# Enriched coends which preserve equivalences

Although this question might be formulated in higher generality, let me try to be concrete:

Let $$(\mathbf{Top},\times,*)$$ be the monoidal category of compactly generated weak Hausdorff spaces; and let $$\mathbf{C}$$ be a small and $$\mathbf{Top}$$-enriched category (in the easiest case I have in mind: a topological group).

Now let $$A\colon \mathbf{C}^{\mathrm{op}}\to \mathbf{Top}$$, $$A'\colon\mathbf{C}^{\mathrm{op}}\to\mathbf{Top}$$, and $$B\colon\mathbf{C}\to \mathbf{Top}$$ be three enriched functors, and let $$\vartheta\colon A\Rightarrow A'$$ be a natural transformation. I am looking at the induced map between the enriched coends $$\int^{c\in\mathbf{C}}\vartheta_c\times\mathrm{id}_{Bc}\colon \int^{c\in\mathbf{C}}Ac\times Bc\to \int^{c\in\mathbf{C}}A'c\times Bc.$$ Assume that $$\vartheta$$ is an equivalence, in the sense that each $$\vartheta_c\colon Ac\to A'c$$ is a weak equivalence. Under which extra conditions can I conclude that also the induced map $$\int^c\vartheta_c\times\mathrm{id}_{Bc}$$ is a weak equivalence?

Some conditions I could imagine and which I would be happy with could be:

• $$A$$ and $$A'$$ are free in the sense that for each $$c\in\mathbf{C}$$, the group $$\mathrm{Aut}_{\mathbf{C}}(c)$$ acts freely on $$Ac$$ and $$A'c$$,
• for each morphism $$f\colon c\to c'$$ in $$\mathbf{C}$$, the map $$Bf\colon Bc\to Bc'$$ is an (equivariant?) cofibration.

Let me finish the question with some examples I had in mind:

• If $$\mathbf{C}=G$$ is just a topological group, and we write $$A$$, $$A'$$ and $$B$$ for the corresponding $$G$$-spaces, then we just look for the induced map $$A\times_G B\to A'\times_G B$$, and here if is e.g. suffient that $$G$$ acts freely on both $$A$$ and $$A'$$.

• If $$\mathbf{C}$$ is the semisimplex category $$\mathbf{\Delta}^+$$ of finite ordered sets $$[n]=\{0<\dotsb and injective monotone maps, and if $$B([n])=\Delta^n$$, then it is well-known that the coend construction, which agrees with the ‘fat’ geometric realisation of $$A$$ resp. $$A'$$, turns levelwise weak equivalences into weak equivalences, see Ebert–Randal-Williams, Thm. 2.2

• If $$\mathbf{C}$$ is the simplex category $$\mathbf{\Delta}$$, then we additionally need that both $$A$$ and $$A'$$ are proper.

• If $$\mathbf{Inj}$$ is the category of non-negative numbers $$r\ge 0$$ together with injective maps $$\{1,\dotsc,r\}\to \{1,\dotsc,r'\}$$ and $$\mathscr{O}=(\mathscr{O}(r))_{r\ge 0}$$ is an operad with a prefered nullary in $$\mathscr{O}(0)$$, then $$\mathscr{O}$$ gives rise to a functor $$\mathscr{O}\colon\mathbf{Inj}^{\mathrm{op}}\to \mathbf{Top}$$. On the other hand, each based space $$X$$ gives rise to a functor $$X\colon\mathbf{Inj}\to \mathbf{Top}, r\mapsto X^r$$, and the free $$\mathscr{O}$$-algebra over $$X$$ is calculated as $$\int^{r\in\mathbf{Inj}}\mathscr{O}(r)\times X^r$$. Now if $$\varphi\colon \mathscr{O}\to\mathscr{O}'$$ is a morphism of operads, we get a morphism of free algebras $$\int^r\mathscr{O}(r)\times X^r\to \int^r\mathscr{O}'(r)\times X^r$$, which should be the unit of the base-change adjunction $$\varphi_!\dashv \varphi^*$$. Now if $$\mathscr{O}$$ and $$\mathscr{O'}$$ are $$\Sigma$$-cofibrant and each $$\varphi_r$$ is a weak equivalence, then the induced map on free algebras is a weak equivalence if $$X$$ is cofibrant, see in much higher generality Berger–Moerdijk, Prop. 5.7.

One sufficient condition is that either $$B$$ or both $$A$$ and $$A'$$ are cofibrant in the projective model structure.
If $$C$$ is a Reedy category, possibly in the generalized sense defined by Berger and Moerdijk, and $$A$$, $$A'$$ and $$B$$ are cofibrant in the Reedy model structure, this is sufficient too.