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In category theory, a branch of mathematics, a dagger category (also called involutive category or category with involutioncite-ref-burgin-1-0[1]cite-ref-lambek-2-0[2]) is a category equipped with a certain structure called dagger or involution. The name dagger category was coined by Peter Selinger.cite-ref-selinger-3-0[3]

• Formal definition

• Examples

• Remarkable morphisms

• See also

• References

A dagger category is a category C {\displaystyle {\mathcal {C}}} equipped with an involutive contravariant endofunctor † † {\displaystyle \dagger } which is the identity on objects.cite-ref-4[4]

In detail, this means that:

• for all morphisms f : A → → B {\displaystyle f:A\to B} , there exists its adjoint f † † : B → → A {\displaystyle f^{\dagger }:B\to A}

• for all morphisms f {\displaystyle f} , ( f † † ) † † = f {\displaystyle (f^{\dagger })^{\dagger }=f}

• for all objects A {\displaystyle A} , i d A † † = i d A {\displaystyle \mathrm {id} _{A}^{\dagger }=\mathrm {id} _{A}}

• for all f : A → → B {\displaystyle f:A\to B} and g : B → → C {\displaystyle g:B\to C} , ( g ∘ ∘ f ) † † = f † † ∘ ∘ g † † : C → → A {\displaystyle (g\circ f)^{\dagger }=f^{\dagger }\circ g^{\dagger }:C\to A}

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 sourcescite-ref-springer-5-0[5] 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 {\displaystyle a<b} implies a ∘ ∘ c < b ∘ ∘ c {\displaystyle a\circ c<b\circ c} for morphisms a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} whenever their sources and targets are compatible.

• The category Rel of sets and relations possesses a dagger structure: for a given relation R : X → → Y {\displaystyle R:X\rightarrow Y} in Rel, the relation R † † : Y → → X {\displaystyle R^{\dagger }:Y\rightarrow X} is the relational converse of R {\displaystyle R} . 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 Hilb of Hilbert spaces also possesses a dagger structure: Given a bounded linear map f : A → → B {\displaystyle f:A\rightarrow B} , the map f † † : B → → A {\displaystyle f^{\dagger }:B\rightarrow A} 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 (definition below).

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

• unitary if f † † = f − − 1 , {\displaystyle f^{\dagger }=f^{-1},}

• self-adjoint if f † † = f . {\displaystyle f^{\dagger }=f.}

The latter is only possible for an endomorphism f : : A → → A {\displaystyle 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.

• *-algebra

• Dagger symmetric monoidal category

• Dagger compact category

cite-note-burgin-11. ↑ 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

cite-note-lambek-22. ↑ J. Lambek, Diagram chasing in ordered categories with involution, Journal of Pure and Applied Algebra 143 (1999), No.1–3, 293–307

cite-note-selinger-33. ↑ 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.

cite-note-44. ↑ "Dagger category in nLab".

cite-note-springer-55. ↑ citereftsalenko2001Tsalenko, M.Sh. (2001) [1994], "Category with involution", Encyclopedia of Mathematics, EMS Press

• Dagger category at the nLab