# 'Quotient' of indexed categories and pushforward congruence relations

If $$R\subset C$$ is a vector subspace of $$C$$, and $$F: C \to D$$ is a linear map, then we have a natural linear map $$C/R\to D/F(R)$$. I was wondering if this can be also generalized to categorical settings. For example, let $$\mathscr F:\mathscr C\to\mathscr D$$ be a functor and let $$\mathcal R$$ be a congruence relation on $$\mathscr C$$ which gives rise to a quotient category $$\mathscr C/\mathcal R$$.

Question 1: Under what conditions can we get a congruence relation $$\mathcal S$$ which somehow represents ''$$\mathscr F(\mathcal R)$$'' so that whenever $$f\sim_{\mathcal R}g$$, we have $$\mathscr F(f)\sim_{\mathcal S} \mathscr F(g)$$? If so, do we have a natural functor like $$\mathscr C/\mathcal R\to \mathscr D/\mathscr F(\mathcal R)$$?

EDIT: It seems that there is a nice situation where a sort of pushforward' could be reasonable:

For simplicity, let's assume everything is strict. Take a $$\mathscr S$$-indexed category, namely, a 2-functor $$\pmb C:\mathscr S^{op}\to \mathsf{Cat}$$. Explicitly, we have a category $$\pmb C(I)$$ for each object $$I$$ of $$\mathscr S$$ and a functor $$f^*:\pmb C(I) \to \pmb C(J)$$ for every arrow $$f:J\to I$$ satisfying $$(\mathrm{id}_I)^*=\mathrm{id}_{\pmb C(I)}$$ and $$(g\circ f)^*=f^*g^*$$. Then it seems that every congruence relation $$\mathcal R$$ on $$\mathscr S$$ gives rise to a congruence relation' for $$\pmb C$$ by requiring $$f^*\approx g^*$$ whenever $$f\sim_{\mathcal R} g$$. Then this relation '$$\approx$$' is sort of analogous to a congruence relation in that if $$f_1^*\approx f_2^*$$ and $$g_1^*\approx g_2^*$$, i.e. $$f_1\sim_{\mathcal R} g_2$$ and $$g_1\sim_{\mathcal R} g_2$$, then by definition $$g_1\circ f_1\sim_{\mathcal R} g_2\circ f_2$$ and therefore we see $$f_1^*g_1^*=(g_1\circ f_1)^*\approx (g_2\circ f_2)^* =f_2^*g_2^*$$

So, it is very interesting to ask:

Question 2: How do we develop 'quotients' of indexed categories?

For example, suppose $$\pmb C:\mathscr S^{op}\to \mathsf {Cat}$$ is an indexed category. As we discussed above, a reasonable quotient of $$\pmb C$$ might come from a quotient of $$\mathscr S$$. In view of another post, suppose there was indeed a reasonable definition of quotient indexed category, what concept of fibered categories was related to it under the Grothendieck construction?

There is no need for bringing in indexed categories here. Just define the congruence relation $$\mathcal S$$ on $$\mathcal D$$ to be the least congruence relation so that $$f\mathcal S g$$ if $$f=F(f')$$ and $$f'=F(g')$$ with $$g\mathcal Rg'$$. This exists since intersections of congruence relations are congruence relations. Note that a slightly more elaborate construction would also work of $$\mathcal R$$ was an actual congruence, which might identify some objects. This would work in much more general categories than the category of (small) categories, namely in any Barr-exact category.
• Thank you. Maybe the indexed category thing is just another question. In this case, does the least congruence relation agree with the relation $\approx$ I introduce here?