nLab
sifted colimit

Contents

Definition

A sifted colimit is a colimit of a diagram DCD \to C where DD 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 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) \,.

Properties

See at distributivity of products and colimits.

See also

References

Last revised on June 14, 2018 at 05:51:26. See the history of this page for a list of all contributions to it.