[Note from moderator: Several messages to categories apparently hung in a mail system for several days. With apologies to posters, I am about to post four from late Decemeber in what should have been their posting order. Sorry about the delay, Bob] A dagger structure on a category should not really be considered evil at all. If you have a functor F: C^op -> C and ask whether it is a dagger structure, then this is (taken literally) an evil question; the answer is yes iff F^2 = 1 and F is the identity on objects, both evil conditions. More precisely, two isomorphic functors may have different answers. (A non-evil version is to ask whether F is isomorphic to a dagger structure.) However, it's not necessary to define a dagger-category as a category C equipped with a functor F: C^op -> C such that F satisfies these conditions. In lower-level language, we ask instead that C be equipped with an operation that takes each morphism f: x -> y to a morphism f^\dag: y -> x such that id^\dag = id, (f g)^\dag = g^\dag f^\dag, and (f^\dag)^\dag = f. Nothing here refers to equality of objects; it can be formulated in a language that (like FOLDS) does not have this concept. Given a dagger structure on C, defined in this elementary way, we can construct a functor \dag: C \to C^op that satisfies the evil property. (Of course, it also satisfies the non-evil version of that property.) But that is neither here nor there as to whether dagger structures are evil. There is some new discussion on the nLab: http://ncatlab.org/nlab/show/evil#daggers In particular, Mike Shulman shows how to translate dagger structures along equivalences of categories, proving that they are not evil. My previous post on this subject should probably be ignored. While any concept ~can~ be de-evilled in the way shown there, this does not necessarily give you the concept that you want, and indeed it need not even preserve already non-evil concepts. (And in this case specifically, it does not seem to be correct, as others have already argued here.) --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I have been following with some amusement the discussion of 'evil'. I am mostly amused because long ago, I think in my grad student days standing in Peter Freyd's office, I had made the suggestion that to avoid overloading adjectives (normal and regular being particularly abominable examples of the phenomenon) mathematicians should resort to using adjectives that usually have a moral denotation. Once one realizes that 'evil' exists not just for objects in categories, but for 0- and 1-arrows in bicategories, 0-, 1- and 2-arrows in tricategories, . . . it seems to me the proper attitude to take toward 'evil' (or strictness) is given by Saunders' dictum about generality: "good general theory does not search for the maximum generality, but for the right generality". So a good (higher) categorical structure should not search for the maximum weakness, but for the right weakness. (Or, if you want, not search for the minimum 'evil', but the the right amount of 'evil'.) For instance, it seems to me that structure of the category of framed tangles in which arrows are ambient isotopy classes (rel boundary) of framed tangles, which Shum's beautiful coherence theorem tells us is monoidally equivalent to the free ribbon (née tortile) category one one object generator is marred and made less useful (certainly for application to knot theory and 3- and 4-manifold topology) by deciding one should work instead with a (2,infinity)-category with one object, framed point sets as 1-arrows, framed tangles as 2-arrows, isotopies as 3-arrows, isotopies of isotopies as 4-arrows, and so ad infinitum. Or, maybe not, but only if one has an application for which the (2,infinity)-category is the right level of weakness. Best Thoughts, David Yetter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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David Yetter -
Toby Bartels