Dagger category#Remarkable morphisms

A dagger category is a category $\backslash mathcal\{C\}$ equipped with an involutive contravariant endofunctor $\backslash dagger$ which is the identity on objects.{{cite web | url=https://ncatlab.org/nlab/show/dagger+category#with_a_contravariant_endofunctor | title=Dagger category in nLab }}

In detail, this means that:

• for all morphisms $f:\; A\; \backslash to\; B$, there exists its adjoint $f^\backslash dagger:\; B\; \backslash to\; A$
• for all morphisms $f$, $(f^\backslash dagger)^\backslash dagger\; =\; f$
• for all objects $A$, $\backslash mathrm\{id\}\_A^\backslash dagger\; =\; \backslash mathrm\{id\}\_A$
• for all $f:\; A\; \backslash to\; B$ and $g:\; B\; \backslash to\; C$, $(g\; \backslash circ\; f)^\backslash dagger\; =\; f^\backslash dagger\; \backslash circ\; g^\backslash dagger:\; C\; \backslash to\; A$

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 sources 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 implies

• The category Rel of sets and relations possesses a dagger structure: for a given relation $R:X\; \backslash rightarrow\; Y$ in Rel, the relation $R^\backslash dagger:Y\; \backslash rightarrow\; X$ 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 $f:A\; \backslash rightarrow\; B$, the map $f^\backslash dagger:B\; \backslash rightarrow\; A$ 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 $\backslash mathcal\{C\}$, a morphism $f$ is called

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

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

{{Portal|Mathematics}}

• *-algebra
• Dagger symmetric monoidal category
• Dagger compact category

P. Selinger, [http://www.mscs.dal.ca/~selinger/papers.html#dagger Dagger compact closed categories and completely positive maps], Proceedings of the 3rd International Workshop on Quantum Programming Languages, Chicago, June 30–July 1, 2005.

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

{{SpringerEOM| title=Category with involution | id=Category_with_involution | oldid=16991 | first=M.Sh. | last=Tsalenko }}

• {{nlab|id=dagger-category|title=Dagger category}}