es/fmuro/htag.pdf">pdf</a>)</p> </li> <li id="Schwede"> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <em>Triangulated categories</em> (<a href="https://math.berkeley.edu/~aaron/atf/triangulated-categories.pdf">pdf</a>)</p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory">Introduction to Stable homotopy theory</a> – <a href="Introduction+to+Stable+homotopy+theory+--+1-1#TriangulatedStructure">Triangulated structure</a></em></p> </li> </ul> <p>A survey of formalisms used in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> to present the triangulated <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of a <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>Axiomatic stable homotopy - a survey</em> (<a href="http://front.math.ucdavis.edu/0307.5143">arXiv:math.AT/0307143</a>)</li> </ul> <p>Discussion of the redundancy in the traditional definition of triangulated category is in</p> <ul> <li id="May"><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em>The additivity of traces in triangulated categories</em>, (<a href="http://www.math.uchicago.edu/~may/PAPERS/AddJan01.pdf">pdf</a>)</li> </ul> <p>There was also some discussion at the <a href="http://nforum.mathforge.org/discussion/3638/triangulated-category">nForum</a>.</p> <p>On <a class="existingWikiWord" href="/nlab/show/localization">localization</a> of triangulated categories:</p> <ul> <li id="Krause08"><a class="existingWikiWord" href="/nlab/show/Henning+Krause">Henning Krause</a>, <em>Localization theory for triangulated categories</em>, in proceedings of <em><a href="https://www.commalg.org/2006/08/13/workshop-on-triangulated-categories-leeds/">Workshop on Triangulated Categories, Leeds 2006</a></em> [<a href="https://arxiv.org/abs/0806.1324">arXiv:0806.1324</a>]</li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/triangulated+categories">triangulated categories</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on February 23, 2024 at 19:18:36. 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