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 $𝒞$ equipped with an involutive contravariant endofunctor $†$ which is the identity on objects.

In detail, this means that:

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

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 $R:X\to Y$ in Rel, the relation $R^{†}:Y\to X$ 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 $f:A\to B$, the map $f^{†}:B\to A$ 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 $𝒞$, a morphism $f$ is called

• unitary if $f^{†}=f^{-1},$
• self-adjoint if $f^{†}=f.$

The latter is only possible for an endomorphism $f:A\to A$. 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.

See also

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

References

• Dagger category at the nLab