Construction of synthetic spectraFreely adjoining colimits

Theorem 1.1

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

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

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

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

Theorem 1.2

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

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

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

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

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

Theorem 1.3 ([1, Corollary 5.3.5.4])

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$LX_j = L \operatorname*{colim}y(P_\alpha ) = \operatorname*{colim}y(P_\alpha ),$

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

Proof