In mathematics, an adhesive category is a

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

where

pushout A ''pushout'' is a student who leaves their school before graduation, through the encouragement of the school. A student who leaves of their own accord (e.g., to work or care for a child), rather than through the action of the school, is consider ...

s 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, bolts, ammo, projectiles, darts, or javelins. 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 trad ...

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: i ...

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 monomorphism ...

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 mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative ''up to'' an isomor ...

span Span may refer to: Science, technology and engineering * Span (unit), the width of a human hand * Span (engineering), a section between two intermediate supports * Wingspan, the distance between the wingtips of a bird or aircraft * Sorbitan ester ...

s 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 (mathematics), category that behaves like the category of Sheaf (mathematics), sheaves of Set (mathematics), sets on a topological space (or more generally: on a Site (math ...

preserving pullbacks and preserving pushouts of monomorphisms.

References

* Steve Lack and Pawel Sobocinski, tp://ftp.daimi.au.dk/BRICS/RS/03/31/BRICS-RS-03-31.pdf ''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 {{categorytheory-stub