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]

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 $g\colon B\to C$ ,

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

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.

Examples

The category Rel of sets and relations possesses a dagger structure i.e. for a given relation $R:X\rightarrow Y$ in Rel, the relation $R^{\dagger }:Y\rightarrow X$ 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 FdHilb of finite dimensional Hilbert spaces also possesses a dagger structure: Given a linear map $f:A\rightarrow B$ , the map $f^{\dagger }:B\rightarrow A$ 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.

Remarkable morphisms

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.

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", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

Dagger category in nLab

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Dagger category

Formal definition

Examples

Remarkable morphisms

See also

References