Existence of abelianization functor

Question

When does the abelianzation functor exists? I am reading Quillen's paper "On the (co-)homology of commutative rings" and in it he states that when $\mathscr{C}$ is an algebraic category (closed under inductive limits and having small projective generators) then the inclusion functor from the category $(\mathscr{C}/X)_{ab}$ of abelian group objects in $\mathscr{C}/X$ to $\mathscr{C}/X$ admits a left adjoint $\mathbf{Ab}$. Why is this true? If $\mathscr{C}$ is a model category instead is it still true?

Answer 1

I'll flesh out the details here, if only for my own sake since I wasn't remembering to use the key theorem cited below before Zhen's hint. The key observation is that an "algebraic category" in Quillen's sense must be locally presentable.

Indeed, more is true, because the projectivity is a red herring: Theorem 6.9 in Adamek-Rosicky-Vitale's Algebraic Theories proves that an algebraic category in Quillen's sense for which the projective generators are finitely presentable is precisely the category of finite-product-preserving functors into $\mathbf{Set}$ out of some small category with finite products (that is, produces of less than $\kappa$ objects.) Such categories are a strictly special case of locally finitely presentable categories. The situation with algebraicity when the small generators are not finitely presentable is more complicated.

Anyway, all we properly need is Theorem 1.20 of Adamek-Rosicky's Locally Presentable and Accessible Categories, which shows that Quillen's algebraic categories are locally presentable since they're cocomplete with a set of small strong generators. Again, the projectivity is irrelevant here.

From here, it's all standard results about locally presentable categories. If $\mathcal C$ is any complete (for instance, locally presentable) category, then the forgetful functor $U:\mathcal C_{\mathrm{ab}}\to \mathcal C$ preserves limits. If $\mathcal C$ is locally $\kappa$-presentable, then finite limits in $\mathcal C$ commute with $\kappa$-filtered colimits (because they do so in $\mathbf{Set},$ thus in presheaf categories, and thus in subcategories of presheaf categories closed under $\kappa$-filtered colimits, which is one equivalent definition of $\kappa$-locally presentable categories), so $U$ also preserves $\kappa$-filtered colimits. Thus $U$ is a continuous, accessible functor between locally presentable categories, so it has (by, say, 1.66 of Adamek-Rosicky) a left adjoint.