nLab sifted colimit

Context

Limits and colimits

limits and colimits

Contents

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

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