# nLab finitely complete category

category theory

## Applications

#### Limits and colimits

limits and colimits

# Contents

## 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:

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 $F\left({t}_{C}\right)$ is terminal in $D$ whenever ${t}_{C}$ is terminal in $C$;

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

$F\left(c\right)\stackrel{F\left({\pi }_{1}\right)}{←}F\left(c×d\right)\stackrel{F\left({\pi }_{2}\right)}{\to }F\left(d\right)$F(c) \stackrel{F(\pi_1)}{\leftarrow} F(c \times d) \stackrel{F(\pi_2)}{\to} F(d)

exhibits $F\left(c×d\right)$ as a product of $F\left(c\right)$ and $F\left(d\right)$, where ${\pi }_{1}:c×d\to c$ and ${\pi }_{2}:c×d\to d$ are the product projections in $C$;

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

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

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

## References

Section A1.2 in

Revised on January 13, 2012 21:13:56 by Mike Shulman (71.136.235.72)