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href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+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='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#comonadicity'>Comonadicity</a></li> <li><a href='#relation_to_codomain_fibration'>Relation to codomain fibration</a></li> <li><a href='#OnSlices'>Slicing of adjoint functors</a></li> <li><a href='#RelWithPresheaves'>Presheaves on over-categories and over-categories of presheaves</a></li> <li><a href='#LimitsAndColimits'>Limits and colimits</a></li> <li><a href='#InitialAndTerminalObjects'>Initial and terminal objects</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>The <strong>slice category</strong> or <strong>over category</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">/</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\mathbf{C}/c</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> over an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">c \in \mathbf{C}</annotation></semantics></math> has</p> <ul> <li> <p>objects that are all arrows <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">f \in \mathbf{C}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>cod</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">cod(f) = c</annotation></semantics></math>, and</p> </li> <li> <p>morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>′</mo><mo>∈</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">g: X \to X' \in \mathbf{C}</annotation></semantics></math> from <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>c</mi></mrow><annotation encoding="application/x-tex">f:X \to c</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>:</mo><mi>X</mi><mo>′</mo><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f': X' \to c</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>∘</mo><mi>g</mi><mo>=</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">f' \circ g = f</annotation></semantics></math>.</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>c</mi><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>′</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mi>f</mi></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mrow><mi>f</mi><mo>′</mo></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>c</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> C/c = \left\lbrace \array{ X &&\stackrel{g}{\to}&& X' \\ & {}_f \searrow && \swarrow_{f'} \\ && c } \right\rbrace </annotation></semantics></math></div> <p>The slice category is a special case of a <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a>.</p> <p>There is a <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>c</mi></msub><mo>:</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">/</mo><mi>c</mi><mo>→</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">U_c: \mathbf{C}/c \to \mathbf{C}</annotation></semantics></math> which maps an object <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>c</mi></mrow><annotation encoding="application/x-tex">f:X \to c</annotation></semantics></math> to its domain <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> and a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>′</mo><mo>∈</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">/</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">g: X \to X' \in \mathbf{C}/c</annotation></semantics></math> (from <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>c</mi></mrow><annotation encoding="application/x-tex">f:X \to c</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>:</mo><mi>X</mi><mo>′</mo><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f': X' \to c</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>∘</mo><mi>g</mi><mo>=</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">f' \circ g = f</annotation></semantics></math>) to the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g: X \to X'</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/duality">dual</a> notion is an <a class="existingWikiWord" href="/nlab/show/under+category">under category</a>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo>=</mo><mstyle mathvariant="bold"><mi>P</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C} = \mathbf{P}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/partial+order">poset</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>P</mi></mstyle></mrow><annotation encoding="application/x-tex">p \in \mathbf{P}</annotation></semantics></math>, then the slice category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>P</mi></mstyle><mo stretchy="false">/</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\mathbf{P}/p</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/down+set">down set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\downarrow (p)</annotation></semantics></math> of elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>P</mi></mstyle></mrow><annotation encoding="application/x-tex">q \in \mathbf{P}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>≤</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">q \leq p</annotation></semantics></math>.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">/</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\mathbf{C}/1</annotation></semantics></math> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math>.</p> </li> <li> <p>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> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> then the <a class="existingWikiWord" href="/nlab/show/category+of+covering+spaces">category of covering spaces</a> over <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> is a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the slice category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Top_{/X}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/category+of+topological+spaces">category of topological spaces</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+topos+theory">fundamental theorem of topos theory</a> states that the slice category over any object in a topos is itself a topos.</p> </li> <li> <p>For a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> the slice category over any <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> is monoidal.</p> <p id="MonoidalTopos"> For instance, the <a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a> of a given <a class="existingWikiWord" href="/nlab/show/topos">topos</a> over any <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> is canonically a <em><a class="existingWikiWord" href="/nlab/show/monoidal+topos">monoidal topos</a></em> (see the Example <a href="monoidal+topos#SliceToposOverAMonoidObject">there</a>).</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="comonadicity">Comonadicity</h3> <p>If <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> admits binary coproducts with the fixed object <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>, then the forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>c</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">C/c \to C</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/comonadic">comonadic</a>. See <a class="existingWikiWord" href="/nlab/show/coreader+comonad">coreader comonad</a> for more details.</p> <h3 id="relation_to_codomain_fibration">Relation to codomain fibration</h3> <p>The assignment of overcategories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">C/c</annotation></semantics></math> to objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> extends to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> C/(-) : C \to Cat </annotation></semantics></math></div> <p>Under the <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a> this functor corresponds to the <a class="existingWikiWord" href="/nlab/show/codomain+fibration">codomain fibration</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>cod</mi><mo>:</mo><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> cod : [I,C] \to C </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> of <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>. (Note that unless <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 <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>, this functor is not actually a <a class="existingWikiWord" href="/nlab/show/Grothendieck+fibration">fibration</a>, though it is always an opfibration.)</p> <div> <h3 id="OnSlices">Slicing of adjoint functors</h3> <p> <div class="num_prop" id="SliceAdjoints"> <h6>Proposition</h6> <p><strong>(sliced adjoints)</strong> <br /> Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>R</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></munder><mover><mo>⟵</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>L</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover></munderover><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> \mathcal{D} \underoverset {\underset{\;\;\;\;R\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C} </annotation></semantics></math></div> <p>be a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> (<a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functors">adjoint ∞-functors</a>), where the <a class="existingWikiWord" href="/nlab/show/category">category</a> (<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">∞-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> has all <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> (<a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a>).</p> <p>Then:</p> <ol> <li> <p>For every <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">b \in \mathcal{C}</annotation></semantics></math> there is induced a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> between the <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a> (<a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-categories">slice ∞-categories</a>) of the form</p> <div class="maruku-equation" id="eq:SlicedAdjointFunctorsOverLb"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒟</mi> <mrow><mo stretchy="false">/</mo><mi>L</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msub><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>R</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></munder><mover><mo>⟵</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>L</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover></munderover><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow></mpadded></mrow><annotation encoding="application/x-tex"> \mathcal{D}_{/L(b)} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/b} \mathrlap{\,,} </annotation></semantics></math></div> <p>where:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{/b}</annotation></semantics></math> is the evident induced functor (applying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> to the entire triangle <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> in <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> which represent the morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/b}</annotation></semantics></math>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex">R_{/b}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/composition">composite</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒟</mi> <mrow><mo stretchy="false">/</mo><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></msub><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>R</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mrow><mo stretchy="false">(</mo><mi>R</mi><mo>∘</mo><mi>L</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mrow></msub><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>η</mi> <mi>b</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> R_{/b} \;\colon\; \mathcal{D}_{/{L(b)}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{(R \circ L(b))}} \overset{\;\;(\eta_{b})^*\;\;}{\longrightarrow} \mathcal{C}_{/b} </annotation></semantics></math></div> <p>of</p> <ol> <li> <p>the evident functor induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>;</p> </li> <li> <p>the (<a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy</a>) <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> along the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \dashv R)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> (i.e. the <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>b</mi></msub></mrow><annotation encoding="application/x-tex">\eta_b</annotation></semantics></math>).</p> </li> </ol> </li> </ul> </li> <li> <p>For every <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">b \in \mathcal{D}</annotation></semantics></math> there is induced a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> between the <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a> of the form</p> <div class="maruku-equation" id="eq:SlicedAdjointFunctorsOverRb"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒟</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>R</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></munder><mover><mo>⟵</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>L</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover></munderover><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>R</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msub><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow></mpadded></mrow><annotation encoding="application/x-tex"> \mathcal{D}_{/b} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/R(b)} \mathrlap{\,,} </annotation></semantics></math></div> <p>where:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex">R_{/b}</annotation></semantics></math> is the evident induced functor (applying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> to the entire triangle <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> which represent the morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒟</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{D}_{/b}</annotation></semantics></math>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{/b}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/composition">composite</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒟</mi> <mrow><mo stretchy="false">/</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></msub><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>L</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>∘</mo><mi>R</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mrow></msub><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>ϵ</mi> <mi>b</mi></msub><msub><mo stretchy="false">)</mo> <mo>!</mo></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> L_{/b} \;\colon\; \mathcal{D}_{/{R(b)}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{C}_{/{(L \circ R(b))}} \overset{\;\;(\epsilon_{b})_!\;\;}{\longrightarrow} \mathcal{C}_{/b} </annotation></semantics></math></div> <p>of</p> <ol> <li> <p>the evident functor induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \dashv R)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> (i.e. the left <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>b</mi></msub></mrow><annotation encoding="application/x-tex">\epsilon_b</annotation></semantics></math>).</p> </li> </ol> </li> </ul> </li> </ol> <p></p> </div> The first statement appears, in the generality of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, as <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. 5.2.5.1</a>. For discussion in <a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a> see at <em><a href="slice+model+structure#SlicedQuillenAdjunction">sliced Quillen adjunctions</a></em>. <div class="proof"> <h6>Proof</h6> <p>(in <a class="existingWikiWord" href="/nlab/show/1-category">1-</a><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>)</p> <p>Recall that (<a href="adjoint+functor#GeneralAdjunctsInTermsOfAdjunctionUnitCounit">this Prop.</a>) the hom-isomorphism that defines an adjunction of functors (<a href="adjoint+functor#AdjointFunctorsInTermsOfNaturalBijectionOfHomSets">this Def.</a>) is equivalently given in terms of <a class="existingWikiWord" href="/nlab/show/composition">composition</a> with</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>η</mi> <mi>c</mi></msub><mo lspace="verythinmathspace">:</mo><mi>c</mi><mover><mo>→</mo><mspace width="thickmathspace"></mspace></mover><mi>R</mi><mo>∘</mo><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\;\;\eta_c \colon c \xrightarrow{\;} R \circ L(c)</annotation></semantics></math></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>ϵ</mi> <mi>d</mi></msub><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>∘</mo><mi>R</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mspace width="thickmathspace"></mspace></mover><mi>d</mi></mrow><annotation encoding="application/x-tex">\;\;\epsilon_d \colon L \circ R(d) \xrightarrow{\;} d</annotation></semantics></math></p> </li> </ul> <p>as follows:</p> <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="740.569" height="56.358" viewBox="0 0 740.569 56.358"> <defs> <g> <g id="8EQecrw6-tBYftONzZEwW2XEHg0=-glyph-0-0"> </g> <g id="8EQecrw6-tBYftONzZEwW2XEHg0=-glyph-0-1"> <path d="M 5.484375 -9.046875 C 5.609375 -9.625 5.65625 -9.765625 6.96875 -9.765625 C 7.375 -9.765625 7.484375 -9.765625 7.484375 -10.046875 C 7.484375 -10.203125 7.3125 -10.203125 7.265625 -10.203125 C 6.953125 -10.203125 6.625 -10.171875 6.3125 -10.171875 L 4.3125 -10.171875 C 4.03125 -10.171875 3.703125 -10.203125 3.421875 -10.203125 C 3.296875 -10.203125 3.140625 -10.203125 3.140625 -9.921875 C 3.140625 -9.765625 3.265625 -9.765625 3.5 -9.765625 C 4.40625 -9.765625 4.40625 -9.65625 4.40625 -9.484375 C 4.40625 -9.453125 4.40625 -9.359375 4.34375 -9.140625 L 2.328125 -1.109375 C 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stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 238.339344 0.00125 L 349.741688 0.00125 " transform="matrix(1, 0, 0, -1, 370.938, 40.345)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.486526 2.868437 C -2.033401 1.145781 -1.021682 0.333281 0.001755 0.00125 C -1.021682 -0.334688 -2.033401 -1.147188 -2.486526 -2.869844 " transform="matrix(1, 0, 0, -1, 720.92012, 40.345)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FLFOHmHuGjQpIITisQPKKjy6zCY=-glyph-4-3" x="660.363" y="35.072"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FLFOHmHuGjQpIITisQPKKjy6zCY=-glyph-7-1" x="664.643" y="36.8295"></use> </g> </svg> <p>Using this, consider the following transformations of morphisms in slice categories, for the <strong>first case</strong>:</p> <p><strong>(1a)</strong></p> <svg 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xlink:href="#A49YqQBNyxe-JulOlMpnNsOACTA=-glyph-8-2" x="326.598338" y="141.72"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 21.607281 -34.013094 L 200.193219 -34.013094 " transform="matrix(1, 0, 0, -1, 318.006, 74.108)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.486205 2.8703 C -2.03308 1.147644 -1.021361 0.335144 -0.00183 -0.00079375 C -1.021361 -0.332825 -2.03308 -1.149231 -2.486205 -2.867981 " transform="matrix(1, 0, 0, -1, 518.43933, 108.1203)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#A49YqQBNyxe-JulOlMpnNsOACTA=-glyph-3-3" x="417.888" y="115.925"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#A49YqQBNyxe-JulOlMpnNsOACTA=-glyph-7-3" x="422.16675" y="118.0975"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#A49YqQBNyxe-JulOlMpnNsOACTA=-glyph-9-1" x="428.4905" y="118.0975"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#A49YqQBNyxe-JulOlMpnNsOACTA=-glyph-7-2" x="432.088" y="118.0975"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#A49YqQBNyxe-JulOlMpnNsOACTA=-glyph-9-2" x="436.1805" y="118.0975"></use> </g> </svg> <p>Here:</p> <ul> <li> <p>(1a) and (1b) are equivalent expressions of the same morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒟</mi> <mrow><mo stretchy="false">/</mo><mi>L</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{D}_{/L(b)}</annotation></semantics></math>, by (at the top of the diagrams) the above expression of <a class="existingWikiWord" href="/nlab/show/adjuncts">adjuncts</a> between <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> and (at the bottom) by the <a class="existingWikiWord" href="/nlab/show/triangle+identity">triangle identity</a>.</p> </li> <li> <p>(2a) and (2b) are equivalent expression of the same morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/b}</annotation></semantics></math>, by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>.</p> </li> </ul> <p>Hence:</p> <ul> <li> <p>starting with a morphism as in (1a) and transforming it to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2)</annotation></semantics></math> and then to (1b) is the identity operation;</p> </li> <li> <p>starting with a morphism as in (2b) and transforming it to (1) and then to (2a) is the identity operation.</p> </li> </ul> <p>In conclusion, the transformations (1) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\leftrightarrow</annotation></semantics></math> (2) consitute a <a href="adjoint+functor#InTermsOfHomIsomorphism">hom-isomorphism</a> that witnesses an adjunction of the first claimed form <a class="maruku-eqref" href="#eq:SlicedAdjointFunctorsOverLb">(1)</a>.</p> <p><br /></p> <p>The <strong>second case</strong> follows analogously, but a little more directly since no pullback is involved:</p> <p><strong>(1a)</strong></p> <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="216.388" height="96.554" viewBox="0 0 216.388 96.554"> <defs> <g> <g id="azn9nXJHvQSRuF8G_pi5FFUrJNc=-glyph-0-0"> </g> <g id="azn9nXJHvQSRuF8G_pi5FFUrJNc=-glyph-0-1"> <path d="M 5.84375 -5.609375 C 5.5625 -5.609375 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stroke-opacity="1" stroke-miterlimit="10" d="M -2.484755 2.867863 C -2.03163 1.149113 -1.019911 0.336613 -0.00038 0.000675 C -1.019911 -0.335262 -2.03163 -1.147762 -2.484755 -2.870419 " transform="matrix(1, 0, 0, -1, 498.40663, 108.1413)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#rC84F5MF9ATo8-dj_lWpaTyaUDw=-glyph-3-2" x="430.073" y="119.129"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#rC84F5MF9ATo8-dj_lWpaTyaUDw=-glyph-6-1" x="438.10675" y="119.129"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#rC84F5MF9ATo8-dj_lWpaTyaUDw=-glyph-3-4" x="442.223" y="119.129"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#rC84F5MF9ATo8-dj_lWpaTyaUDw=-glyph-4-3" x="446.503" y="120.88525"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#rC84F5MF9ATo8-dj_lWpaTyaUDw=-glyph-6-2" x="451.218" y="119.129"></use> </g> </svg> <p>In conclusion, the transformations (1) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\leftrightarrow</annotation></semantics></math> (2) consitute a <a href="adjoint+functor#InTermsOfHomIsomorphism">hom-isomorphism</a> that witnesses an adjunction of the second claimed form <a class="maruku-eqref" href="#eq:SlicedAdjointFunctorsOverRb">(2)</a>.</p> </div> </p> <p> <div class="num_remark" id="LeftAdjointOfSlicedAdjunctionFormsAdjuncts"> <h6>Remark</h6> <p><strong>(left adjoint of sliced adjunction forms adjuncts)</strong> <br /> The sliced adjunction (Prop. <a class="maruku-ref" href="#SliceAdjoints"></a>) in the second form <a class="maruku-eqref" href="#eq:SlicedAdjointFunctorsOverRb">(2)</a> is such that the sliced <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> sends slicing morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math> to their <a class="existingWikiWord" href="/nlab/show/adjuncts">adjuncts</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>τ</mi><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{\tau}</annotation></semantics></math>, in that (again by <a href="adjoint+functor#GeneralAdjunctsInTermsOfAdjunctionUnitCounit">this Prop.</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mo stretchy="false">/</mo><mi>d</mi></mrow></msub><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>c</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mi>τ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>R</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mover><mi>τ</mi><mo>˜</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd><mi>b</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒟</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> L_{/d} \, \left( \array{ c \\ \big\downarrow {}^{\mathrlap{\tau}} \\ R(b) } \right) \;\; = \;\; \left( \array{ L(c) \\ \big\downarrow {}^{\mathrlap{\widetilde{\tau}}} \\ b } \right) \;\;\; \in \; \mathcal{D}_{/b} </annotation></semantics></math></div> <p></p> </div> </p> <p>The two adjunctions in <a class="maruku-ref" href="#SliceAdjoints"></a> admit the following joint generalisation, which is proven <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, lem. 5.2.5.2</a>. (Note that the statement there is even more general and here we only use the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo><msup><mi>Δ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">K = \Delta^0</annotation></semantics></math>.)</p> <p> <div class="num_prop" id="SliceAdjointsGeneralized"> <h6>Proposition</h6> <p><strong>(sliced adjoints)</strong> <br /> Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><munderover><mo>⊥</mo><munder><mo>⟵</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>R</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></munder><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>L</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover></munderover><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C} \underoverset {\underset{\;\;\;\;R\;\;\;\;}{\longleftarrow}} {\overset{\;\;\;\;L\;\;\;\;}{\longrightarrow}} {\bot} \mathcal{D} </annotation></semantics></math></div> <p>be a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functors">adjoint ∞-functors</a>, where the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">∞-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> has all <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a>. Suppose further we are given objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">c \in \mathcal{C}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">d \in \mathcal{D}</annotation></semantics></math> together with a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>c</mi><mo>→</mo><mi>R</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha: c \to R(d)</annotation></semantics></math> and its adjunct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>:</mo><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">\beta:L(c) \to d</annotation></semantics></math>.</p> <p>Then there is an induced a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functors">adjoint ∞-functors</a> between the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-categories">slice ∞-categories</a> of the form</p> <div class="maruku-equation" id="eq:SlicedAdjointFunctorsGeneral"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>c</mi></mrow></msub><munderover><mo>⊥</mo><munder><mo>⟵</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>R</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></munder><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>L</mi> <mrow><mo stretchy="false">/</mo><mi>b</mi></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover></munderover><msub><mi>𝒟</mi> <mrow><mo stretchy="false">/</mo><mi>d</mi></mrow></msub><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow></mpadded></mrow><annotation encoding="application/x-tex"> \mathcal{C}_{/c} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longleftarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longrightarrow}} {\bot} \mathcal{D}_{/d} \mathrlap{\,,} </annotation></semantics></math></div> <p>where:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mo stretchy="false">/</mo><mi>c</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{/c}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/composition">composite</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mo stretchy="false">/</mo><mi>c</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>c</mi></mrow></msub><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>L</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msub><mi>𝒟</mi> <mrow><mo stretchy="false">/</mo><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mrow></msub><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>β</mi> <mo>!</mo></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msub><mi>𝒟</mi> <mrow><mo stretchy="false">/</mo><mi>d</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> L_{/c} \;\colon\; \mathcal{C}_{/{c}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{D}_{/{L(c)}} \overset{\;\;\beta_!\;\;}{\longrightarrow} \mathcal{D}_{/d} </annotation></semantics></math></div> <p>of</p> <ol> <li> <p>the evident functor induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>:</mo><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">\beta:L(c) \to d</annotation></semantics></math> (i.e. the left <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> along <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>).</p> </li> </ol> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mrow><mo stretchy="false">/</mo><mi>d</mi></mrow></msub></mrow><annotation encoding="application/x-tex">R_{/d}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/composition">composite</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mrow><mo stretchy="false">/</mo><mi>d</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒟</mi> <mrow><mo stretchy="false">/</mo><mi>d</mi></mrow></msub><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>R</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></mrow></msub><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msup><mi>α</mi> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>c</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> R_{/d} \;\colon\; \mathcal{D}_{/{d}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{R(d)}} \overset{\;\;(\alpha^*\;\;}{\longrightarrow} \mathcal{C}_{/c} </annotation></semantics></math></div> <p>of</p> <ol> <li> <p>the evident functor induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>c</mi><mo>→</mo><mi>R</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha:c \to R(d)</annotation></semantics></math> (i.e. the <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> along <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>).</p> </li> </ol> </li> </ul> <p></p> </div> </p> </div> <h3 id="RelWithPresheaves">Presheaves on over-categories and over-categories of presheaves</h3> <p>See <a class="existingWikiWord" href="/nlab/show/slice+of+presheaves+is+presheaves+on+slice">slice of presheaves is presheaves on slice</a>.</p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/category">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> an <a class="existingWikiWord" href="/nlab/show/object">object</a> of <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> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">C/c</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/over+category">over category</a> of <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> over <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>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">PSh(C/c) = [(C/c)^{op}, Set]</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">C/c</annotation></semantics></math> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(C)/Y(c)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/over+category">over category</a> of <a class="existingWikiWord" href="/nlab/show/presheaf">presheaves</a> on <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> over the presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y(c)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y : C \to PSh(c)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>There is an <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a> of categories</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>c</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> takes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F \in PSh(C/c)</annotation></semantics></math> to the presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>′</mo><mo>:</mo><mi>d</mi><mo>↦</mo><msub><mo>⊔</mo> <mrow><mi>f</mi><mo>∈</mo><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F' : d \mapsto \sqcup_{f \in C(d,c)} F(f)</annotation></semantics></math> which is equipped with the natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><mi>F</mi><mo>′</mo><mo>→</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta : F' \to Y(c)</annotation></semantics></math> with component map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>d</mi></msub><mo>:</mo><msub><mo>⊔</mo> <mrow><mi>f</mi><mo>∈</mo><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_d: \sqcup_{f \in C(d,c)} F(f) \to C(d,c)</annotation></semantics></math>.</p> <p>A weak inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> is given by the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>e</mi><mo stretchy="false">¯</mo></mover><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \bar e : PSh(C)/Y(c) \to PSh(C/c) </annotation></semantics></math></div> <p>which sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><mi>F</mi><mo>′</mo><mo>→</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \eta : F' \to Y(C)) </annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F \in PSh(C/c)</annotation></semantics></math> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mo stretchy="false">(</mo><mi>f</mi><mo>:</mo><mi>d</mi><mo>→</mo><mi>c</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>F</mi><mo>′</mo><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mi>c</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> F : (f : d \to c) \mapsto F'(d)|_c \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>′</mo><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">F'(d)|_c</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</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>F</mi><mo>′</mo><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mi>c</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>F</mi><mo>′</mo><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>η</mi> <mi>d</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd><mi>pt</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example">Example</h6> <p>Suppose the presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F \in PSh(C/c)</annotation></semantics></math> does not actually depend on the morphisms to <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>, i.e. suppose that it factors through the forgetful functor from the <a class="existingWikiWord" href="/nlab/show/over+category">over category</a> to <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F : (C/c)^{op} \to C^{op} \to Set \,. </annotation></semantics></math></div> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>′</mo><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>⊔</mo> <mrow><mi>f</mi><mo>∈</mo><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>⊔</mo> <mrow><mi>f</mi><mo>∈</mo><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F'(d) = \sqcup_{f \in C(d,c)} F(f) = \sqcup_{f \in C(d,c)} F(d) \simeq C(d,c) \times F(d) </annotation></semantics></math> and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>′</mo><mo>=</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">F ' = Y(c) \times F</annotation></semantics></math> with respect to the <a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a>.</p> </div> <p>See also <a class="existingWikiWord" href="/nlab/show/functors+and+comma+categories">functors and comma categories</a>.</p> <p>For the analogous statement in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a> see at <em><span class="newWikiWord">(∞,1)-category of (∞,1)-presheaves – Interaction with overcategories<a href="/nlab/new/infinity1-category%2Bof%2Binfinity1-presheaves">?</a></span></em>.</p> <h3 id="LimitsAndColimits">Limits and colimits</h3> <div class="num_prop" id="ColimitInSliceAreReflectedByColimitsInPlainCategory"> <h6 id="proposition_2">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> in an <a class="existingWikiWord" href="/nlab/show/over+category">over category</a> is computed as a colimit in the underlying category.</p> <p>Precisely: let <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> be a <a class="existingWikiWord" href="/nlab/show/category">category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">t \in \mathcal{C}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/object">object</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">/</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}/t</annotation></semantics></math> the corresponding <a class="existingWikiWord" href="/nlab/show/overcategory">overcategory</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo stretchy="false">/</mo><mi>t</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">p \colon \mathcal{C}/t \to \mathcal{C}</annotation></semantics></math> the obvious projection.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>D</mi><mo>→</mo><mi>𝒞</mi><mo stretchy="false">/</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">F \colon D \to \mathcal{C}/t</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/functor">functor</a>. Then, if it exists, the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">p \circ F</annotation></semantics></math> in <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 the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> of the colimit over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mi>F</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mo>⟶</mo></munder><mo stretchy="false">(</mo><mi>p</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> p \big( \underset{\longrightarrow}{\lim} F \big) \;\simeq\; \underset{\longrightarrow}{\lim} (p \circ F) </annotation></semantics></math></div> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>⟶</mo></munder><mi>F</mi></mrow><annotation encoding="application/x-tex">\underset{\longrightarrow}{\lim} F</annotation></semantics></math> is uniquely characterized by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>⟶</mo></munder><mo stretchy="false">(</mo><mi>p</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{\longrightarrow}{\lim} (p \circ F)</annotation></semantics></math> this way.</p> </div> <p>This statement, and its proof, is the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> to the corresponding statement for <a class="existingWikiWord" href="/nlab/show/undercategories">undercategories</a>, see <a href="under+category#LimitsAndColimits">there</a>.</p> <div class="num_prop" id="LimitsInSliceViaLimitsOfCoconedDiagram"> <h6 id="proposition_3">Proposition</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/category">category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒟</mi><mo>⟶</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/X}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a> (the over-category if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> is the point) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>K</mi><mo>→</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">F \;\colon\; K \to \mathcal{C}_{/X}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in the <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a>, then the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>←</mo></munder><mi>F</mi></mrow><annotation encoding="application/x-tex">\underset{\leftarrow}{\lim} F</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/X}</annotation></semantics></math> coincides with the limit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>←</mo></munder><mi>F</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\underset{\leftarrow}{\lim} F/X</annotation></semantics></math> in <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>.</p> </div> <p>For a proof see at <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a> <em><a href="limit+in+a+quasi-category#InOvercategories">here</a></em>.</p> <h3 id="InitialAndTerminalObjects">Initial and terminal objects</h3> <p>As a special case of the above discussion of limits and colimits in a slice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/X}</annotation></semantics></math> we obtain the following statement, which of course is also immediately checked explicitly.</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <ul> <li> <p>If <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> has an initial object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi></mrow><annotation encoding="application/x-tex">\emptyset</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/X}</annotation></semantics></math> has an <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>∅</mi><mo>→</mo><mi>X</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle \emptyset \to X\rangle</annotation></semantics></math>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/X}</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">id</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathrm{id}_X</annotation></semantics></math>.</p> </li> </ul> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>over-category</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+over+category">enriched over category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/under+category">under category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+topos">over topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/slice+2-category">slice 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+%28%E2%88%9E%2C1%29-category">over (∞,1)-category</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+an+over+category">model structure on an over category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></p> </li> </ul> </li> </ul> <h2 id="references">References</h2> <p>Formalization in <a class="existingWikiWord" href="/nlab/show/cubical+Agda">cubical Agda</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/1lab">1lab</a>: <em><a href="https://1lab.dev/Cat.Instances.Slice.html">Slice categories</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 11, 2024 at 13:05:52. 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