Subject: Re: Equality again Newsgroups: gmane.science.mathematics.categories Date: Tuesday 1st June 2010 14:38:04 UTC (over 6 years ago) Dear Marco, We could use of the dotted-equality symbol only when the canonical isomorphism under consideration is part of a contractible network of isomorphisms. The network does not need to be explicitly identified if the context is clear enough. For example, the dotted equality (A times B)times C =. A times (B times C) is refering to the associativity constraint. The dotted equality A times B =. B times A is refering to the symmetry constraint. But the dotted equality A times A =. A times A is ambiguous and should be excluded (actually, it is not ambiguous, since the identity of A times A is denoted A times A = A times A ). I am proposing a rule of thumb, not a new formalism. Mathematics is as much an art as it is an exact science. Best, André -------- Message d'origine-------- De: [email protected] de la part de Marco Grandis Date: mar. 01/06/2010 02:36 À: Prof. Peter Johnstone; [email protected] Objet : categories: Re: Equality again On 27 May 2010, at 13:30, Prof. Peter Johnstone wrote: > > TeX provides a command \doteq for an equality sign with a dot over it; > this is used in other areas of mathematics to mean "is approximately > equal to", but as far as I know it hasn't yet been used by category- > theorists. Perhaps we could use it to mean "is canonically > isomorphic to"? > > I'd also like to use it (or something like it) between pairs of > morphisms, meaning that (they are not equal but) they become equal > when composed with the appropriate canonical isomorphisms (to which > I can't be bothered to give names) in order to match up their domains > and codomains. (Of course, this is simply saying that they are > canonically isomorphic as objects of the functor category [2,C], > where C is the category in which they live.) > > Peter Johnstone Dear Peter, Isn't this very dangerous? 1. First, I think you are referring to some (specified) *coherent* (contractible) system of isomorphisms, otherwise you can easily prove that 1 = - 1 (see an example below). 2. Even in that case, we know that coherence can be a delicate thing. Let us take the cartesian product in Set (or the tensor product in a symmetric monoidal category). Would you write XxY =. YxX for the symmetry isomorphism s? Then by XxX =. XxX do you mean s or the identity? For XxXxX =. XxXxX we have six permutations of variables, generated by sxX and Xxs; and so on. I hope nobody will suggest some complicated trick to account for this; transpositions and permutations are already there, known to everybody; but we have to name them. 3. Coming back to point 1, "canonical" isomorphisms need not be coherent. There are a lot of such situations; I like to refer to the induced isomorphisms in homological algebra, because much of my early work was linked with that. ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ] |
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