Dagger category

Dagger category

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].

Contents

Formal definition

A dagger category is a category \mathbb{C} equipped with an involutive, identity-on-object functor \dagger\colon \mathbb{C}^{op}\rightarrow\mathbb{C}.

In detail, this means that it associates to every morphism f\colon A\to B in \mathbb{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 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 a\circ c<b\circ c for morphisms a, b, c whenever their sources and targets are compatible.

Examples

Remarkable morphisms

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

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

References

  1. ^ M. Burgin, Categories with involution and correspondences in g-categories, IX All-Union Algebraic Colloquium, Gomel (1968), pp.34–35
  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. ^ Springer's Encyclopaedia of Mathematics. Category with involution

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