Dagger category

In category theory, a branch of mathematics, a dagger category (also called involutive category or category with involution ) is a category equipped with a certain structure called dagger or involution. The name dagger category was coined by Peter Selinger.

Formal definition
A dagger category is a category $$\mathcal{C}$$ equipped with an involutive contravariant endofunctor $$\dagger$$ which is the identity on objects.

In detail, this means that:
 * for all morphisms $$f: A \to B$$, there exist its adjoint $$f^\dagger: B \to A$$
 * for all morphisms $$f$$, $$(f^\dagger)^\dagger = f$$
 * for all objects $$A$$, $$\mathrm{id}_A^\dagger = \mathrm{id}_A$$
 * for all $$f: A \to B$$ and $$g: B \to C$$, $$(g \circ f)^\dagger = f^\dagger \circ g^\dagger: C \to A$$

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 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 $$a < b$$ 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 $$R:X \rightarrow Y$$ 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 Hilb of Hilbert spaces also possesses a dagger structure: Given a bounded 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 (definition below).

Remarkable morphisms
In a dagger category $$\mathcal{C}$$, a morphism $$f$$ is called 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.
 * unitary if $$f^\dagger = f^{-1},$$
 * self-adjoint if $$f^\dagger = f.$$