nLab
dagger-category

Terminology and wording

In Wikipedia dagger category is said to be the same as involutive category or category with involution, but Springer’s Encyclopedy requires for a category with involution additional conditions namely a partial order on the set of morphisms and that the order is compatible with the composition of morphisms.

Idea

The definition of a category effectively enforces an ordering on the “0-faces” – the source and target objects – of every 1-cell (every morphism). In many cases this is essential, in that there is no way to regard the generic morphism $a\stackrel{f}{\to }b$ in the generic category as a morphism from $b$ to $a$ instead.

But there are many categories for which this is not the case, where every morphism naturally only comes with the information of an unordered pair $\{a,b\}$ of objects, without any prejudice on which is to be regarded as source and which as target. An important general example is:

• the category $\mathrm{Spans}(C)$ of spans in a category $C$ with pullbacks, or dually, the category $\mathrm{CoSpans}(C)$ of cospans in a category $C$ with pushouts.

More concrete examples are:

• categories of cobordisms (but notice that cobordisms are naturally regarded as cospans which makes this a special case of the above example);

• the category Hilb of Hilbert spaces, where for every linear map $f:H_1\to H_2$ we also have the adjoint map (in the sense of Hilbert spaces, not in the categorical sense) $f^{†}:H_2\to H_1$ (but notice that according to groupoidification this is also essentially to be regarded as a special case of categories of spans).

A dagger structure on a category is extra structure which encodes the idea of removing the ordering information on the 0-faces of 1-cells in a category: it is is a contravariant functor which sends every morphism $f:a\to b$ to a morphims going the other way, $f^{†}:b\to a$.

The notation and terminology here is motivated from the example Hilb of Hilbert spaces, where $f^{†}$ is traditionally the notion for the adjoint of a linear map $f$. The canonical dagger-structure on Hilb and on nCob is crucial in quantum field theory where it is used to encode the idea of unitarity:

a unitary functorial QFT of dimension $n$ is supposed to be a functor $n\mathrm{Cob}\to \mathrm{Hilb}$ which respects the dagger-structure on both sides.

Definition

A dagger category is a category $C$ equipped with a contravariant functor

$$†:C\to C$$\dagger : C \to C

which is the identity on objects, and which satisfies $†\circ †={\mathrm{id}}_C$.

Note that regarded as an extra structure on categories, a dagger structure is evil, since it imposes equations on objects.

Examples

• The category Rel of sets and relations is a dagger category, taking dagger as relational converse.

• More generally, let $C$ be a category with pullbacks and let ${\mathrm{Span}}_1(C)$ be the 1-category of spans up to isomorphism: its morphisms are spans with one leg labeled as source, the other labeled as target. Then the functor $†:{\mathrm{Span}}_1(C{)}^{\mathrm{op}}\to {\mathrm{Span}}_1(C)$ which just exchanges this labeling is a dagger-structure on ${\mathrm{Span}}_1(C)$.

• $ℛ(G)$, the category of unitary representations of a (discrete) group $G$ and intertwining maps, is a dagger category. For an intertwiner $\varphi :R\to S$, let ${\varphi }^{†}:S\to R$ be the adjoint of $\varphi$ in Hilb.

Underlying groupoid

For the purpose of considering what makes two objects of a $†$-category equivalent, one should not consider all isomorphisms (invertible morphisms) but rather all unitary isomorphisms: those morphisms $f$ whose adjoint is their inverse.

For example, in $\mathrm{Hilb}$, there are many invertible linear operators, but only those of norm $1$ (the invertible isometries) are unitary.

The unitary isomorphisms form a groupoid, which may be regarded as the core of the $†$-category.

Model structure on dagger-categories

the following is based on a remark by Andre Joyal, posted to the CategoryTheory mailing list on Jan 6, 2010

Consider dagger-categories from the point of view of homotopy theory.

Recall that the category Cat of small categories naturally admits the model category structure called the folk model structure on Cat.

The category of small dagger categories $\mathrm{DCat}$ also admits a “natural” model category structure:

• dagger functor $f:A\to B$ is a weak equivalence iff it is

  • full and faithful;

  • and unitary surjective, meaning that every object of $B$ is unitary isomorphic to an object in the image of the functor $f$;

• the cofibrations and the trivial fibrations are as in Cat;

• fibrations are the unitary isofibration: maps having the right lifting property for unitary isomorphisms.

The forgetful functor $\mathrm{DCat}\to \mathrm{Cat}$ is a right adjoint but it is not a right Quillen functor with respect to the natural model structures on these categories.

Moreover, a forgetful functor $\mathrm{XStruc}\to \mathrm{Cat}$ should reflect weak equivalences in addition to preserving them. The forgetful functor $\mathrm{DCat}\to \mathrm{Cat}$ preserves weak equivalences but it does not reflect them. Because two objects in a dagger category can be isomorphic without been unitary isomorphic.

In other words the forgetful functor $\mathrm{DCat}\to \mathrm{Cat}$ is wrong. This may explains why a dagger category cannot be regarded as a category equipped a homotopy invariant structure, as discussed in more detail in the example sections of the entry evil.

But the notion of dagger category is perfectly reasonable from an homotopy theoretic point of view. This is because the model category $\mathrm{DCat}$ is a combinatorial model category. It follows, by a general result, that the notion of of dagger category is homotopy essentially algebraic There a homotopy limit sketch whose category of models (in spaces) is Quillen equivalent to the model category $\mathrm{DCat}$. This is true also for the model category Cat.

$(\mathrm{\infty }\mathrm{,1})$-dagger-categories

the following is based on a remark by Andre Joyal, posted to the CategoryTheory mailing list on Jan 6, 2010

There should be a notion of dagger quasi-category.

A dagger simplicial set can be defined to be a simplicial set $X$ equipped with an involutive isomorphism $†:X\to X^{\mathrm{op}}$ which is the identity on 0-cells. The category of dagger simplicial sets (and dagger preserving maps) is the category of presheaves on the category whose objects are the ordinals $[n]$, but where the maps $[m]\to [n]$ are order reversing or preserving.

References

The concept of $†$-category is discussed here:

• S. Abramsky and B. Coecke, A categorical semantics of quantum protocols, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04), IEEE Computer Science Press (2004). arXiv

and further abstracted in:

• 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. web

The importance of $†$-categories in quantum theory is discussed here:

• John Baez, Quantum quandaries: a category-theoretic perspective, in Structural Foundations of Quantum Gravity, eds. Steven French, Dean Rickles and Juha Saatsi, Oxford U. Press, 2006, pp. 240–265. See especially Section 3: The $\star$-category of Hilbert spaces. (web)

Note that in older literature, the term ”$\star$-category” or “star-category” is sometimes used instead of $†$-category.

Certain specially nice $†$-categories, such as $C^{*}$-categories and modular tensor categories, play an important role in topological quantum field theory and the theory of quantum groups:

• Jürg Fröhlich and Thomas Kerler, Quantum Groups, Quantum Categories, and Quantum Field Theory, Springer Lecture Notes in Mathematics 1542, Springer-Verlag, Berlin, 1991.

• Bojko Bakalov and Alexander Kirillov, Jr., Lectures on Tensor Categories and Modular Functors, American Mathematical Society, Providence, Rhode Island, 2001. (web)