If $X, Y$ are simplicial sets, and $f, g: X\to Y$ are two simplicial maps between them, we can define a homotopy of simplicial sets $h: X\times I\to Y$ or, equivalently, $h': X\to Y^I$.
The latter definition can be broken down into something which is quite combinatorial, a mess of morphisms; it makes sense in any category, so for any category $\mathcal{A}$, we have if I let $\mathbf{S}(A)$ denote the category of simplicial objects in $\mathcal{A}$, we can define a homotopy between any two maps $f,g: X\to Y$ in $\mathbf{S}(\mathcal{A})$. This definition is the one given in May's Simplicial Objects in Algebraic Topology.
This definition is certainly useful in an arbitrary category, because if $Y_\bullet$ is a simplicial object, then if we come up with any functor $\mathcal{A}\to \mathbf{Set}$ (for instance, a Hom functor) then we can get a simplicial set. If $\mathcal{A}$ is Abelian, we can do homological algebra in $\mathcal{A}$ directly; or find a functor from $\mathcal{A}$ into an Abelian category.
I am interested in something else. An augmented cosimplicial object is the "free diagram generated by a monoid in a monoidal category." So there are many cosimplicial objects in monoidal categories that are not necessarily Abelian. And we can develop some basic notions of homological algebra in these settings, or more generally when there is a monoidal category $\mathcal{A}$ that acts on an ordinary category $\mathcal{C}$ as a "module", we can develop some notions of homological algebra in that setting relative to a choice monoid in $\mathcal{A}$. In particular taking $\mathcal{A}= [\mathcal{C},\mathcal{C}]$ we get homological algebra in $\mathcal{C}$ relative to a monad.
So, on this note, I have two questions.
- If $\mathcal{A}$ is an arbitrary category - possibly monoidal but not necessarily additive, with whatever limits and colimits you want - is there any inherent meaning to a simplicial homotopy of maps between simplicial objects in $\mathcal{A}$? Any kind of useful purpose which can be expressed without talking about passing to $\mathbf{Sets}$ or an Abelian category?
- What is the most efficient, concise way to write down the definition of a simplicial homotopy in an arbitrary category? It is no longer legitimate to speak of a map $X\to Y^I$ when $I$ does not necessarily exist and the category of simplicial objects is no longer necessarily Cartesian closed. Is there any good definition shorter than the mess of morphisms in May?