Dagger category

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In mathematics, a dagger category (also called involutive category or category with involution [1][2]) is a category equipped with a certain structure called dagger or involution. The name dagger category was coined by Selinger.[3]

Formal definition

A dagger category is a category equipped with an involutive, identity-on-objects functor .

In detail, this means that it associates to every morphism in its adjoint such that for all and ,

Note that in the previous definition, the term adjoint is used in the linear-algebraic sense, not in the category theoretic sense.

Some reputable sources [4] additionally require for a category with involution 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.


  • 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.

Remarkable morphisms

In a dagger category , a morphism is called

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.

See also



  1. 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
  2. J. Lambek, Diagram chasing in ordered categories with involution, Journal of Pure and Applied Algebra 143 (1999), No.1–3, 293–307
  3. 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.
  4. {{#invoke:citation/CS1|citation |CitationClass=citation }}