Evaluating a colimit of functors

Question

This is my first question here. I am working on some categories and this question regarding functors seemed very natural:

Let $\{G_\alpha \rightarrow F\}_{\alpha \in \Gamma}$ with $F$ and $\{G_\alpha\}_{\alpha \in \Gamma} $ functors in $Sets^{C}$ be an universal cone, i.e., $F = co\varinjlim_{\alpha\in \Gamma} \ G_\alpha$, and $\Gamma$ a filtered category. Then is it true that

$F(C) = co\varinjlim_{\alpha\in \Gamma} \ G_\alpha(C)$ ?

I am not asking for a solution just a general tip to solve it! Thanks :)

Answer 1

(I misread your question and wrote the answer about limits; but everything below holds with colimiting cocones in place of limiting cones).

Given a functor category $[\mathcal E,\mathcal C]$, a diagram $\mathcal D\xrightarrow{J}[\mathcal E,\mathcal C]$, and a limiting cone $F\overset{\phi_A}\Rightarrow JA$ over $J$, it is not in general true that for any object $X$ in $\mathcal E$, $F(X)\xrightarrow{\phi_{A,X}}JA(X)$ is a limiting cone for $\mathcal D\xrightarrow{J-(X)}\mathcal C$.

Rather, the conclusion you want can be proven if the limiting cones $F(X)\xrightarrow{\phi_{A,X}}JA(X)$ for $\mathcal D\xrightarrow{J-(X)}\mathcal C$ exist for every object $X$ in $\mathcal E$. Explicitly, fix a choice of such limiting cones for each object $X$. Then using the universal properties of all the limiting cones you can prove

1. the vertices $FX$ of those limiting cones extend to a functor $\mathcal E\xrightarrow{F}\mathcal C$ as follows. A morphism $X\xrightarrow{f} Y$, determines morphisms $JA(X)\xrightarrow{JA(f)}JA(Y)$, one for each $A$. If we compose them with the projection morphisms $FX\xrightarrow{\phi_{A,X}}JA(X)$ of the cone over $\mathcal D\xrightarrow{J-(X)}\mathcal C$, we will get morphisms$FX\xrightarrow{\phi_{A,X}}JA(X)\xrightarrow{JA(f)}JA(Y)$. These from a cone over $\mathcal D\xrightarrow{J-(Y)}$ because each $JA\overset{Ja}\Rightarrow JB$ for a morphism $A\xrightarrow{a}B$ in $\mathcal D$ is a natural transformation. Explicitly, in the diagrams $$\require{AMScd}\begin{CD} FX @>\phi_{A,X}>> JA(X) @>JA(f)>> JA(Y)\\ @| @V(Ja)_XVV @V(Ja)_YVV\\ FX @>\phi_{B,X}>> JB(X) @>JB(f)>> JB(Y) \end{CD}(1)$$ the right square commutes by naturality of $JA\overset{Ja}\Rightarrow JB$, and the left square by $FX\xrightarrow{\phi_{A,X}}JA(X)$ being the projection morphisms of a cone. Hence the whole diagram commutes, so the composites $FX\xrightarrow{\phi_{A,X}}JA(X)\xrightarrow{JA(f)}JA(Y)$ are indeed the projection morphisms of a cone over $\mathcal D\xrightarrow{J-(Y)}\mathcal C$. Therefore, there must be a unique morphism $FX\xrightarrow{Ff}FY$ so that all the projections $FX\xrightarrow{\phi_{A,X}}JA(X)\xrightarrow{JA(f)}JA(Y)$ factor as $FX\xrightarrow{Ff}FY\xrightarrow{\phi_{A,Y}}JA(Y)$. In other words, a unique morphism $FX\xrightarrow{Ff}FY$ so that all the diagrams $$\begin{CD} FX @>\phi_{A,X}>> JA(X)\\ @VFfVV @VJA(f)VV\\ FY @>\phi_{A,Y}>> JA(Y) \end{CD}(2)$$ commute.

2. You should check that $FX\xrightarrow{F\mathrm{id}_X}FX$ is the identity morphism and that $FX\xrightarrow{Ff}FY\xrightarrow{Fg}FZ$ is $FX\xrightarrow{Fg\circ Ff}FZ$, i.e. that we have extended the choice of vertices of limiting cones $FX$ into a functor $\mathcal E\xrightarrow{F}\mathcal C$. Knowing the functor, since $FX\xrightarrow{Ff}FY$ were defined so that all all diagrams (2) commute, we can see that the projection morphisms $F(X)\xrightarrow{\phi_{A,X}}JX$ are the components of natural transformations $F\overset{\phi_A}\Rightarrow JA$.

3. The resulting collection of natural transformations $F\overset{\phi_A}\Rightarrow JA$, one for each $A$ in $\mathcal E$, is a cone over $\mathcal D\xrightarrow{J}[\mathcal E,\mathcal C]$. Indeed, for every morphism $A\xrightarrow{a}B$, the diagram $$\begin{CD} F @>\phi_A>> JA\\ @| @VJaVV\\ F @>\phi_B>> JB \end{CD}$$ of natural transformations commutes because the left squares in diagrams (1) are the components of these natural transformations, and these commute.

4. It remains to check that the cone over $\mathcal D\xrightarrow{J}[\mathcal E,\mathcal C]$ given by the projection morphisms $F\overset{\phi_A}\Rightarrow JA$ is a limiting cone. Consider another cone cone over $\mathcal D\xrightarrow{J}[\mathcal E,\mathcal C]$ with projection morphisms $G\overset{\psi_A}\Rightarrow JA$. Then for each $X$ in $\mathcal E$, the components $GX\xrightarrow{\psi_{A,X}}JA(X)$ are themselves projection morphisms of a cone over $\mathcal D\xrightarrow{J-(X)}\mathcal C$ (evaluation at an object is functor). Since the $FX\xrightarrow{\phi_{A,X}}JA(X)$ were the projection morphisms of a limiting cone, it follows that there exist unique morphisms $GX\xrightarrow{\chi_X}FX$ so that $\psi_{X,A}=\phi_{X,A}\circ\chi_X$. In particular, if there were a natural transformation $G\Rightarrow F$ through which $G\overset{\psi_A}\Rightarrow JA$ factor as $G\Rightarrow F\overset{\phi_A}\Rightarrow JA$, its components would have to be the $GX\xrightarrow{\chi_X}FX$ we defined.

5. As soon as we check that $GX\xrightarrow{\chi_X}FX$ are the components of a natural transformation, we'll be done. Add these to diagrams (2), to obtain diagrams $$\begin{CD} GX @>\chi_X>> FX @>\phi_{A,X}>> JA(X)\\ @VGfVV @VFfVV @VJB(f)VV\\ GY @>\chi_Y>> FY @>\phi_{A,Y}>> JA(Y) \end{CD}$$ The right square commutes because we have natural transformations $F\overset{\phi_A}\Rightarrow JA$, and the outer rectangle commutes because $GX\xrightarrow{\chi_X}FX\xrightarrow{\phi_{A,X}}JA(X)$ are the components of the natural transformation $G\overset{\psi_A}\Rightarrow JA$. It follows that the composites $\phi_{A,Y}\circ(Ff\circ\chi_X)$ and $\phi_{A,Y}\circ(\chi_Y\circ Gf)$ are equal for each object $Y$. Since $FY\xrightarrow{\phi_{A,Y}}JA(Y)$ are the projection morphisms of a limiting cone over $\mathcal D\xrightarrow{J-(Y)}\mathcal C$, by the uniqueness property of limits it follows that $\chi_Y\circ Gf$ and $Ff\circ\chi_X$ are the same morphism.

Note that since limiting cones are unique up to isomorphism, then if $F\overset{\phi_A}\Rightarrow JA$ is a limiting cone, it must be isomorphic to a limiting cone built out of a choice of limiting cones for the diagrams $\mathcal D\xrightarrow{J-X}\mathcal C$.

In particular, if $\mathcal C$ has all limits of diagrams of shape $\mathcal D$, the functor category $[\mathcal E,\mathcal C]$ also does, and the evaluation functors $[\mathcal E,\mathcal C]\xrightarrow{\mathrm{ev}_X}\mathcal C$ at object $X$ in $\mathcal E$ preserve those limits. More generally, the same argument works for arbitrary Kan extensions, not just limits.