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Given a category, one is often interested in the category of (abelian) group and (commutative) ring objects in it. I would like to know what exactness properties such categories and their simplicial analogues have, e.g what can be said about the simplicial abelian groups and commutative ring objects in some category $\mathsf C$ depending on the completeness and exactness properties of $\mathsf C$.

I am also interested in the behavior of objects and arrows defined by lifting properties, e.g injectives, strong epis, etc. For instance, if $\mathsf C$ has enough injectives, when does this hold for (simplicial) algebraic objects in $\mathsf C$?

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    $\begingroup$ @მამუკაჯიბლაძე, what do you mean? $\endgroup$ – Noah Schweber Aug 11 '16 at 22:27
  • $\begingroup$ There's the Dold-Kan correspondence, which might be useful. $\endgroup$ – user97187 Aug 14 '16 at 5:57
  • $\begingroup$ For an example involving weak hypotheses and a weak conclusion which is specifically relevant to commutative ring objects, see here. $\endgroup$ – Tim Campion Aug 15 '16 at 2:32
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If $C$ is a regular or (Barr-)exact category and $T$ is a Lawvere algebraic theory, then the category $Alg_T$ of $T$-algebras is also regular or exact, respectively. Reference: Exact Categories by Michael Barr, theorem 5.11 (pdf). In particular, if $C$ is exact, then the category of abelian group objects $Ab(C)$ is additive and exact, i.e., is an abelian category.

Also if $C$ is regular/exact, then so is any functor category $C^D$ for $D$ small. So if $C$ is exact, then so is for example $Ab(C^{\Delta^{op}})$, the category of simplicial abelian groups in $C$.

A convenient set of axioms on an abelian category that guarantees enough injectives is given by the notion of Grothendieck category (after the famous Tohoku paper): a cocomplete abelian category with exact filtered colimits and a generator. Suppose that $C$ is a category such that

  • $C$ is Barr-exact and cocomplete,

  • filtered colimits commute with finite limits.

Then $Ab(C)$ is a cocomplete category with exact filtered colimits. To see this, it helps to know the following facts:

  • Filtered colimits exist in $Ab(C)$ and are computed just as they are in $C$ (this is true for any category of algebras $Alg_T$).

  • $Ab(C)$ has arbitrary coproducts (just take a filtered colimit over a system of finite coproducts), hence is a cocomplete abelian category.

About the condition that $Ab(C)$ has a generator: well, if $C$ has a generator $G$ and the underlying functor $U: Ab(C) \to C$ is monadic, then we can construct a free abelian group object $F(G)$ and this will be a generator in $Ab(C)$. Under the relatively strong exactness condition that products distribute over colimits in $C$, one can prove this monadicity (a more general result is given here), but otherwise I'm not sure what general conditions on $C$ guarantee existence of generators in $Ab(C)$. At least this tells us that if $C$ is a Grothendieck topos, then $Ab(C)$ is a Grothendieck category (but maybe you knew that already).

On constructing injective hulls in categories, you may find additional enlightenment in this paper by Barr.

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