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I’m interested in definition of a homology of a map in model category $C$, as an example let’s take $C = \mathrm{sGrp}$.

Let $\Gamma$ be a discrete group, its Quillen homology groups defined as $H_n \Gamma = \pi_{n-1} (Q\Gamma)_{ab}$. Here $Q\Gamma$ is a cofibrant replacement (free resolution) of $\Gamma$ in $\mathrm{sGrp}$. Of course, they coincide with usual group homology, since we can take as $Q\Gamma=G\bar W \Gamma$ and for any (reduced) simplicial set $X$ $\pi_{n-1}(GX)_{ab}=\bar H_n X$.

Now I want to associate a similar invariant to a homomorphism of groups $ G \xrightarrow f H $. Traditional way to do so is to take (hyperhomology, so to speak) $H_n f = \pi_{n-1} C_{ab}$, here $C=\mathrm{hocofib_{sGrp}}\{QG\to QH\}$. Defined this way, $H_n f$ fits into a long exact sequence

$$ \dots\to H_n G\to H_n H\to H_n f\to\dots $$

This definition looks okay, but I am just curious, is it possible to define homology of $f$ as a «true» Quillen homology, perhaps, in another model category. For example, if we treat $f$ as an object in $G\downarrow \mathrm{sGrp}$, then we can define homology of $f$ as a Quillen homology in this category: $H^{\downarrow}_n f = \pi_{n-1} (Qf)_{ab}$. Here $Qf$ is a cofibrant replacement of $f$ in $G\downarrow \mathrm{sGrp}$ which actually comes from functorial factorization of $f$ into cofibration and acyclic fibration $$ G\to Qf\to H $$

Now we can compare $H_n^{\downarrow} f$ and $H_n f$. Immediately for a trivial morphism from a free group $F\xrightarrow f 1$ we can see that $H_1^{\downarrow} f\neq H_1 f$.

Question Is it possibly to define Quillen homology of $f$ in a way that it will coincide with a «natural» definition, i.e. fit into long exact sequence.

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