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 morphism 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.
In Wikipedia a dagger category is said to be the same as involutive category or category with involution. However, Springer’s Encyclopedia of Mathematics requires that a “category with involution” is also compatibly enriched over posets.
In enriched category theory, involutive categories have also been called symmetric categories. Sometimes the involution is required to be strict in the sense that the dagger is an equality rather than an isomorphism: in line with horizontal categorification, one could argue these should be called commutative categories, since a one-object category with such an involution is a commutative monoid.
A dagger category or $\dagger$-category $C$ is a category with a function $(-)^\dagger: Hom_C(A,B) \to Hom_C(B,A)$ for every object $A,B \in Ob(C)$, such that
Given a category $C$ with a type or class of objects $Ob(C)$ and a set of morphisms $Mor(C)$ with source and target functions $s:Mor(C) \to Ob(C)$ and $t:Mor(C) \to Ob(C)$, $C$ is a dagger category if it has a function $(-)^\dagger:Mor(C) \to Mor(C)$ such that
A dagger 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 $\dagger^{op} \circ \dagger = \mathrm{id}_C$.
This definition, which is perhaps the most concise, does rely on a notion of identity-on-objects functor. This is no problem in most foundations for mathematics, although it violates the principle of equivalence (except for strict categories). In homotopy type theory (or more generally intensional type theory) it doesn’t make sense to talk about “being the identity on objects” as a property of a given functor, but one can still define an “identity-on-objects endofunctor” (covariant or contravariant) as a basic notion; see identity-on-objects functor for details. When unwound for $\dagger$-categories, this yields the above “family of functions” definition.
A morphism $f$ in a †-category is called a 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
Given two $\dagger$-categories $A$ and $B$, a $\dagger$-functor $F : A \to B$ consists of a function $F_0 : Ob(A) \to Ob(B)$ with a function $F_{a,b}:Hom_A(a,b) \to Hom_B(F a,F b)$ for every object $a,b:Ob(A)$, where $F_{a,b}$ is generally also denoted as $F$, such that
More concisely, one can say that a $\dagger$-functor $(A,\dagger) \to (B,\ddagger)$ is a functor $F : A \to B$ of the underlying categories that commutes with the $\dagger$-structures in that $F \circ \dagger = \ddagger \circ F^{op}$. This appears to violate the principle of equivalence, since it is a strict equality of functors; however, once $\dagger$ and $\ddagger$ are known to be the identity on objects (whatever that means), the two functors $F\circ \dagger$ and $\ddagger \circ F^{op}$ automatically agree on objects.
A natural transformation between $\dagger$-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.
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 category of couplings between probability spaces.
Every symmetric proset is a thin †-category.
strict dagger category?
A $\dagger$-2-poset is a $\dagger$-category that is also a 2-poset. Examples of $\dagger$-2-posets include allegories and bicategories of relations.
the following is based on a remark by Andre Joyal, posted to the CategoryTheory mailing list on Jan 5, 2010, with a follow-up on Jan 6.
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 †-category is homotopy essentially algebraic. There is 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 horizontal categorification of an anti-involutive monoid is a $\dagger$-category
Every symmetric proset is a thin $\dagger$-category, and the groupoidal categorification of a symmetric proset is a $\dagger$-category, with the $\dagger$ operation being the groupoidal categorification of the symmetric property.
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.
The finite-dimensional Hilbert spaces of interest in quantum information theory and quantum computation form $\dagger$-categories that are also compact closed categories (dagger compact categories), a fact that allow a string diagram-calculus for quantum mechanics in terms of †-compact categories which motivates much of the contemporary discussion of dagger-categories, starting with Abramsky & Coecke 2004.
Motivated by the example of finite-dimensional Hilbert spaces, the concept of $\dagger$-categories appears in the form of strongly compact closed categories in
then named dagger compact closed categories in
and as such became the foundation of quantum information theory via dagger-compact categories.
Exposition in this context of quantum physics:
and of quantum information theory via dagger-compact categories:
Chris Heunen, Jamie Vicary, §2.3 in: Categories for Quantum Theory, Oxford University Press 2019 [ISBN:9780198739616]
based on:
Chris Heunen, Jamie Vicary, §2.3 in: Lectures on categorical quantum mechanics (2012) [pdf, pdf]
But the notion of plain dagger categories has been considered much earlier, reinvented many times under different names (see Karvonen 2018 §1.2 for more historical references) such as under the name “categories with involution”:
Saunders Mac Lane, pp. 1045 of: An Algebra of Additive Relations, Proceedings of the National Academy of Sciences of the United States of America 47 7 (1961) 1043-1051 [jstor:71127]
Dieter Puppe, §1 in Korrespondenzen in abelschen Kategorien, Mathematische Annalen 148 (1962) 1–30 [doi:10.1007/BF01438388]
Mark S. Burgin, Categories with involution, and correspondences in $\gamma$-categories, Trudy Moskovskogo Matematicheskogo Obshchestva 22 (1970) 161–228 [mathnet:mmo235]
Vaguely related is the notion of “symmetric category” is (analogous to symmetric proset or symmetric bicategory)
Still, dedicated discussion of these notions only appears later under the name of dagger-categories:
concerning limits:
concerning monads (such as the Frobenius quantum reader monad):
concerning abstract characterization:
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, see the references there for more.
On dagger higher categories:
Three definitions of dagger $(\infty,1)$-categories have been proposed by:
Generalization to dagger $(\infty,n)$-categories:
Theo Johnson-Freyd, Dagger Higher Categories, workshop (2023) [web]
Theo Johnson-Freyd, Higher Dagger Categories, talk at CQTS (08 Nov 2023) [slides:pdf, pdf, video:YT]
Last revised on January 25, 2024 at 15:52:33. See the history of this page for a list of all contributions to it.