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.
The following conditions on a category $C$ are equivalent. When they are satisfied, we say that $C$ is adhesive.
$C$ has pullbacks and pushouts of monomorphisms, and pushout squares of monomorphisms are also pullback squares and are stable under pullback.
$C$ has pullbacks, and pushouts of monomorphisms, and the latter are also (bicategorical) pushouts in the bicategory of spans in $C$.
(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.
$C$ 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.
In an adhesive category, suppose given a pushout square
such that $m$ is a monomorphism. Then:
(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.
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.
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.
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 $m$).
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 stable under pushouts in toposes plays a central role for Cisinski model structures such as notably the standard model structure on simplicial sets, where the monomorphisms are cofibrations and as such required to be closed under pushout (in particular).
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”.