nLab sind-object

Idea

A sind-object of a category $\mathcal{C}$ is a formal sifted colimit of objects of $\mathcal{C}$. Here “formal” means that the colimit is taken in the category of presheaves of $\mathcal{C}$ (the free cocompletion of $\mathcal{C}$). The category of sind-objects of $\mathcal{C}$ is written $sind$-$\mathcal{C}$ or $SInd(\mathcal{C})$. It is the sifted colimit completion of $\mathcal{C}$.

There is an analogy here with ind-objects, which are formal filtered colimits, and where the category of ind-objects $Ind(\mathcal{C})$ is the filtered colimit completion of $\mathcal{C}$.

Typically the main case of interest here is the situation where $\mathcal{C}$ is small with finite coproducts. If a small category $\mathcal{C}$ has finite coproducts, the sifted colimit completion comprises the finite-product-preserving presheaves:

$$Sind(\mathcal{C})\cong FP(\mathcal{C}^{op},Set)$$

The analogy here is that if a small category $\mathcal{C}$ has finite colimits, the filtered colimit completion comprises the finite limit preserving presheaves:

$$Ind(\mathcal{C})\cong Lex(\mathcal{C}^{op},Set)$$

One reason this is often relevant because if $\mathcal{C}^\op$ has finite products, it can be regarded as an algebraic theory, and then the sifted colimit completion comprises the category of models of the theory.

Another point of relevance is that this is a canonical way of building a cartesian closed category out of a distributive category.

Definition

If $\mathcal{C}$ has finite coproducts then consider the category of finite product preserving functors

$$Sind(\mathcal{C})\cong FP(\mathcal{C}^{op},Set)$$

The representable objects always preserves finite products, and so the Yoneda embedding $\mathcal{C}\to [\mathcal{C}^{op},Set]$ factors through $Sind(\mathcal{C})$. This exhibits $Sind(\mathcal{C})$ as the free sifted colimit completion of $\mathcal{C}$.

Proposition

If $\mathcal{D}$ is has sifted colimits then the Yoneda embedding induces a natural bijection between functors $\mathcal{C}\to \mathcal{D}$ and sifted-colimit-preserving functors $Sind(\mathcal{C})\to \mathcal{D}$.

The construction has another universal property, as the free colimit completion of a category respecting the existing finite coproducts.

Proposition

If $\mathcal{D}$ is has colimits then the Yoneda embedding induces a natural bijection between finite-coproduct-preserving functors $\mathcal{C}\to \mathcal{D}$ and colimit-preserving-functors $Sind(\mathcal{C})\to \mathcal{D}$.

Distributive categories and cartesian closure

The sifted-colimit-completion also gives a way of full and faithfully embedding a distributive category in a cartesian closed category, while preserving the structure. From a type-theoretic perspective, this shows that function types are a conservative extension of finite type theory.

Proposition

Let $\mathcal{C}$ be a small category with finite coproducts and products. The following are equivalent.

• $\mathcal{C}$ is a distributive category.

• $Sind(\mathcal{C})$ is cartesian closed.

References

• Jiri Adamek, Jiri Rosicky, Enrico Vitale, What are sifted colimits?, TAC 23 (2010) pp. 251–260. (web)

• Jiri Adamek, Jiri Rosicky, On sifted colimits and generalized varieties, TAC 8 (2001) pp 33–53. (web)

The idea of product-preserving-functors as a cartesian closed category is in various sources, including:

• Marcelo Fiore, “Enrichment and representation theorems for categories of domains and partial functions”. 1996. web.

• John Power, “Generic models for computational effects”, Theor. Comput. Sci. 364(2): 254-269 (2006)

• Younesse Kaddar, “Ideal distributors”, Section 1. 2020. report.