nLab sifted category

Definition

Definition

A category $D$ is called sifted if colimits of diagrams of shape $D$ commute with finite products in Set: for every diagram

$$F : D \times S \to Set \,,$$

where $S$ is a finite discrete category the canonical morphism

$$( \underset{\underset{d \in D}{\longrightarrow}}{\lim} \prod_{s \in S} F(d,s)) \to \prod_{s \in S} \underset{\underset{d \in D}{\longrightarrow}}{\lim} F(d,s)$$

is an isomorphism.

Dually, $D$ is called cosifted if the opposite category $D^{op}$ is sifted.

A colimit over a sifted diagram is called a sifted colimit.

Properties

Characterizations

Proposition

An inhabited small category $D$ is sifted precisely if the diagonal functor

$$D \to D \times D$$

is a final functor.

This is due to (GabrielUlmer)

More explicitly this means that:

Proposition

An inhabited small category is sifted if for every pair of objects $d_1,d_2\in D$, the category $Cospan_D(d_1,d_2)$ of cospans from $d_1$ to $d_2$ is connected.

Corollary

Every category with finite coproducts is sifted.

Proof

Since a category with finite coproducts is nonempty (it has an initial object) and each category of cospans has an initial object (the coproduct).

We make this special case more explicit below in Example .

Examples

Example

The diagram category for reflexive coequalizers, $\{ 0 \stackrel{\overset{d_0}{\to}}{\stackrel{\overset{s_0}{\leftarrow}}{\underset{d_1}{\to}}} 1\}^{op}$ with $s_0 \circ d_0 = s_0 \circ d_1 = id$, is sifted.

Example

The presence of the degeneracy map $s_0 \colon 1 \to 0$ in example is crucial for the statement to work: the category $\{0 \stackrel{\overset{d_0}{\to}}{\underset{d_1}{\to}} 1\}^{op}$ is not sifted; there is no way to connect the cospan $(d_0,d_0)$ to the cospan $(d_1,d_1)$.

Example may be thought of as a truncation of:

Example

The opposite category of the simplex category is sifted.

Example

Every filtered category is sifted.

Proof

Since filtered colimits commute even with all finite limits, they in particular commute with finite products.

Example

(categories with finite products are cosifted)

Let $\mathcal{C}$ be a small category which has finite products. Then $\mathcal{C}$ is a cosifted category, equivalently its opposite category $\mathcal{C}^{op}$ is a sifted category, equivalently colimits over $\mathcal{C}^{op}$ with values in Set are sifted colimits, equivalently colimits over $\mathcal{C}^{op}$ with values in Set commute with finite products, as follows:

For $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ to functors on the opposite category of $\mathcal{C}$ (hence two presheaves on $\mathcal{C}$) we have a natural isomorphism

$$\underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( \mathbf{X} \times \mathbf{Y} \right) \;\simeq\; \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{X} \right) \times \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{Y} \right) \,.$$

Related concepts

• distributivity of products and colimits

• sifted category, sifted (∞,1)-category

• filtered category, filtered (∞,1)-category

• direct category

References

• Pierre Gabriel, Fritz Ulmer, Lokal präsentierbare Kategorien , LNM

  221, Springer Heidelberg 1971.

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

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

• Jiri Adamek, Jiri Rosicky, Enrico Vitale, Algebraic Theories - a Categorical Introduction to General Algebra , Cambrige UP 2010. (ch. 2) (draft)