nLab
sifted category

Definition

Definition

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

$$F:D\times S\to \mathrm{Set},$$F : D \times S \to Set \,,

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

$$({\underrightarrow{\lim }}_{d\in D}\prod _{s\in S}F(d,s))\to \prod _{s\in S}{\underrightarrow{\lim }}_{d\in D}F(d,s)$$({\lim_\to}_{d \in D} \prod_{s \in S} F(d,s)) \to \prod_{s \in S} {\lim_\to}_{d \in D} F(d,s)

is an isomorphism.

$D$ is called cosifted if the opposite category $D^{\mathrm{op}}$ 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

$$D\to D\times D$$D \to D \times D

is a final functor.

This is due to (GabrielUlmer)

More explicitly this means that:

Proposition

A category is sifted if for every pair of objects $d_1,d_2\in D$, the category ${\mathrm{Cospan}}_D(d_1,d_2)$ 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).

Examples

Example

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

Example

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

Example 1 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.

Related concepts

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

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

• direct category

References

• P. Gabriel and F. Ulmer, Lokal präsentierbare Kategorien , Springer LNM 221, Springer-Verlag 1971

• Jiri Adamek, Jiri Rosicky, On sifted colimits and generalized varieties, TAC (pdf)

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