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I know of two ways things called "homology" are constructed in mathematics, and would like to know how, if at all, these two types of constructions are one in the same. To keep things simple, I'll only work with $\mathbb Z$ coefficients. $\newcommand\Ab{\mathrm{Ab}}\newcommand\sSet{\mathrm{sSet}}\newcommand\sAb{\mathrm{sAb}}\newcommand\Ch{\mathrm{Ch}}\DeclareMathOperator{\im}{im}\newcommand\Top{\mathrm{Top}}\DeclareMathOperator\Hom{Hom}$

One way to construct a homology theory is to start with an arbitrary category $\mathcal C$ and a functor $N:\mathcal C\to\mathrm{sSet}$ to the category of simplicial sets. Now, given any object in $\mathcal C$, you can produce homology groups by composing $N$ with the functor $F:\mathrm{sSet}\to\mathrm{sAb}=\mathrm{Ab}^{\Delta^\mathrm{op}}$ sending a simplicial set $X$ to the simplicial abelian group whose $n$-simplices are the free abelian group $F(X)_n=\mathbb Z^{\oplus X_n}$ on $X_n$. Then, you compose with the Dold-Kan functor $D:\mathrm{sAb}\to\mathrm{Ch}_\bullet(\Ab)$ to get some chain complex, and take homology of that. In summary, start with some functor $N:\mathcal C\to\sSet$, perform the composition

$$\mathcal C\xrightarrow N\sSet\xrightarrow F\sAb\xrightarrow D\Ch_\bullet(\Ab),$$

and then take $\ker/\im$ of the result. This is how e.g. singular homology is formed with $N:\mathrm{Top}\to\sSet$ taking a space $X$ to its singular set $N(X)_n=\Hom_\Top(\Delta^n,X)$.

Alternatively, you can start with an abelian category $\mathcal A$ and a right-exact functor $F:\mathcal A\to\Ab$. Then, you automatically get left-derived functors $L^kF:\mathcal A\to\Ab$ formed by starting with a projective resolution $P_\bullet\to X\to0$ of some $X\in\mathcal A$, and then taking homology of the chain complex $N(P_\bullet)$ obtained by applying $N$ to $P_\bullet$ objectwise. This is e.g. how one constructs Tor by starting with $\mathcal A=\Ab$ and $F=-\otimes_{\mathbb Z}B$ where $B$ is some fixed abelian group.

My main question is the following: is there a general framework of constructing homology theories that subsumes both of these? I would be particularly interested in knowing if in fact, one of these already subsumes the other (e.g. maybe the formation of derived functors always passes through the category of simplicial sets).

One quick indication that the two are related is the case of group homology. Given a group $G$, you can form group homology $H_n(G;-):\mathbb Z[G]\mathrm{-Mod}\to\Ab$ as the left derived functors of taking $G$-coinvariants $M\rightsquigarrow M_G=M\otimes_{\mathbb Z[G]}\mathbb Z$. However, if $M=\mathbb Z$ then you get the same homology groups via the first construction by starting with $N:\mathrm{Grp}\to\sSet$ the functor sending a group $G$ to its nerve whose $n$-simplices are $N(G)_n=\Hom_{\mathrm{Cat}}([n],G)$, the set of functors from the poset $[n]=(0\to1\to\dots\to n)$ to the group $G$ viewed as a one-object category, because now the first construction boils down to computing group homology using the bar resolution.

Any comments/answers mentioning other homology theories that could be constructed in both ways, or mentioning how to construct $H_n(G;M)$ by starting with a functor to $\sSet$ in cases where $G$ acts on $M$ non-trivially would be appreciated even if they do not answer the main question.

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  • $\begingroup$ What should I understand a 'homology theory' to be? $\endgroup$
    – Tyrone
    Feb 5 '20 at 11:58
  • $\begingroup$ I was intentionally vague about that because I do not know a good formal definition off the top of my head. At the very least, it should be a functor $\mathcal C\to\mathrm{Ch}_\bullet(\mathcal A)$ from some category $\mathcal C$ to the category of chain complexes on some abelian category $\mathcal A$ (Maybe actually the homotopy category of chain complexes), but this alone feels too permissive. I want the way you end up with a chain complex to be natural in some sense that I do not know how to put in words. $\endgroup$
    – Niven
    Feb 7 '20 at 9:46
  • $\begingroup$ You can relate the two concepts by the notion of ``simplicial resolutions''. $\endgroup$
    – Pedro Tamaroff
    Feb 9 '20 at 17:18
  • $\begingroup$ Is the idea (something like) the following: I start with a category $\mathcal C$ (one for which the simplicial category $s\mathcal C$ has a model structure) with a final element $*$, a (somehow nice?) functor $F:\mathcal C\to\mathrm{Ab}$, and a functor $R:\mathcal C\to[\Delta^{\mathrm{op}},\mathcal C]=s\mathcal C$ with the property that there are (functorial) maps $R(c)\to c$ realizing $R(c)$ as a fibrant resolution (I view $c\in s\mathcal C$ as the simplicial object with $c$ are its 0-vertices and $*$ as its $n$-vertices for all $n\neq0$)... $\endgroup$
    – Niven
    Feb 11 '20 at 4:03
  • $\begingroup$ ...Given this, I compose $F,R$ to get a functor $F:\mathcal C\to\mathrm{sAb}$ and then Dold-Kan + ker/image that? If I have completely missed the mark, would you mind helping me correct course? $\endgroup$
    – Niven
    Feb 11 '20 at 4:08

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