nLab
finitely complete category

Definition

A finitely complete category (which the Elephant calls a cartesian category ) is a category $C$ which admits all finite limits, that is all limits for any diagrams $F:J\to C$ with $J$ 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.

Variants

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:

• It has a terminal object and admits all binary products and equalizers; or
• It has a terminal object and admits all binary pullbacks.

An appropriate notion of morphism between finitely complete categories $C$, $D$ 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:

• it preserves terminal objects, binary products, and equalizers; or
• it preserves terminal objects and binary pullbacks.

Since these conditions frequently come up individually, it may be worthwhile listing them separately:

• $F:C\to D$ preserves terminal objects if $F(t_C)$ is terminal in $D$ whenever $t_C$ is terminal in $C$;

• $F:C\to D$ preserves binary products if the pair of maps

  $$F(c)\stackrel{F({\pi }_1)}{\leftarrow }F(c\times d)\stackrel{F({\pi }_2)}{\to }F(d)$$F(c) \stackrel{F(\pi_1)}{\leftarrow} F(c \times d) \stackrel{F(\pi_2)}{\to} F(d)

  exhibits $F(c\times d)$ as a product of $F(c)$ and $F(d)$, where ${\pi }_1:c\times d\to c$ and ${\pi }_2:c\times d\to d$ are the product projections in $C$;

• $F:C\to D$ preserves equalizers if the map

  $$F(i):F(e)\to F(c)$$F(i): F(e) \to F(c)

  is the equalizer of $F(f),F(g):F(c)\overrightarrow{\to }F(d)$, whenever $i:e\to c$ is the equalizer of $f,g:c\overrightarrow{\to }d$ in $C$.

Related concepts

• cartesian category, cartesian functor, cartesian logic, cartesian theory

• regular category, regular functor, regular logic, regular theory, regular coverage, regular topos

• coherent category, coherent functor, coherent logic, coherent theory, coherent coverage, coherent topos

• geometric category, geometric functor, geometric logic, geometric theory

References

Section A1.2 in

• Peter Johnstone, Sketches of an Elephant