category theory

# Contents

## 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

$F : D \times S \to Set \,,$

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

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

is an isomorphism.

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

$D \to D \times D$

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 $d_1,d_2\in D$, the category $Cospan_D(d_1,d_2)$ 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, $\{ 0 \stackrel{\overset{d_0}{\to}}{\stackrel{\overset{s_0}{\leftarrow}}{\underset{d_1}{\to}}} 1\}^{op}$ with $s_0 \circ d_0 = s_0 \circ d_1 = id$, is sifted.

###### Example

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

## References

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

Revised on June 27, 2015 04:13:23 by Thomas Holder (89.15.236.4)