nLab
sifted category

Contents

Definition

Definition

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

F:D×SSet, F : D \times S \to Set \,,

where SS is a finite discrete category the canonical morphism

(limdD sSF(d,s)) sSlimdDF(d,s) ( \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, DD is called cosifted if the opposite category D opD^{op} is sifted.

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

Properties

Characterizations

Proposition

An inhabited small category DD is sifted precisely if the diagonal functor

DD×D 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 2Dd_1,d_2\in D, the category Cospan D(d 1,d 2)Cospan_D(d_1,d_2) of cospans from d 1d_1 to d 2d_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 5.

Examples

Example

The diagram category for reflexive coequalizers, {0d 1s 0d 01} op\{ 0 \stackrel{\overset{d_0}{\to}}{\stackrel{\overset{s_0}{\leftarrow}}{\underset{d_1}{\to}}} 1\}^{op} with s 0d 0=s 0d 1=id s_0 \circ d_0 = s_0 \circ d_1 = id, is sifted.

Example

The presence of the degeneracy map s 0:10s_0 \colon 1 \to 0 in example 1 is crucial for the statement to work: the category {0d 1d 01} op\{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)(d_0,d_0) to the cospan (d 1,d 1)(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.

Example

(categories with finite products are cosifted

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

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

lim𝒞 op(X×Y)(lim𝒞 opX)×(lim𝒞 opY). \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) \,.

References

  • Pierre Gabriel, Fritz Ulmer, Lokal präsentierbare Kategorien , LNM 221, Springer Heidelberg 1971.

Last revised on June 14, 2018 at 06:19:26. See the history of this page for a list of all contributions to it.