Construction of synthetic spectraFreely adjoining colimits

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

Theorem 1.1

[1, Theorem 5.1.5.6] Let C{\mathcal{C}} be a small \infty -category. Then the Yoneda embedding y ⁣:CP(C)=Fun(Cop,Spc)y\colon {\mathcal{C}}\to P({\mathcal{C}}) = \operatorname{Fun}({\mathcal{C}}^{\mathrm{op}}, {\mathrm{Spc}}) is the free co-completion of C{\mathcal{C}}. That is, for any co-complete category D{\mathcal{D}}, pre-composition with the Yoneda embedding gives an equivalence

FunL(P(C),D)Fun(C,D), \operatorname{Fun}^L(P({\mathcal{C}}), {\mathcal{D}}) \to \operatorname{Fun}({\mathcal{C}}, {\mathcal{D}}),

where FunL(P(C),D)\operatorname{Fun}^L(P({\mathcal{C}}), {\mathcal{D}}) is the category of co-continuous functors P(C)DP({\mathcal{C}})\to {\mathcal{D}}.

If we only want to add some colimits, we restrict to a subcategory of P(C)P(C).

Theorem 1.2

[1, Proposition 5.3.6.2] Let C{\mathcal{C}} be a small \infty -category and K\mathcal{K} a collection of simplicial sets. Let PK(C)P^{\mathcal{K}}({\mathcal{C}}) be the full subcategory of P(C)P({\mathcal{C}}) generated by representables under K\mathcal{K}-indexed colimits. Then for any category D{\mathcal{D}} with K\mathcal{K}-indexed colimits, pre-composition with the Yoneda embedding gives an equivalence

FunK(PK(C),D)Fun(C,D), \operatorname{Fun}_{\mathcal{K}}(P^{\mathcal{K}}({\mathcal{C}}), {\mathcal{D}}) \to \operatorname{Fun}({\mathcal{C}}, {\mathcal{D}}),

where FunK(PK(C),D)\operatorname{Fun}_{\mathcal{K}}(P^{\mathcal{K}}({\mathcal{C}}), {\mathcal{D}}) is the category of K\mathcal{K}-indexed colimit-preserving functors PK(C)DP^{\mathcal{K}}({\mathcal{C}}) \to {\mathcal{D}}.

While the category PK(C)P^{\mathcal{K}}({\mathcal{C}}) 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 PK(C)P^{\mathcal{K}}({\mathcal{C}}). Consequently, it is also difficult to prove categorical properties of PK(C)P^{\mathcal{K}}({\mathcal{C}}), e.g. if it is presentable.

Thankfully, in certain cases of interest, we can describe PK(C)P^{\mathcal{K}}({\mathcal{C}}) 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 P(C)P({\mathcal{C}}), and in particular presentable. There are two main such examples:

Theorem 1.3 ([1, Corollary 5.3.5.4])

Let C{\mathcal{C}} be a small \infty -category with finite colimits and K\mathcal{K} be the collection of filtered simplicial sets. Then PK(C)P^{\mathcal{K}}({\mathcal{C}}) is the full subcategory of P(C)P({\mathcal{C}}) consisting of finite limit-preserving presheaves. That is, it sends finite colimits in C{\mathcal{C}} to finite limits in Spc{\mathrm{Spc}}. Moreover, the Yoneda embedding CPK(C){\mathcal{C}}\hookrightarrow P^{\mathcal{K}}({\mathcal{C}}) preserves all finite colimits.

In this case, we refer to PK(C)P^{\mathcal{K}}({\mathcal{C}}) as Ind(C)\operatorname{Ind}({\mathcal{C}}).

Theorem 1.4 ([1, Lemma 5.5.8.14, Proposition 5.5.8.10])

Let C{\mathcal{C}} be a small \infty -category with finite coproducts and K\mathcal{K} be the collection of filtered simplicial sets and Δop\Delta ^{\mathrm{op}}. Then PK(C)P^{\mathcal{K}}({\mathcal{C}}) is the full subcategory of P(C)P({\mathcal{C}}) consisting of (finite) product-preserving presheaves. Moreover, the Yoneda embedding CPK(C){\mathcal{C}}\hookrightarrow P^{\mathcal{K}}({\mathcal{C}}) preserves all finite coproducts.

In this case, we refer to PK(C)P^{\mathcal{K}}({\mathcal{C}}) as PΣ(C)P_\Sigma ({\mathcal{C}}).

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

  1. filtered colimits commute with finite limits in Spc{\mathrm{Spc}}; and

  2. every colimit is a filtered colimit of finite colimits.

Remark 1.5

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 C{\mathcal{C}} is generated freely by its compact objects Cω{\mathcal{C}}^\omega . That is, we have C=Ind(Cω){\mathcal{C}}= \operatorname{Ind}({\mathcal{C}}^\omega ). Instead of freely adding filtered colimits to Cω{\mathcal{C}}^\omega , then freely adding geometric realizations, a better strategy is to start with CωC^\omega and freely add filtered colimits and geometric realizations in one go. The restricted Yoneda functor CPΣ(Cω){\mathcal{C}}\to P_\Sigma ({\mathcal{C}}^\omega ) is easily seen to be fully faithful and preserve filtered colimits. Since Cω{\mathcal{C}}^\omega is usually essentially small, this also lets us avoid size issues.

Proof
We prove the case of Ind(C)\operatorname{Ind}({\mathcal{C}}). The proof for PΣ(C)P_\Sigma ({\mathcal{C}}) is similar. To disambiguate, let Ind(C)P(C)\operatorname{Ind}({\mathcal{C}}) \subseteq P({\mathcal{C}}) be the category of finite limit-preserving sheaves.

We first show that y ⁣:CInd(C)y\colon {\mathcal{C}}\to \operatorname{Ind}({\mathcal{C}}) preserves finite colimits. This follows from a straightforward calculation

Hom(colimy(Pα),X)=limHom(y(Pα),X)=limX(Pα)=X(colimPα)=Hom(colimy(Pα),X). \begin{aligned} \operatorname{Hom}\left(\operatorname*{colim}y(P_\alpha ), X\right) & = \lim \operatorname{Hom}(y(P_\alpha ), X)\\ & = \lim X(P_\alpha )\\ & = X\left(\operatorname*{colim}P_\alpha \right) \\ & = \operatorname{Hom}\left(\operatorname*{colim}y(P_\alpha ), X\right). \end{aligned}

Since filtered colimits of spaces commute with finite limits, we know that Ind(C)\operatorname{Ind}({\mathcal{C}}) is closed under filtered colimits. Since representables preserve finite limits, we know that PK(C)Ind(C)P^{\mathcal{K}}({\mathcal{C}}) \subseteq \operatorname{Ind}({\mathcal{C}}).

To show the other inclusion, let XP(C)X \in P({\mathcal{C}}). Then we can write

X=colimjJXj, X = \operatorname*{colim}_{j \in \mathcal{J}} X_j,

where J\mathcal{J} is filtered and each XjX_j is a finite colimit of representables. Now suppose that XInd(C)X \in \operatorname{Ind}({\mathcal{C}}). Our goal is to write XX as a filtered colimit of representables.

Let ι ⁣:Ind(C)P(C)\iota \colon \operatorname{Ind}({\mathcal{C}}) \hookrightarrow P({\mathcal{C}}) be the inclusion, and L ⁣:P(C)Ind(C)L \colon P({\mathcal{C}}) \to \operatorname{Ind}({\mathcal{C}}) its left adjoint. Then they both preserve filtered colimits, and

X=ιLX=colimjJιLXj. X = \iota LX = \operatorname*{colim}_{j \in \mathcal{J}} \iota L X_j.

So it suffices to show that ιLXj\iota L X_j is representable. Let Xj=colimy(Pα)X_j = \operatorname*{colim}y(P_\alpha ). Then we have

LXj=Lcolimy(Pα)=colimy(Pα), LX_j = L \operatorname*{colim}y(P_\alpha ) = \operatorname*{colim}y(P_\alpha ),

where the second colimit is taken inside Ind(C)\operatorname{Ind}({\mathcal{C}}). But y ⁣:CInd(C)y \colon {\mathcal{C}}\to \operatorname{Ind}({\mathcal{C}}) preserves finite colimits, so the right-hand side is simply given by y(colimPα)y\left(\operatorname*{colim}P_\alpha \right).

Proof