\to M$, $$g\circ f := (\Delta[n] \times K \stackrel{diag \times K}{\longrightarrow} \Delta[n] \times\Delta[n] \times K\stackrel{\Delta[n] \times f}{\longrightarrow }\Delta[n] \times L \stackrel{g}{\to} M);$$ Identity : $id_K : \Delta[0] \times K \stackrel{\cong}{\to} K$,

\(ii) $\mathcal{T}op$, ‘the’ category of spaces (of course, there are numerous variants but you can almost pick whichever one you like as long as the constructions work): $$\mathcal{T}op(X,Y)_n := Top(\Delta^n \times X, Y)$$ Composition and identities are defined analogously to in (i).

\(iii) For each $X$, $Y \in Cat$, the category of small categories, then we similarly get $\mathcal{C}at(X,Y)$, $$\mathcal{C}at(X,Y)_n = {Cat}([n] \times X, Y).$$ We leave the other structure up to the reader.

\(iv) $\mathcal{C}rs$, the category of crossed complexes: see for background and other references, and Tonks, [@andythesis] for a more detailed treatment of the simplicially enriched category structure; $$\mathcal{C}rs(A,B) := Crs(\pi(n)\otimes C, D)$$ Composition has to be defined using an approximation to the identity, again see [@andythesis].

\(v) $\mathcal{C}h^+_K$, the category of positive chain complexes of modules over a commutative ring $K$. (Details are left to the reader, or follow from the Dold-Kan theorem and example (vi) below.)

\(vi) $\mathcal{S}(Mod_K)$, the category of simplicial $K$-modules. The structure uses tensor product with the free simplicial $K$-module on 