1 Freely adjoining colimits
We start with the problem of freely adjoining colimits. The simplest case is to freely adjoin all small colimits, which gives
[1, Theorem 5.1.5.6] Let be a small -category. Then the Yoneda embedding is the free co-completion of . That is, for any co-complete category , pre-composition with the Yoneda embedding gives an equivalence
where is the category of co-continuous functors .
If we only want to add some colimits, we restrict to a subcategory of .
[1, Proposition 5.3.6.2] Let be a small -category and a collection of simplicial sets. Let be the full subcategory of generated by representables under -indexed colimits. Then for any category with -indexed colimits, pre-composition with the Yoneda embedding gives an equivalence
where is the category of -indexed colimit-preserving functors .
While the category exists, it is pretty difficult to reason about in general. Given a presheaf, there is no clear criterion one can use to check whether it is in . Consequently, it is also difficult to prove categorical properties of , e.g. if it is presentable.
Thankfully, in certain cases of interest, we can describe as the category of presheaves that preserve certain limits. Combining [1, Lemmas 5.5.4.16-18], we learn that such categories are accessible localizations of , and in particular presentable. There are two main such examples:
Let be a small -category with finite colimits and be the collection of filtered simplicial sets. Then is the full subcategory of consisting of finite limit-preserving presheaves. That is, it sends finite colimits in to finite limits in . Moreover, the Yoneda embedding preserves all finite colimits.
Let be a small -category with finite coproducts and be the collection of filtered simplicial sets and . Then is the full subcategory of consisting of (finite) product-preserving presheaves. Moreover, the Yoneda embedding preserves all finite coproducts.
The is that finite colimits are “complementary” to filtered colimits, while coproducts are “complementary” to (filtered colimits + geometric realization). Specifically, the first result follows from the facts that
filtered colimits commute with finite limits in ; and
every colimit is a filtered colimit of finite colimits.
Recall that our original motivation was to freely add cokernels to an abelian category. In a non-abelian setting, we would want to add coequalizers, or rather their derived analogues — geometric realizations. This approach is taken by [2], but results in a less pretty category. Our approach here is slightly different, and is based on the ideas of [2, Section 6.4].
The main observation is that in most cases, our category is generated freely by its compact objects . That is, we have . Instead of freely adding filtered colimits to , then freely adding geometric realizations, a better strategy is to start with and freely add filtered colimits and geometric realizations in one go. The restricted Yoneda functor is easily seen to be fully faithful and preserve filtered colimits. Since is usually essentially small, this also lets us avoid size issues.
We first show that preserves finite colimits. This follows from a straightforward calculation
Since filtered colimits of spaces commute with finite limits, we know that is closed under filtered colimits. Since representables preserve finite limits, we know that .
To show the other inclusion, let . Then we can write
where is filtered and each is a finite colimit of representables. Now suppose that . Our goal is to write as a filtered colimit of representables.
Let be the inclusion, and its left adjoint. Then they both preserve filtered colimits, and
So it suffices to show that is representable. Let . Then we have
where the second colimit is taken inside . But preserves finite colimits, so the right-hand side is simply given by .