We introduce adhesive categories, which are categories with structure ensuring that pushouts along monomorphisms are well-behaved. Many types of graphical structures used in computer science are shown to be examples of adhesive categories. Double-pushout graph rewriting generalises well to rewriting on arbitrary adhesive categories. We also introduce the more general quasiadhesive categories, in which only pushouts along regular monomorphisms are required to be well-behaved.
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