A category $D$ is called sifted if colimits of diagrams of shape $D$ commute with finite products in Set: for every diagram
where $S$ is a finite discrete category the canonical morphism
is an isomorphism.
Dually, $D$ is called cosifted if the opposite category $D^{op}$ is sifted.
A colimit over a sifted diagram is called a sifted colimit.
An inhabited small category $D$ is sifted precisely if the diagonal functor
is a final functor.
This is due to (GabrielUlmer)
More explicitly this means that:
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.
Every category with finite coproducts is sifted.
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 .
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.
The presence of the degeneracy map $s_0 \colon 1 \to 0$ in example 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 may be thought of as a truncation of:
The opposite category of the simplex category is sifted.
Every filtered category is sifted.
Since filtered colimits commute even with all finite limits, they in particular commute with finite products.
(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
sifted category, sifted (∞,1)-category
221, Springer Heidelberg 1971.
Jiri Adamek, Jiri Rosicky, On sifted colimits and generalized varieties, TAC 8 (2001) pp.33-53. (tac)
Jiri Adamek, Jiri Rosicky, Enrico Vitale, What are sifted colimits?, TAC 23 (2010) pp. 251–260. (tac)
Jiri Adamek, Jiri Rosicky, Enrico Vitale, Algebraic Theories - a Categorical Introduction to General Algebra , Cambrige UP 2010. (ch. 2) (draft)
Last revised on February 21, 2024 at 15:15:08. See the history of this page for a list of all contributions to it.