nLab sifted colimit

Definition

A sifted colimit is a colimit of a diagram $D \to C$ where $D$ is a sifted category (in analogy with a filtered colimit, involving diagrams of shape a filtered category). Such colimits commute with finite products in Set, by definition.

Examples

Example

A motivating example is a reflexive coequalizer. In fact, sifted colimits can “almost” be characterized as combinations of filtered colimits and reflexive coequalizers. (Adamek-Rosicky-Vitale 10)

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

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

Properties

See at distributivity of products and colimits.

See also

• sind-object, an object in the free cocompletion under sifted colimits

• sifted $(\infty,1)$-colimit

• finite product

References

• Pierre Gabriel, Friedrich Ulmer, Lokal präsentierbare Kategorien, Springer LNM

  221, Springer-Verlag 1971

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