Molecular Hamiltonian
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 equipped with an involutive, identity-on-object functor .
In detail, this means that it associates to every morphism in its adjoint such that for all and ,
Note that in the previous definition, the term adjoint is used in the linear-algebraic sense, not in the category theoretic sense.
Some reputable sources [4] additionally require for a category with involution 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 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 in Rel, the relation is the relational converse of .
- A self-adjoint morphism is a symmetric relation.
- The category FdHilb of finite dimensional Hilbert spaces also possesses a dagger structure: Given a linear map , the map is just its adjoint in the usual sense.
Remarkable morphisms
In a dagger category , a morphism is called
- unitary if ;
- self-adjoint if (this is only possible for an endomorphism ).
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
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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.
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