# nLab exactness property

### Context

#### Regular and Exact categories

∞-ary regular and exact categories

regularity

exactness

#### Limits and colimits

limits and colimits

category theory

# Exactness properties

## Idea

An exactness property of a category asserts the existence of certain limits and colimits, and moreover that the limits and colimits interact in a certain way. Frequently, this includes stability of the colimits under pullback, and also a condition expressing that some of the input data can be recovered from the colimit.

Many types of exactness can be expressed in terms of “colimits in the left-exact world”.

Exactness properties of a functor refer to preservation of limits or colimits of certain kind, existence of adjoints and possibly their exactness properties.