is called cosifted if the opposite category is sifted.
A colimit over a sifted diagram is called a sifted colimit.
This is due to (GabrielUlmer)
More explicitly this means that:
An inhabited small category is sifted if for every pair of objects , the category of cospans from to 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).
The diagram category for reflexive coequalizers, with , is sifted.
The presence of the degeneracy map in example 1 is crucial for the statement to work: the category is not sifted; there is no way to connect the cospan to the cospan .
Example 1 may be thought of as a truncation of:
- Pierre Gabriel, Fritz Ulmer, Lokal präsentierbare Kategorien , LNM 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)