Re: Equality again

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

On Mon, 24 May 2010, Joyal, André wrote:

> I love the equality symbol more than an isomorphism symbol,
> and an isomorphism symbol more than an equivalence symbol.
> I always try to use the equality symbol whenever possible.
> I often use the equality symbol for a canonical isomorphism.
> Is there a special symbol for canonical isomorphism? (as oppose
> to a plain isomorphism). I would love to write something like
>
>  A times (B times C) =' (A times B) times C
>
> André
>

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



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