Equivalence of diagrams in terms of their blah-limits

Question

Pick a category where all the following makes sense (I'm going to imagine the category of simplicial sets, for no real reason beyond "I like it"). Consider the two diagrams $$ A \xrightarrow{f} X \xleftarrow{g} B $$ and $$ A\times B \xrightarrow{\langle f,g\rangle} X\times X \xleftarrow{\Delta} X $$ where $\Delta$ is the diagonal. They are equivalent in one sense: their limits are isomorphic (namely, the pullback). But they're equivalent in another sense: their homotopy limits are weakly equivalent (namely, the homotopy pullback).

I'd like to be able to say that these two diagrams are "blah-limit equivalent", in the sense that if I pick some notion of equivalence ("blah-equivalence"), and consider the notion of limit where the universal property holds up to this notion of equivalence ("blah-limits"), then these two diagrams have blah-equivalent blah-limits.

1. Tiny question. I know that for "blah = isomorphism" and "blah = homotopy equivalence" the two specific diagrams above are blah-equivalent, as described above. Is this true more generally, for other choices of blah? (For example, a choice that recovers other notions of 2-limits).

2. Main question. Is there a more formal way of saying what I'm trying to say here? Given a suitable notion of "equivalence" (maybe e.g. something satisfying 2-out-of-3), is there a structure on the category of diagrams with which I can say two diagrams are somehow equivalent "up to 'limit'", where the 'limit' satisfies the universal property "only up to this notion of equivalence"?

• Z. M: This is an instance of (co)initial functors: ncatlab.org/nlab/show/final+%28infinity%2C1%29-functor
• Naïm Camille Favier: @Z.M Should I assume what you mean is that, if I add enough product projections to the second diagram for it to make sense as a "standalone" category, then the inclusion of the cospan $A \to X \leftarrow B$ into it is initial? Could you spell this out as an answer?
• Zhen Lin: I'm not sure this is as simple as coinitiality. The relationship between the two diagrams is not precomposition by a functor but something more complicated. Personally I would rather explain this in terms of weighted limits.
• Tim: I was guessing that an answer might involve words like "initial functor" and "sketch", but I couldn't figure out anything more concrete than that. I'd be interested to hear what you're thinking about weighted limits @ZhenLin

Answer 1

For simplicity (i.e. to avoid too many ${}^\textrm{op}$) let me discuss colimits instead of limits.

Let $\mathcal{C}$ be a small category. It is well known that $\textbf{Psh} (\mathcal{C})$, the category of presheaves on $\mathcal{C}$, is the category with colimits of small diagrams freely generated by $\mathcal{C}$, i.e. for any category $\mathcal{D}$ with colimits of small diagrams and any functor $\mathcal{C} \to \mathcal{D}$, there is a unique (up to unique isomorphism) functor $\textbf{Psh} (\mathcal{C}) \to \mathcal{D}$ extending the given one (along the Yoneda embedding) and preserving colimits of small diagrams. In particular, any "purely formal" fact about colimits of small diagrams where the "variables" are labelled by $\mathcal{C}$ can be proved by considering what happens in $\textbf{Psh} (\mathcal{C})$ when the "variables" have their canonical values.

For example, $\mathcal{C}$ could be $\{ A \overset{f}{\leftarrow} X \overset{g}{\rightarrow} B \}$. Consider the following pushout diagrams in $\textbf{Psh} (\mathcal{C})$, $$\require{AMScd} \begin{CD} h_X @>{h_g}>> h_B \\ @V{h_f}VV @VVV \\ h_A @>>> P \end{CD} \qquad \begin{CD} h_X \amalg h_X @>{h_f \amalg h_g}>> h_A \amalg h_B \\ @V{\nabla}VV @VVV \\ h_X @>>> Q \end{CD}$$ where $h_\bullet$ denotes the Yoneda embedding $\mathcal{C} \to \textbf{Psh} (\mathcal{C})$. There are evident commutative diagrams of the form below, $$ \begin{CD} h_X @>{h_g}>> h_B \\ @V{h_f}VV @VVV \\ h_A @>>> Q \end{CD} \qquad \begin{CD} h_X \amalg h_X @>{h_f \amalg h_g}>> h_A \amalg h_B \\ @V{\nabla}VV @VVV \\ h_X @>>> P \end{CD}$$ and it can be shown that we obtain an isomorphism $P \cong Q$. Applying the universal property of $\textbf{Psh} (\mathcal{C})$, it follows that in any category with colimits of small diagrams, the pushout of $A \overset{f}{\leftarrow} X \overset{g}{\rightarrow} B$ is canonically isomorphic to $X \overset{\nabla}{\leftarrow} X \amalg X \overset{f \amalg g}{\rightarrow} A \amalg B$. A more sophisticated argument (using the Yoneda embedding) allows us to conclude the same without even assuming the existence of all pushouts – one pushout exists if and only if the other does, and they are the same.

Given a presheaf $W$ on $\mathcal{C}$ and a diagram $D : \mathcal{C} \to \mathcal{D}$, the value of the colimit-preserving functor $\textbf{Psh} (\mathcal{C}) \to \mathcal{D}$ obtained by extending $D$ along the Yoneda embedding is the weighted colimit $W \star_\mathcal{C} D$. (Again, it is possible to define weighted colimits without assuming that all colimits exist, but since it does not clarify matters I will skip that story.) For example, $h_X \star_\mathcal{C} D$ is isomorphic to $D (X)$, of course. The usual colimit $\varinjlim_\mathcal{C} D$ is isomorphic to 1ドル \star_\mathcal{C} D$, where 1ドル$ denotes the presheaf with constant value 1ドル$. A functor $C : \mathcal{C}' \to \mathcal{C}$ is cofinal if and only if $\varinjlim_{\mathcal{C}'} h_C \cong 1$ in $\textbf{Psh} (\mathcal{C})$, because $\varinjlim_{\mathcal{C}'} (h_C \star_\mathcal{C} D) \cong \varinjlim_{\mathcal{C}'} (D C)$. And so on.

Hopefully the above illustrates how the yoga of weighted colimits can be used to make sense of "purely formal" facts about colimits.

• Tim: I'm not sure I really get how the last paragraph (about weighted colimits) follows on from/generalises the previous one (where you show how these diagrams are "equivalent" in the sense that they have the same pushout in $\mathbf{Psh}(\mathcal{C})$). But maybe I just need to reread this a few times.
• Tim: When it comes to different notions of equivalence (e.g. to recover the story about $\mathrm{holim}$ (or, here, $\mathrm{hocolim}$), is the idea that we use the fact that e.g. homotopy (co)limits can be computed by (enriched) weighted (co)limits? And in some sense maybe any notion of "limit, where the universal property holds only up to some notion of equivalence" can also be computed by a weighted limit?
• Zhen Lin: Everything I said applies in (∞, 1)-categories and enriched categories. I would not say that one follows from the other, however.
• Naïm Camille Favier: I don't understand why the first part of your answer involves presheaves. It seems to me that we can prove directly that the two diagrams have the same pushout in any category where they make sense. Why work in $\mathbf{PSh}(\mathcal{C})$ at that point?
• Zhen Lin: It is the right level of generality. Why work in a polynomial ring over the integers when you could just directly prove the identity you want in every ring?