Dear Peter and all, I cannot resist adding my grain of salt to the ongoing discussion on dagger categories. I will take the point of view of a homotopy theorist. Recall that the category of small categories Cat admits a "natural" model structure (called the "folk" model structure for the wrong reason by the folks). The category of small dagger categories DCat also admits a "natural" model structure. A dagger functor f:A-->B is a weak equivalence iff it is fully faithful and unitary surjective (this last condition means 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. A fibrations is a unitary isofibration (a map having the lifting property for unitary isomorphisms). The forgetful functor DCat ---> Cat is a right adjoint but it is not a right Quillen functor with respect to the natural model structures on these categories. In other words the forgetful functor DCat ---> Cat is wrong. This may explains why a dagger category cannot be regarded as a category equipped a homotopy invariant structure. But I claim that the notion of dagger category is perfectly reasonable from an homotopy theoretic point of view. This is because the model category DCat is combinatorial. 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 DCat. This is true also for the model category Cat. 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 dag:X-->X^o 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]-->[n] are order reversing or preserving. Finally, the (homotopy) trace of a category (resp. quasi-category) has the structure of a cyclic set in the sense of Connes. I conjecture that the (homotopy) trace of a dagger category (resp. dagger quasi-category) has the structure of a dihedral set in the sense of Fiedorowicz and Loday. Happy New Year to all! andré PS: I will be quiet during the next few weeks. -------- Message d'origine-------- De: categories@mta.ca de la part de Peter Selinger Date: dim. 03/01/2010 02:23 À: Categories List Objet : categories: the definition of "evil" Dear all, sorry for sending yet another message on the topic of "evil" structures on categories. After some interesting private replies, as well as Dusko's latest message (which should have appeared on the list by the time you read this), I noticed that not everyone is agreeing on the technical meaning of the term "evil". I will therefore attempt to state a more precise technical definition of the term as I have used it. Perhaps 2-category theorists already have another name for this. The information definition I had used is that a structure is "evil" if it does not "transport along equivalences of categories". I thought it was reasonably obvious what was meant by "transport along", but there is actually a lot of variation in what people understand this phrase to mean. John Baez gave a pointer to a website containing a technical definition of "evil": http://ncatlab.org/nlab/show/evil. Unfortunately, this site only speaks of properties, not structures. It is easy to state what it means for a property of categories to be transported along equivalences: namely, if C has the property, and C and C' are equivalent, then C' has the property. Structures are more tricky. Certainly, it should not just mean that if C admits such a structure, and C' is a category equivalent to C, then C' admits such a structure. (Then "admitting a structure" would merely be a property). This seems to be the definition Dusko has used. If we used this definition, there would be almost no evil structures; in particular, the original (strict) notion of dagger category is not evil in this sense. Dagger structure is reflected by full and faithful functors, and therefore by one half of an equivalence. The point is that the other half won't respect it. At least to me, "transported" suggests that the given equivalence respects the structure in some sense. So here is my attempt at a definition. DEFINITION. Let X be some structure on categories. By this, I mean that there is a given 2-category called X-Cat, whose objects are called X-categories, whose morphisms are called X-functors, and whose 2-cells are called X-transformations, and for which there is a given 2-functor U to Cat, called the forgetful functor. We say that X is "transported along equivalences of categories" if the following holds. Given an X-category D', with underlying category D = U(D'), and a category C, and an equivalence (F,G,e,h) of categories D and C, where F: D -> C, G: C -> D, and e: FG -> id_C, h: GF -> id_D, it is then possible to find: (1) an X-category C' whose underlying category U(C') is isomorphic [not equivalent!] to C. Let c : U(C') -> C be the isomorphism (i.e., an invertible functor in Cat) with inverse c': C -> U(C'); (2) an X-equivalence of X-categories (F',G',e',h'), where F': D' -> C', G': C' -> D', e': F'G' -> id_C', and h': G'F' -> id_D' [the concept of equivalence makes sense in any 2-category]; such that (3) (U(F'), U(G'), U(e'), U(h')) = (cF, Gc', cec', h). Here, cF and Gc' denotes composition of functors, and cec' denotes whiskering. The structure X is called "evil" iff it is not transported along equivalences of categories. This finishes the definition. More informally, "transported along equivalences" therefore means that if D and C are equivalent, and D has an X-structure, then there is a way to equip C with an X-structure and to lift the original equivalence to an X-equivalence. There was a need for the isomorphism c in the definition, because the forgetful functor U : X-Cat -> Cat may not be strictly speaking surjective onto 0-cells in some real-life examples (and in any case, this forgetful functor may sometimes only be well-defined up to isomorphism). It is important that c is an isomorphism, rather than an equivalence, because else the definition becomes vacuous (and we are precisely interested in notions that are not well-defined up to equivalence). Also note that I didn't require the data (C',F',G',e',h') to be unique, not even up to equivalence in X-Cat. Although in practice, it will often be unique in this sense. So my definition allows for a given structure to be transported "in essentially more than one way" along a given equivalence. I am open to strengthening the definition to forbid this. It is clear that the definition generalizes to any 2-category instead of Cat, so one might for example speak of structures on monoidal categories, or on categories-with-a-distinguished-subcategory, or even on dagger categories, as being evil or not. Here are some examples of structures: * monoidal structure on categories is non-evil (for concreteness, taken with strong monoidal functors and monoidal natural transformations). * strict monoidal structure is evil, when taken with strict monoidal functors. With strong monoidal functors, I think it is still evil, but I am not sure at this late hour. * dagger structure is evil. More generally, any structure X with which one can equip FHilb (the category of finite dimensional Hilbert spaces), and which allows a definition of unitary map that includes all identities and that coincides with the usual one on FHilb, and for which the full and faithful X-functors preserve and reflect unitary maps, is evil. Here is the technical argument again, as it seems to have been misunderstood. The forgetful functor F : FHilb -> FVect induces an equivalence, whose other half G : FVect -> FHilb requires a choice of inner product on each finite dimensional vector space. Define such a G in some way. Fix some X-structure on FVect. Let V be some non-trivial vector space, and let i and j be two different inner products on V. Then (V,i) and (V,j) are Hilbert spaces, so different objects of FHilb. Consider the morphism (in FHilb) f:(V,i) -> (V,j) given by f(v) = v. It is evidently not unitary. However, we have F(f) = id_V: V -> V, which is unitary, no matter the X structure that was chosen on FVect. So F does not reflect unitary maps. QED. Note that it is F, not G, that is causing problems. As remarked above, since G is full and faithful, it is possible to successfully reflect the dagger structure along G to FVect. This amounts to arbitrarily choosing some inner product on each vector space. But it won't be compatible with F. Also note that this argument is independent of the definition of the 2-cells of X-Cat. So it is even valid for some weaker definitions of "evil", for example, if one only requires F and G to lift to X-functors, rather than to an X-equivalence. I will argue that any structure X that claims to be a "weak" version of dagger structure should at least satisfy the conditions I listed as preconditions for the argument above. This is the basis for my claim that no construction such as Toby's or Dusko's can succeed in producing a non-evil equivalent of dagger structure. * the structure of "being equipped with a chosen Frobenius structure on each object" is evil, relative to monoidal categories. * the structure of "being equipped with an identity-on-objects covariant functor" is evil. * more generally, the structure of "being equipped with a chosen subcategory" is evil, unless the subcategory is required, as part of the structure, to contain all isomorphisms (in which case it is not evil). * poset-enrichment (with composition f o g monotone in f and g) is non-evil. * The following structure is evil: equip a category with a partial order on each hom-set, so that composition f o g is monotone in g, but not necessarily in f. Proof: Given such a structure on any category, define g:A->B to be "monotone" if (X,g) : (X,A) -> (X,B) is monotone for all X. Consider the category whose objects are partially-ordered sets, and whose morphisms are *all* functions (thanks to Fred Linton for this example). It can be equipped with the aforementioned structure, by giving the pointwise ordering to the functions in each hom-set. As a category, it is equivalent to Set. The rest of the argument proceeds as above for Hilbert spaces, with "monotone" instead of "unitary": take some non-trivial set with two different partial orders, then the identity is non-monotone, etc. The last example is almost an enrichment in Poset, but instead of the usual cartesian product on Poset, we have used another bifunctor on Poset, given by cartesian product P x Q of the underlying sets, with the non-standard order defined by (p,q) <= (p',q') iff p=p' and q<=q'. This operation is bifunctorial and associative, but not quite monoidal, because it lacks a right unit. More generally, taking "enrichments" in such almost-monoidal categories often yields evil structures. An analogous example works for the category of finite abelian groups and all set-theoretic functions. Happy new year to all, -- Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]