# $V$-cat and $V$-graph: coequalizers in the category of enriched functors

This question is regarding the 1974 JPAA paper $$V$$-cat and $$V$$-graph by Harvey Wolff.

To be precise, I don't understand a certain step in the proof of Corollary 2.9, which (the corollary) is crucial for the main theorem of the paper.

Let $$V$$ be a closed symmetric monoidal category with coequalizers, $$A$$ an $$V$$-enriched category and $$U$$ the forgetful functor from $$V$$-enriched categories to $$V$$-enriched graphs. A pair $$F,G\colon D\to U(A)$$ is called a pre-$$V$$-congruence in $$A$$ if $$Ob(D) = Ob(A)$$ ($$D$$ is a $$V$$-graph and $$A$$ is a $$V$$-category). Given a pre-$$V$$-congruence $$(F,G)$$, there is an associated $$V$$-graph $$E: Ob(E) = Ob(A)$$ and $$E(A,B)$$ is the coequalizer of $$F_{A,B},G_{A,B}\colon D(A,B)\to A(A,B)$$ in $$V$$. It comes with an associated morphism $$L$$ of $$V$$-graphs where $$L_{A,B}$$ is the coequalizer map. This graph is a coequalizer of $$F$$ and $$G$$ in the category $$V$$-graphs. A pre-$$V$$-congruence in a $$V$$-congruence if $$E$$ is a $$V$$-category is a way that $$L$$ is a $$V$$-enriched functor.

Now Corollary 2.9.(ii) says that if $$F_1,F_2\colon A\to B$$ are $$V$$-functors such that $$(U(F_1),U(F_2))$$ is a pre-$$V$$-congruence, for which there exists a $$V$$-functor $$H\colon B\to A$$ satisfying $$F_1H = 1$$, then it is a $$V$$-congruence.

Now for the step I don't understand: the author goes from $$M_B\circ (1\otimes (F_2H))$$ to $$F_2H\circ M_B$$ where $$M_B$$ is the composition of morphisms in $$B$$. I've got a hunch that either I'm missing something obvious and making a fool of myself posting this or there is a mistake, but an enriched functor should satisfy $$M_B\circ((F_2H)\otimes (F_2H)) = (F_2H)\circ M_B$$, and not what is written there.

If I indeed am missing something trivial, please, be gentle: I'm not great at enriched category theory, and my only interest of these questions is understanding why the category of small dg-categories is cocomplete.

Edit. If the proof is indeed incorrect, can it still be salvaged? Alternatively, even if it's not the question per se, I would still be satisfied with the excplicit construction in the case $$V = Ch(k)$$, in the case of dg-categories and dg-functors.

• (Aside: somewhere in the paper it's presumably important that $V$ is symmetric monoidal closed with coequalizers -- or at least that $v\otimes (-)$ preserves coequalizers for each $v \in V$.) I haven't gone into the details, but 2.9(ii) is a strange statement. Rarely is such a diagram relevant. I would think that Wolff needs it to show that $VCat$ is monadic over $VGph$ using the Beck monadicity theorem, in which case he probably only needs it for a split coequalizer, which has an extra equation. Feb 5, 2022 at 21:01
• To be more explicit, if the point is to verify the hypotheses of the Beck monadicity theorem, it's probably sufficient to assume the additional equation $GHF = GHG$, making $G,H,F$ into what's called a "contractible pair" on the above-linked nlab page. The resulting data can be thought of as a 1-truncated simplicial object with extra degeneracies. Feb 5, 2022 at 21:12
• Ah, sorry, "split coequalizer" or "contractible pair" in the terminology of the nlab page I linked to is different from what Wolff calls a "split coequalizer" in 2.9(i) (which I would rather call a "reflexive coequalizer). Any coequalizer arising from an adjunction will almost surely be contractible in the sense of the nlab page I linked to. It's funny that Wolff should structure his argument that way -- the fact that $VCat$ has coequalizers follows anyway from monadicity (since $VGph$ is cocomplete), and nowhere in proving monadicity does one need to understand general coequalizers. Feb 5, 2022 at 21:31
• As alluded to, there's a completely general formula for any category of algebras of a monad. Unfortunately, for $VCat$, coequalizers in general are nasty, so to get anywhere useful one generally wants to take into account more information about whatever particular coequalizer one is interested in. Also, coproducts + pushouts + filtered colimits also give all colimits -- I find the pushout formula in $VCat$ slightly easier to grok than the coequalizer one. Feb 5, 2022 at 21:39
• Filtered colimits are created by the forgetful functor to $VGph$. This follows from monadicity + the fact that the free $V$-category monad preserves filtered colimits. That is, a filtered colimit in $VCat$ is computed by taking the filtered colimit of the underlying graphs, and then defining composition in the obvious way. Feb 7, 2022 at 13:08