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From what I understand, the singular homology groups of a topological space are defined like so:

Topological Particulars. There's a covariant functor $F : \mathbb{\Delta} \rightarrow \mathbf{Top}$ that assigns to each natural number $n$ the corresponding $n$-simplex. This yields a functor $$\mathbf{Top}(F-,-) : \Delta^{op} \times \mathbf{Top} \rightarrow \mathbf{Set}.$$ Hence to each topological space $X$, we can assign a simplicial set $\mathbf{Top}(F-,X) : \Delta^{op} \rightarrow \mathbf{Set}.$

General nonsense. We observe that every simplicial set induces a simplicial abelian group; that every simplicial abelian group induces a chain complex; and that chain complexes have homology and cohomology groups. Ergo, simplicial sets have homology/cohomology groups.

Putting these together, we may speak of the homology and cohomology groups of a topological space $X$. However, the topological particulars don't seem too important. In fact, for any category $\mathbf{C}$ and any functor $F : \Delta \rightarrow \mathbf{C}$, there's a simplicial set $\mathbf{C}(F-,X)$ attached to each $X \in \mathbf{C}$, and therefore $X$ has homology and cohomology.

For example, the underlying set functor $U : \mathbf{CMon} \rightarrow \mathbf{Set}$ has a left-adjoint $F : \mathbf{Set} \rightarrow \mathbf{CMon}$. But since $\Delta \subseteq \mathbf{Set}$ and $\mathbf{CMon} \subseteq \mathbf{Mon}$, this yields a functor $F : \Delta \rightarrow \mathbf{Mon}$. This should in turn allow us to attach homology and cohomology groups to each monoid $M$, by studying the simplicial set $\mathbf{Mon}(F-,M)$.

Question. Is this a thing? If not, why not?

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    $\begingroup$ Not all such things will satisfy the Eilenberg-Steenrod axioms. $\endgroup$ – Ittay Weiss Nov 5 '16 at 5:34
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    $\begingroup$ @IttayWeiss, yes, sure; and, the homotopy axiom doesn't even make sense at this level of generality. But, still, we can study these groups. Is there any reason not to? $\endgroup$ – goblin Nov 5 '16 at 5:36
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    $\begingroup$ I'm not sure what 'Is this a thing?' is really asking. I guess one of: Is it possible? Is it interesting? Is it studied? The first has an answer, so too the last but probably its more difficult. The second one for reasons which are themselves not very interesting, is more or less unanswerable. $\endgroup$ – user259242 Nov 5 '16 at 5:52
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    $\begingroup$ @user259242 And therein lies your mistake. $\endgroup$ – Alex Youcis Nov 5 '16 at 6:08
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    $\begingroup$ @user259242 OK. I was merely commenting on the fact that '...just because it's more advanced (and therefore probably interesting)...' can be a bad perspective. It was just a harmless jab, hopefully taken in good spirits :) $\endgroup$ – Alex Youcis Nov 5 '16 at 7:28
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If I understood correctly, you have a cosimplicial object $F^\bullet \in \mathsf{cC}$ (AKA a functor $F : \Delta \to \mathsf{C}$), and an object $X \in \mathsf{C}$; and you're considering the simplicial set $\operatorname{Hom}_{\mathsf{C}}(F^\bullet, X) \in \mathsf{sSet}$. Sure, people use constructions like this from time to time, it's a very general construction... But since it's so general it's hard to get more specific than that. It occurs in tons of different settings.

I don't think it's really fair to call that "the homology of $X$", either; it heavily depends on what $F^\bullet$ is. For example when you have a category tensored over $\mathsf{sSet}$, given two objects $X$ and $Y$, you can build the mapping space $$\operatorname{Map}_{\mathsf{C}}(X,Y) = \operatorname{Hom}_{\mathsf{C}}(X \otimes \Delta^\bullet, Y) \in \mathsf{sSet}$$ which is used very, very often, satisfying among other things $\pi_0 \operatorname{Map}_{\mathsf{C}}(X,Y) = [X,Y]$ is the set of homotopy classes of map $X \to Y$.

Even more specifically the singular simplicial set $S_\bullet(X)$ is given by $\operatorname{Map}_{\mathsf{Top}}(*, X)$ (where $\mathsf{Top}$ is tensored over simplicial sets in the standard fashion). So homology is really a special case of a special case.

What you're considering is very general. Homology is interesting because it satisfies things like the Eilenberg–Steenrod axioms, we have theorems like the UCT, Künneth's theorem... You can prove a great deal about homology using the setting you're considering (for example $\operatorname{Hom}_{\mathsf{C}}(F^\bullet, X \times Y) = \operatorname{Hom}_{\mathsf{C}}(F^\bullet, X) \times \operatorname{Hom}_{\mathsf{C}}(F^\bullet, Y)$ is obvious, and then you have the Eilenberg–Zilber theorem and finally Künneth's formula), but many other properties heavily depend on the specific $F^\bullet = |\Delta^\bullet|$ used.

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