nLab
locally strongly finitely presentable category
Contents
Context
Category theory
Compact objects
objects $d \in C$ such that $C(d,-)$ commutes with certain colimits

Models
Relative version
Contents
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 ):

$C$ is the free cocompletion of a small category with finite coproducts under sifted colimits : see sind-object .
$C$ 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.
$C$ is the category of models for an algebraic theory .
$C$ is the category of models for a finite product sketch .
$C_{sfp}$ 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
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 )

Last revised on August 28, 2024 at 21:18:42.
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