category theory

## Idea

An adhesive category is a category in which pushouts of monomorphisms exist and “behave more or less as they do in the category of sets”, or equivalently in any topos.

## Definition

###### Definition

The following conditions on a category $C$ are equivalent. When they are satisfied, we say that $C$ is adhesive.

1. $C$ has pullbacks and pushouts of monomorphisms, and pushout squares of monomorphisms are also pullback squares and are stable under pullback.

2. $C$ has pullbacks, and pushouts of monomorphisms, and the latter are also (bicategorical) pushouts in the bicategory of spans in $C$.

3. (If $C$ is small) $C$ has pullbacks and pushouts of monomorphisms, and admits a full embedding into a Grothendieck topos preserving pullbacks and pushouts of monomorphisms.

4. $C$ has pullbacks and pushouts of monomorphisms, and in any cubical diagram:

if $X\to Y$ is a monomorphism, the bottom square is a pushout, and the left and back faces are pullbacks, then the top face is a pushout if and only if the front and right face are pullbacks. In other words, pushouts of monomorphisms are van Kampen colimits.

## Properties

###### Proposition

In an adhesive category, the pushout of a monomorphism is again a monomorphism.

E.g. (Lack, prop. 2.1) Notice that generally monomorphisms (as discussed there) are preserved by pullback.

###### Proposition

An adhesive category with a strict initial object is automatically an extensive category.

## Examples

Adhesiveness is an exactness property, similar to being a regular category, an exact category, or an extensive category. In particular, it can be phrased in the language of “lex colimits”.

## References

Revised on May 2, 2013 04:35:16 by John Baez (169.235.156.49)