# nLab finite limit

### Context

#### Limits and colimits

limits and colimits

# Contents

## Definition

A finite limit is a limit over a diagram that is a finite category.

More generally, in higher category theory, a finite limit is a limit of a diagram that is a finite (n,r)-category.

## Properties

###### Proposition

A category that has all finite products and equalizers also has all finite limits.

This is analogous to how a category with all small products and equalizers has all small limits. We also have:

###### Proposition

A category that has all pullbacks and a terminal object also has all finite limits.

More precisely, finite limits are contained in the saturation of the class containing only finite products and equalizers, and also that of the class containing only pullbacks and terminal objects. (The actual saturation is somewhat larger than this — it is the class of L-finite limits.)

A category that has all finite limits is called a finitely complete category or a (finitary) essentially algebraic theory.

A functor that preserves finite limits is called left exact functor, a lex functor, a cartesian functor, or a finitely continuous functor. The 2-category of finitely complete categories, left exact functors and natural transformations is called Lex.

For the analog notion in (∞,1)-category theory see finite (∞,1)-limit.

Revised on August 27, 2012 19:04:03 by Urs Schreiber (89.204.137.217)