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, suppose given a pushout square

$\array{ C & \xrightarrow{m} & A \\ ^f\downarrow && \downarrow^g \\ B & \xrightarrow{n} & D }$

such that $m$ is a monomorphism. Then:

1. $n$ is also a monomorphism.
2. The square is also a pullback square.
3. The square is also a distributivity pullback around $(g,m)$; hence in particular $n = \forall_g m$ is the universal quantification.

(Notice that generally monomorphisms (as discussed there) are preserved by pullback.) For a proof of the above proposition, see (Lack, prop. 2.1) and (Lack-Sobocinski, Lemmas 2.3 and 2.8). The latter Lemma 2.8 states only that $n = \forall_g m$ (a weaker universal property since it refers only to other monomorphisms into $D$), but the proof applies more generally.

###### Proposition

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

We define a pushout complement of $m:C\to A$ and $g:A\to D$ to be a pair of arrows $f:C\to B$ and $n:B\to D$ such that $n f = g m$ and this commutative square is a pushout. The following proposition is crucial in double pushout graph rewriting.

###### Proposition

In an adhesive category, if $m:C\to A$ is mono and $g:A\to D$ is any morphism, then if a pushout complement exists, it is unique up to unique isomorphism.

###### Proof

We give only a sketch; details are in (LS, Lemma 4.5). If $(f,n)$ and $(f',n')$ are two pushout complements, consider the two pushout squares as morphisms in the arrow category with target $g$, and take their pullback. The resulting commutative cube can be viewed as a morphism in the category of commutative squares from the pullback square of $m$ against itself (which is again $m$, since $m$ is mono) to the pullback square of $n$ against $n'$. Denote the vertex of the latter pullback square by $U$. Applying the van Kampen property in two directions, we find that the maps $U\to B$ and $U\to B'$ are both pushouts of $1_C$, hence isomorphisms. This gives an isomorphism between the pushout complements; it is unique since $n$ and $n'$ are mono (being pushouts of the mono $m$).

## 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”.