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 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:
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) $f^\dagger : H_2 \to H_1$ (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 a contravariant functor which sends every morphism $f : a \to b$ to a morphims going the other way, $f^\dagger : b \to a$.
The notation and terminology here is motivated from the example Hilb of Hilbert spaces, where $f^\dagger$ is traditionally the notion for the adjoint of a linear map $f$. The canonical †-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 $n Cob \to Hilb$ which respects the †-structure on both sides.
A dagger category, or †-category, is a category $C$ equipped with a contravariant endofunctor, hence an ordinary functor from the opposite category $C^{op}$ of $C$ to $C$ itself
which
is the identity on objects,
is an involution $\dagger \circ \dagger = \mathrm{id}_C$.
Note that regarded as an extra structure on categories, the concept of †-structure violates the principle of equivalence, since it imposes equations on objects.
A morphism $f$ in a †-category is called unitary morphism if its †-adjoint equals its inverse:
For the purpose of considering what makes two objects of a $\dagger$-category equivalent, one should not consider all isomorphisms (invertible morphisms) but rather all unitary isomorphisms.
The unitary isomorphisms form a groupoid, which may be regarded as the dagger-core of the $\dagger$-category.
For example, in Hilb, there are many invertible linear operators, but only those of norm $1$ (the invertible isometries) are unitary.
A morphism $f$ in a †-category is called a self-adjoint morphism if it equals its †-adjoint
A morphism $F : (C, \dagger) \to (D, \ddagger)$ of †-categories – a †-functor – is a functor $F : C \to D$ of the underlying categories, which commutes with the †-structures in that
A natural transformation between †-functors is just a natural transformation of the underlying functors.
The †-adjoint $\eta^*$ of a natural transformation
between two †-functors $F, G : (C,\dagger) \to (D,\ddagger)$ is given by the componentwise $\ddagger$-adjoint:
To check that $\eta^*$ is indeed a natural transformation $\eta^* : G \to F$ consider $f : a \to b$ any morphism in $C$ and $f^\dagger : b \to a$ its $\dagger$-adjoint and let
be the corresponding naturality square of $\eta$. Taking the $\ddagger$-adjoint of the entire diagram yields
by the fact that $F$ and $G$ are †-functors. This is the naturality square over $f$ of $\eta^* : G \to F$.
Write $DagCat$ for the category whose objects are †-categories and whose morphisms are †-functors.
For $(C,\dagger)$ and $(D,\dagger)$ two †-categories, write $([(C,\dagger),(D,\ddagger)]_{dag}, \star) \in DagCat$ for the †-category whose objects are †-functors, whose morphisms are natural transformations, with the †-operation $\star : \eta \mapsto \eta^*$ as above.
The assignment $((C,\dagger),(D,\ddagger)) \mapsto [(C,\dagger),(D,\ddagger)]_{dag}, \star)$ extends to an internal hom-functor
that makes $DagCat$ into a cartesian closed category.
This follows step-by-step the standard proof that Cat is cartesian closed, while observing that each step respects the respect for †-structures.
To indicate the main point, let $C, D$ and $E$ be †-categories and consider a functor $F : C \times D \to E$. For $(f : c_1 \to c_2) \in C$ and $(g : d_1 \to d_2) \in D$ we have natural assignments
that respect daggering all morphisms, in the evident way.
Keeping $d_1$ and $d_2$ fixed, respectively this makes $F(-,d_1), F(-,d_2) : C \to E$ †-functors. We see from the diagrams that $F(-,(d_1 \stackrel{g}{\to}) d_2)$ is a natural transformation between these †-functors, and the fact that $F$ intertwines the dagger operation of $D$ with that of $E$ means $F$ regarded as a functor $D \to [C,E]$ intertwines the †-structures of $D$ and $[D,E]_{dag}$, by the above definition.
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.
The category Rel of sets and relations is a †-category, taking dagger as relational converse.
More generally, let $C$ be a category with pullbacks and let $Span_1(C)$ 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 $\dagger : Span_1(C)^{op} \to Span_1(C)$ which just exchanges this labeling is a †-structure on $Span_1(C)$.
$\mathcal{R}(G)$, the category of unitary representations of a (discrete) group $G$ and intertwining maps, is a †-category. For an intertwiner $\phi : R \rightarrow S$, let $\phi^\dagger : S \rightarrow R$ be the adjoint of $\phi$ in Hilb.
> the following is based on a remark by Andre Joyal, posted to the CategoryTheory mailing list on Jan 6, 2010
Consider †-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 †-categories $DCat$ also admits a “natural” model category structure:
†-functor $f:A \to B$ is a weak equivalence iff it is
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 $DCat \to Cat$ 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 $XStruc \to Cat$ should reflect weak equivalences in addition to preserving them. The forgetful functor $DCat\to Cat$ preserves weak equivalences but it does not reflect them. Because two objects in a †-category can be isomorphic without been unitary isomorphic.
In other words the forgetful functor $DCat\to Cat$ is wrong. This may explains why a †-category cannot be regarded as a category equipped a homotopy invariant structure, as discussed in more detail in the example sections of the entry principle of equivalence.
But the notion of †-category is perfectly reasonable from an homotopy theoretic point of view. This is because the model category $DCat$ is a combinatorial model category. It follows, by a general result, that the notion of of †-category is homotopy essentially algebraic There a homotopy limit sketch whose category of models (in spaces) is Quillen equivalent to the model category $DCat$. This is true also for the model category Cat.
> the following is based on a remark by Andre Joyal, posted to the CategoryTheory mailing list on Jan 6, 2010
A †-simplicial set can be defined to be a simplicial set $X$ equipped with an involutive isomorphism $\dagger :X\to X^{op}$ which is the identity on 0-cells. The category of †-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.
> 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 †-quasi-category based on $\dagger$-simplicial sets as ordinary quasi-categories are based on ordinary simplicial sets.
(…)
The category convolution algebra of a dagger category is naturally a star-algebra. The star-involution is given by pullback of functions along the $\dagger$-functor.
Large parts of quantum mechanics and quantum computation are naturally formulated as the theory of $\dagger$-categories that are also compact closed categories in a compatible way – dagger compact categories.
For more on this see
The concept of $\dagger$-category is discussed here:
and further abstracted in:
The importance of $\dagger$-categories in quantum theory is discussed here:
Note that in older literature, the term “$\star$-category” or “star-category” is sometimes used instead of $\dagger$-category.
Certain specially nice $\dagger$-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)