In mathematics, an adhesive category is a
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
where
pushouts of monomorphisms exist and work more or less as they do in the category of sets. An example of an adhesive category is the category of directed multigraphs, or
quiver
A quiver is a container for holding arrows or Crossbow bolt, bolts. It can be carried on an archer's body, the bow, or the ground, depending on the type of shooting and the archer's personal preference. Quivers were traditionally made of leath ...
s, and the theory of adhesive categories is important in the theory of
graph rewriting
In computer science, graph transformation, or graph rewriting, concerns the technique of creating a new graph out of an original graph algorithmically. It has numerous applications, ranging from software engineering ( software construction and also ...
.
More precisely, an adhesive category is one where any of the following equivalent conditions hold:
* ''C'' has all
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
s, it has pushouts along
monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphis ...
s, and pushout squares of monomorphisms are also pullback squares and are stable under pullback.
* ''C'' has all pullbacks, it has pushouts along monomorphisms, and the latter are also (bicategorical) pushouts in the
bicategory
In category theory in mathematics, a 2-category is a category (mathematics), category with "morphisms between morphisms", called 2-morphisms. A basic example is the category Cat of all (small) categories, where a 2-morphism is a natural transforma ...
of
spans in ''C''.
If ''C'' is small, we may equivalently say that ''C'' has all pullbacks, has pushouts along monomorphisms, and admits a full embedding into a
Grothendieck topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notion ...
preserving pullbacks and preserving pushouts of monomorphisms.
References
* Steve Lack and Pawel Sobocinski
''Adhesive categories'' ''Basic Research in Computer Science series'', BRICS RS-03-31, October 2003.
* Richard Garner and Steve Lack
"On the axioms for adhesive and quasiadhesive categories" ''Theory and Applications of Categories'', Vol. 27, 2012, No. 3, pp 27–46.
* Steve Lack and Pawel Sobocinski
"Toposes are adhesive"
* Steve Lack
''Theory and Applications of Categories'', Vol. 25, 2011, No. 7, pp 180–188.
External links
*
Category theory
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