Local finality condition (for re-indexing parameterized colimits) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T19:01:37Z http://mathoverflow.net/feeds/question/114779 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114779/local-finality-condition-for-re-indexing-parameterized-colimits Local finality condition (for re-indexing parameterized colimits) David Spivak 2012-11-28T15:34:36Z 2013-02-10T23:42:08Z <p>I'm in need of a condition that is analogous to the "finality" condition in the following lemma:</p> <p>Lemma: A functor $F\colon A\to B$ is final if and only if for any functor $x\colon B\to Set$, the natural map $colim (xF)\to colim(x)$ is an isomorphism.</p> <p>This lemma could be taken instead as a definition of <i>final functor</i>, but finality is more easily recognized by whether all slice categories of a certain kind are non-empty and connected. I want a recognition principle for a more general kind of finality, which I'm calling local finality.</p> <p>The more general context requires a bit of notation. If $A$ is a category, write $A-Set$ for the category of functors $A\to Set$. If $F\colon A\to B$ is a functor, write $\Delta_F\colon B-Set\to A-Set$ for the composition with $F$" functor, and write $\Sigma_F$ for its left adjoint and $\Pi_F$ for its right adjoint (these three are also sometimes denoted by $F^*, F_!$, and $F_*$ respectively).</p> <p>The following lemma (obviously) holds for some appropriate definition of <i>locally final</i>. </p> <p>Lemma: Suppose that we have a commutative diagram $A\xrightarrow{F}B\xrightarrow{x}C$ and let $G:=xF$. Then $F$ is locally final if and only if the natural map $\Sigma_G\Delta_F\to\Sigma_x$ is an isomorphism.</p> <p>Is there a nice recognition principle for this kind of local finality"? I have a big messy condition obtained by following my nose, but it's of no use. Any help would be greatly appreciated.</p> <p>Thanks!</p> http://mathoverflow.net/questions/114779/local-finality-condition-for-re-indexing-parameterized-colimits/114799#114799 Answer by Karol Szumiło for Local finality condition (for re-indexing parameterized colimits) Karol Szumiło 2012-11-28T18:25:08Z 2012-11-28T18:25:08Z <p>I assume you meant <code>$\Sigma_G \Delta_F \to \Sigma_x$</code>, otherwise this doesn't parse.</p> <p>Here's my condition. It's not completely straightforward, but it is a generalization of the classical one you mentioned. I guess it's a matter of taste whether it's messy.</p> <p>Consider a triple <code>$(b, f, c)$</code> where <code>$b$</code> is an object of <code>$B$</code>, <code>$c$</code> is an object of <code>$C$</code> and <code>$f : x b \to c$</code> is a morphism in <code>$C$</code>. To each such triple one can associate a kind of "double slice", namely the "category of factorizations of <code>$f$</code> through objects of the form <code>$F a$</code>". Its objects are triples <code>$(g, a, h)$</code> where <code>$a$</code> is an object of <code>$A$</code>, <code>$g : b \to F a$</code> and <code>$h : x F a \to c$</code> such that <code>$h (x g) = f$</code>. The morphisms are morphisms of <code>$A$</code> making two evident triangles commute. My claim is that <code>$\Sigma_G \Delta_F \to \Sigma_x$</code> is a natural isomorphism precisely when all such "categories of factorizations" are connected (which I take to include non-empty).</p> <p>The argument is as follows. For any functor <code>$W : A^\mathrm{op} \to \mathrm{Set}$</code> there is a <code>$W$</code>-weighted colimit functor <code>$\mathrm{colim}^W : \mathrm{Set}^A \to \mathrm{Set}$</code> given by the coend formula <code>$\mathrm{colim}^W Y = \int^{a \in A} Y_a \times W_a$</code> for <code>$Y \in \mathrm{Set}^A$</code>. A transformation <code>$\phi : V \to W$</code> induces a transformation <code>$\phi_* : \mathrm{colim}^V \to \mathrm{colim}^W$</code>. It is easy to see that <code>$\phi_*$</code> is an isomorphism if and only if <code>$\phi$</code> is since we can recover <code>$\phi$</code> form <code>$\phi_*$</code> by evaluating it on representable functors.</p> <p>We have formulas for Kan extensions via coends which say that <code>$(\Sigma_x Y) c = \mathrm{colim}^{C(x -, c)} Y$</code> and similarly <code>$(\Sigma_G \Delta_F Y)_c = \mathrm{colim}^{\int^a B(-, F a) \times C(x F a, c)} Y$</code> and the natural transformation in question is induced by the transformation <code>$\int^a B(-, F a) \times C(x F a, c) \to C(x -, c)$</code> which takes a pair of morphisms <code>$g : b \to F a$</code> and <code>$h : x F a \to c$</code> and sends it to the composite <code>$h (x g)$</code>. We need to check that it is an isomorphism i.e. that the fiber over every point <code>$f : x b \to c$</code> is a singleton. There is an explicit description of this coend as the quotient set of an equivalence relation and it yields a description of the fiber over <code>$f$</code> as the quotient of the set of objects of the above "category of factorizations of <code>$f$</code>" by the relation which turns out to be the relation of being in the same component of this category.</p> http://mathoverflow.net/questions/114779/local-finality-condition-for-re-indexing-parameterized-colimits/120970#120970 Answer by Ricardo Andrade for Local finality condition (for re-indexing parameterized colimits) Ricardo Andrade 2013-02-06T12:56:53Z 2013-02-10T23:42:08Z <p>Since nobody has said so, I will mention that the notion you describe is a particular case of the known --- but perhaps obscure --- concept of Guitart exact square. One can read about it in <a href="http://ncatlab.org/nlab/show/exact+square" rel="nofollow">the nlab page</a> and in <a href="http://arxiv.org/abs/1101.4144" rel="nofollow">an article by Maltsiniotis</a>. Even though the latter article aims to generalize exact squares to a homotopical context, it still gives a good, if somewhat skewed, overview of the concept.</p> <p>To justify the relative usefulness of exact squares, let me state that instances of that notion characterize: fully faithful functors, (co)final functors, initial functors, absolute Kan extensions, among other concepts (including my personal favourite, <a href="http://www.tac.mta.ca/tac/volumes/8/n20/8-20abs.html" rel="nofollow">absolutely dense functors</a>).</p> <p>For completeness, I will summarize some characterizations of the notion of exact square. A square of small categories, functors, and natural transformations: <code>$$\begin{matrix} A &amp; \overset{U}{\longrightarrow} &amp; B \\ \llap{\scriptstyle L}\Big\downarrow &amp; \big\Downarrow\rlap{\scriptstyle\alpha} &amp; \Big\downarrow\rlap{\scriptstyle R} \\ A' &amp; \underset{D}{\longrightarrow} &amp; B' \end{matrix}$$</code> (i.e. a natural transformation $\alpha:R\circ U\to D\circ L$) is called <em>exact</em> if any of the following equivalent conditions hold:</p> <ol> <li>The natural 2-cell induced by $\alpha$ in $$\begin{matrix} \textrm{Set}^A & \overset{\Delta_U}{\longleftarrow} & \textrm{Set}^B \\ \llap{\scriptstyle \Sigma_L}\Big\downarrow & \big\Downarrow & \Big\downarrow\rlap{\scriptstyle \Sigma_R} \\ \textrm{Set}^{\smash{A'}} & \underset{\Delta_D}{\longleftarrow} & \textrm{Set}^{\smash{B'}} \end{matrix}$$ is an isomorphism. That is, the induced natural transformation $\Sigma_L\circ\Delta_U \to \Delta_D\circ\Sigma_R$ is an isomorphism.</li> <li>For every $x\in A'$, the naturally induced functor on over-categories $$A/x=L/x \longrightarrow R/(D(x))=B/(D(x))$$ is (co)final.</li> <li>For every $x\in B$, the naturally induced functor on under-categories $$x/A=x/U \longrightarrow (R(x))/D=(R(x))/A'$$ is initial.</li> <li>For each object $x\in B$ and $y\in A'$, and each arrow $f:R(x) \to D(y)$ in $B'$, the category of factorizations $C_{x,y,f}$ is connected (which I, Karol, and <a href="http://mathoverflow.net/questions/120536/is-the-empty-graph-a-tree" rel="nofollow">a surprisingly large/vocal set of mathematicians</a> take to mean non-empty). Here, the category $C_{x,y,f}$ is defined by: <ul> <li>the objects of $C_{x,y,f}$ are triples $(z,g,h)$ where $z$ is an object of $A$, $g:x\to U(z)$ is a morphism in $B$, and $h:L(z)\to y$ is a morphism of $A'$, such that $D(h)\circ\alpha_z \circ R(g)=f$;</li> <li>a morphism $(z,g,h)\to (z',g',h')$ in $C_{x,y,f}$ is an arrow $k:z\to z'$ such that $g'= U(k)\circ g$ and $h=h'\circ L(k)$.</li> </ul> </li> <li>For all objects $x\in B$ and $y\in A'$, the natural map from the coend $$\int^{a\in A} B(x,U(a))\times A'(L(a),y) \longrightarrow B'(R(x),D(y))$$</li> is an isomorphism of sets. </ol> <p>Before proceeding, observe that condition 4 is simply a restatement of conditions 2 and 3. In fact, the categories of factorizations $C_{x,y,f}$ defined in 4 are exactly the categories whose connectedness must be checked to ensure that the functor in condition 2 is cofinal (respectively, that the functor in condition 3 is initial). More precisely, the categories $C_{x,y,f}$ are the under-categories $a/F$ of the functor $F$ in condition 2 (respectively, the over-categories of the functor in condition 3) for objects $a$ in the codomain of $F$. I feel this both motivates and gives a nice way to remember the definition of $C_{x,y,f}$.</p> <p>Then $A\overset{F}{\rightarrow}B\overset{x}{\rightarrow}C$ is locally final (in the sense David Spivak states) if and only if the the square <code>$$\begin{matrix} A &amp; \overset{F}{\longrightarrow} &amp; B \\ \llap{\scriptstyle G}\Big\downarrow &amp; \big\Downarrow\rlap{\scriptstyle\textrm{id}_G}\ &amp; \Big\downarrow\rlap{\scriptstyle x} \\ C &amp; \underset{\textrm{id}_C}{\longrightarrow} &amp; C \end{matrix}$$</code> (filled by the the identity 2-cell on $G=x\circ F$) is exact. Under this interpretation, condition 4 above is exactly the condition given in Karol Szumiło's answer.</p> <p><strong>Addendum:</strong> To finish off, here are a few further equivalent characterizations of the exactness of the original square ($\alpha:R\circ U\to D\circ L$) drawn at the top of this answer:</p> <ul> <li><p>For any cocomplete category X, the natural 2-cell induced by $\alpha$ in <code>$$\begin{matrix} X^A &amp; \overset{\Delta_U}{\longleftarrow} &amp; X^B \\ \llap{\scriptstyle \Sigma_L}\Big\downarrow &amp; \big\Downarrow &amp; \Big\downarrow\rlap{\scriptstyle \Sigma_R} \\ X^{\smash{A'}} &amp; \underset{\Delta_D}{\longleftarrow} &amp; X^{\smash{B'}} \end{matrix}$$</code> is an isomorphism. Note that for $X=\textrm{Set}$, we recover condition 1 above.</p></li> <li><p>The natural 2-cell induced by $\alpha$ in <code>$$\begin{matrix} \textrm{Set}^A &amp; \overset{\Pi_U}{\longrightarrow} &amp; \textrm{Set}^B \\ \llap{\scriptstyle \Delta_L}\Big\uparrow &amp; \big\Uparrow &amp; \Big\uparrow\rlap{\scriptstyle \Delta_R} \\ \textrm{Set}^{\smash{A'}} &amp; \underset{\Pi_D}{\longrightarrow} &amp; \textrm{Set}^{\smash{B'}} \end{matrix}$$</code> is an isomorphism.</p></li> <li><p>For any complete category X, the natural 2-cell induced by $\alpha$ in <code>$$\begin{matrix} X^A &amp; \overset{\Pi_U}{\longrightarrow} &amp; X^B \\ \llap{\scriptstyle \Delta_L}\Big\uparrow &amp; \big\Uparrow &amp; \Big\uparrow\rlap{\scriptstyle \Delta_R} \\ X^{\smash{A'}} &amp; \underset{\Pi_D}{\longrightarrow} &amp; X^{\smash{B'}} \end{matrix}$$</code> is an isomorphism. Note that for $X=\textrm{Set}$, we recover the preceding condition.</p></li> <li><p>The opposite square <code>$$\begin{matrix} A^{\textrm{op}} &amp; \overset{L^{\textrm{op}}}{\longrightarrow} &amp; {A'}^{\textrm{op}} \\ \llap{\scriptstyle U^{\textrm{op}}}\Big\downarrow &amp; \big\Downarrow\rlap{\scriptstyle\alpha^{\textrm{op}}} &amp; \Big\downarrow\rlap{\scriptstyle D^{\textrm{op}}} \\ B^{\smash{\textrm{op}}} &amp; \underset{R^{\textrm{op}}}{\longrightarrow} &amp; {B'}^{\smash{\textrm{op}}} \end{matrix}$$</code> is exact. Note that the preceding 3 conditions and condition 1 applied to this opposite square give diagrams with categories of presheaves (contravariant functors) on $A$, $B$, $A'$, and $B'$, instead of categories of covariant functors on those categories. In fact, a common (equivalent) definition of exactness is the analog of condition 1 for presheaves: the 2-cell in the square <code>$$\begin{matrix} \widehat{A} &amp; \overset{\Sigma_{U^{\textrm{op}}}}{\longrightarrow} &amp; \widehat{B} \\ \llap{\scriptstyle \Delta_{L^{\textrm{op}}}}\Big\uparrow &amp; \big\Downarrow\rlap{\scriptstyle} &amp; \Big\uparrow\rlap{\scriptstyle \Delta_{R^{\textrm{op}}}} \\ \widehat{A'} &amp; \underset{\Sigma_{D^{\textrm{op}}}}{\longrightarrow} &amp; \widehat{B'} \end{matrix}$$</code> is an isomorphism.</p></li> </ul>