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 $[n]=\{0<\dotsb<n\}$ 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.
