In mathematics, a dagger category (also called involutive category or category with involution ) is a category equipped with a certain structure called dagger or involution. The name dagger category was coined by Selinger.
Note that in the previous definition, the term "adjoint" is used in a way analogous to (and inspired by) the linear-algebraic sense, not in the category-theoretic sense.
Some sources define a category with involution to be a dagger category with the additional property that its set of morphisms is partially ordered and that the order of morphisms is compatible with the composition of morphisms, that is a<b implies for morphisms a, b, c whenever their sources and targets are compatible.
- The category Rel of sets and relations possesses a dagger structure i.e. for a given relation in Rel, the relation is the relational converse of . In this example, a self-adjoint morphism is a symmetric relation.
- The category Cob of cobordisms is a dagger compact category, in particular it possesses a dagger structure.
- The category FdHilb of finite dimensional Hilbert spaces also possesses a dagger structure: Given a linear map , the map is just its adjoint in the usual sense.
- Any monoid with involution is a dagger category with only one object. In fact, every endomorphism hom-set in a dagger category is not simply a monoid, but a monoid with involution, because of the dagger.
- A discrete category is trivially a dagger category.
- A groupoid (and as trivial corollary a group) also has a dagger structure with the adjoint of a morphism being its inverse. In this case, all morphisms are unitary.
In a dagger category , a morphism is called
- unitary if ;
- self-adjoint if
The latter is only possible for an endomorphism .
The terms unitary and self-adjoint in the previous definition are taken from the category of Hilbert spaces where the morphisms satisfying those properties are then unitary and self-adjoint in the usual sense.
- M. Burgin, Categories with involution and correspondences in γ-categories, IX All-Union Algebraic Colloquium, Gomel (1968), pp.34–35; M. Burgin, Categories with involution and relations in γ-categories, Transactions of the Moscow Mathematical Society, 1970, v. 22, pp. 161–228
- J. Lambek, Diagram chasing in ordered categories with involution, Journal of Pure and Applied Algebra 143 (1999), No.1–3, 293–307
- P. Selinger, Dagger compact closed categories and completely positive maps, Proceedings of the 3rd International Workshop on Quantum Programming Languages, Chicago, June 30–July 1, 2005.
- Tsalenko, M.Sh. (2001), "Category with involution", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4