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In mathematics, an adhesive category is a 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 quivers, and the theory of adhesive categories is important in the theory of graph rewriting.

More precisely, an adhesive category is one where any of the following equivalent conditions hold:

C has all pullbacks, it has pushouts along monomorphisms, 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 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 preserving pullbacks and preserving pushouts of monomorphisms.

References

Steve Lack and Pawel Sobocinski, Adhesive categories[permanent dead link], 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, "An embedding theorem for adhesive categories", Theory and Applications of Categories, Vol. 25, 2011, No. 7, pp 180–188.

External links

Adhesive category in nLab

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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