Simplicial orbit realization functor.

Question

In their paper, Singular functors and realization functors, Dwyer and Kan define a "realization functor" $\mathbf{O}\otimes - \colon \mathbf{S}^{\mathbf{O}^{op}}\to \mathbf M$ for a tensored and cotensored simplicial category $\mathbf M$ and a simplicial subcategory $\mathbf{O\hookrightarrow M}$ satisfying certain conditions which (I think) are not directly relevant to my question. Their definition is as follows: I find the definition of the first morphism unclear, I don't see how $X_g$ is defined, since we are not taking $O_e\otimes \Delta[n]$ to be in $\mathbf O$ necessarily, and further, it doesn't seem to have the correct variance. Am I missing something?

Answer 1

I think the first arrow should go from the right to the left. As remarked in the comments, this would give you a simplicially enriched coend.

Since $\mathbf{M}$ is tensored over $\mathbf{S}$, we have an isomorphism of simplicial sets $$\mathbf{S}(\Delta[n],\mathbf{O}(O_e,O_{e'}))\cong \mathbf{O}(\Delta[n]\otimes O_e,O_{e'}).$$ So after taking $0$-simplices we see that a morphism $\Delta[n]\otimes O_e\to O_{e'}$ is the same as an $n$-simplex of $\mathbf{O}(O_e,O_{e'}) = \mathbf{O}^{\mathrm{op}}(O_{e'},O_e)$. Since $X$ is a (contravariant) simplicially enriched functor, it comes equipped with a map of simplicial sets $X\colon \mathbf{O}^{\mathrm{op}}(O_{e'},O_{e})\to \mathbf{S}(XO_{e'},XO_{e})$. The right hand side has $n$-simplices $\mathrm{Hom}_\mathbf{S}(\Delta[n]\times XO_{e'},XO_{e})$. Thus, if $g\colon \Delta[n]\otimes O_e\to O_{e'}$ is an $n$-simplex of the left hand side, it gets sent to a map of simplicial sets $X_g\colon \Delta[n]\times XO_{e'}\to XO_{e}$.