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In category theory, a branch of 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 Peter Selinger.[3]

Formal definition

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 $$g\colon B\to C$$ ,

$${\displaystyle \mathrm {id} _{A}=\mathrm {id} _{A}^{\dagger }\colon A\rightarrow A}$$
$${\displaystyle (g\circ \!f)^{\dagger }=f^{\dagger }\!\circ g^{\dagger }\colon C\rightarrow A}$$
$${\displaystyle f^{\dagger \dagger }\!=f\colon A\rightarrow B}$$

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.
Examples

The category Rel of sets and relations possesses a dagger structure: for a given relation $${\displaystyle R:X\rightarrow Y}$$ in Rel, the relation $${\displaystyle R^{\dagger }:Y\rightarrow X}$$ is the relational converse of 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$$f:A\rightarrow B$$, the map $${\displaystyle f^{\dagger }:B\rightarrow A}$$ 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).

Remarkable morphisms

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

unitary if $${\displaystyle f^{\dagger }=f^{-1},}$$
self-adjoint if $${\displaystyle f^{\dagger }=f.}$$

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.

*-algebra
Dagger symmetric monoidal category
Dagger compact category

References

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) [1994], "Category with involution", Encyclopedia of Mathematics, EMS Presss

Dagger category in nLab

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