nLab
adhesive category

Adhesive categories

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 CC are equivalent. When they are satisfied, we say that CC is adhesive.

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

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

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

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

    cube

    if XYX\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)