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: categories@mta.ca de la part de Marco Grandis Date: mar. 01/06/2010 02:36 À: Prof. Peter Johnstone; categories@mta.ca 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/ ]