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[edit]

A dagger category is a category \mathcal{C} equipped with an involutive functor \dagger\colon \mathcal{C}^{op}\rightarrow\mathcal{C} that is the identity on objects, where \mathcal{C}^{op} is the opposite category.

In detail, this means that it associates to every morphism f\colon A\to B in \mathcal{C} its adjoint f^\dagger\colon B\to A such that for all f\colon A\to B and g\colon B\to C,

  •  \mathrm{id}_A=\mathrm{id}_A^\dagger\colon A\rightarrow A
  •  (g\circ f)^\dagger=f^\dagger\circ g^\dagger\colon C\rightarrow A
  •  f^{\dagger\dagger}=f\colon A\rightarrow B\,

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 [4] 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 a\circ c<b\circ c 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[edit]

In a dagger category \mathcal{C}, a morphism  f is called

  • unitary if f^\dagger = f^{-1};
  • self-adjoint if f^\dagger = f

The latter is only possible for an endomorphism f\colon A \to A.

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[edit]


  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. ^ Tsalenko, M.Sh. (2001), "Category with involution", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4