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

Notice that generally monomorphisms are preserved by pullback (see there) but in a general category they may not need to be preserved by pushout. In an adhesive category, however, they are:

###### Proposition

In an adhesive category, suppose given a pushout square

$\array{ C & \overset{m}{\longrightarrow} & A \\ \mathllap{{}^f}\big\downarrow && \big\downarrow\mathrlap{{}^g} \\ B & \underset{n}{\longrightarrow} & D }$

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.

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 $m$).

###### Proposition

In an adhesive category, every monomorphism is regular. In particular, every adhesive category is balanced.

###### Proof

Let $m: A \to B$ be a monomorphism. By adhesiveness, the cokernel pair (pushout)

$\array{ A & \overset{m}{\longrightarrow} & B \\ \mathllap{{}^m}\big\downarrow && \big\downarrow\mathrlap{{}^u} \\ B & \underset{v}{\longrightarrow} & P \rlap{.} }$

is a pullback. This exhibits $m$ as an equalizer of $u$ and $v$.

## Examples

###### Example

Any topos is adhesive (Lack-Sobocisnki). For Grothendieck toposes this is easy, because $Set$ is adhesive and adhesivity is a condition on colimits and finite limits, hence preserved by functor categories and left-exact localizations. For elementary toposes it is a theorem of Lack and Sobocinski.

The fact that monomorphisms are preserved by pushouts in toposes plays a central role for Cisinski model structures such as notably the classical model structure on simplicial sets, where the class of monomorphisms is identified with the class of cofibrations and as such required to be closed under pushout (in particular).

###### Example

Some counter-examples:

(Lack and Sobocinski, Counterexample 7).

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

Last revised on October 19, 2023 at 15:02:39. See the history of this page for a list of all contributions to it.