nLab locally strongly finitely presentable category

Idea

A locally strongly finitely presentable category is like a locally finitely presentable category, but where the class of filtered colimits (respectively the class of finite limits) is replaced by the class of sifted colimits (respectively the class of finite products).

Locally strongly finitely presentable category are precisely those categories equivalent to varieties of algebras.

Definition

A category $C$ satisfying (any of) the following equivalent conditions is said to be locally strongly finitely presentable (or lsfp):

1. $C$ is the free cocompletion of a small category with finite coproducts under sifted colimits: see sind-object.
2. $C$ has all small colimits, the category $C_{sfp}$ is essentially small, and any object in $C$ is a sifted colimit of the canonical diagram of strongly finitely presentable objects mapping into it.
3. $C$ is the category of models for an algebraic theory.
4. $C$ is the category of models for a finite product sketch.
5. $C_{sfp}$ has finite coproducts, and the restricted Yoneda embedding $C\hookrightarrow [C_{sfp}^{op},Set]$ identifies $C$ with the category of finite-product-preserving functors $C_{fp}^{op} \to Set$.

Related pages

• sind-object
• locally finitely presentable category
• sifted category

 References

• 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)