A finitely complete category is a category which admits all finite limits, that is all limits for any diagrams with a finite category. Finitely complete categories are also called lex categories. They are also called (at least by Johnstone in the Elephant) cartesian categories, although this term more often means a cartesian monoidal category.
Small finitely complete categories form a 2-category, Lex.
There are several well known reductions of this concept to classes of special limits. For example, a category is finitely complete if and only if:
An appropriate notion of morphism between finitely complete categories , is a left exact functor, or a functor that preserves finite limits (also called a lex functor, a cartesian functor, or a finitely continuous functor). A functor preserves finite limits if and only if:
Since these conditions frequently come up individually, it may be worthwhile listing them separately:
preserves terminal objects if is terminal in whenever is terminal in ;
preserves binary products if the pair of maps
exhibits as a product of and , where and are the product projections in ;
preserves equalizers if the map
is the equalizer of , whenever is the equalizer of in .
In any finitely complete category, the kernel pair of the identity morphism on an object is the diagonal morphism of and has a coequalizer isomorphic to itself.
Section A1.2 in
Last revised on February 17, 2024 at 09:25:19. See the history of this page for a list of all contributions to it.