Re: Terminology

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Dear Jean,

My own understanding (superficial and possibly wrong) of the history is that  since Bourbaki there have been definitions of "structure" with the aim of reconciling the algebraic examples (where the homomorphisms preserve structure) with the topological spaces (where the continuous maps have inverse images that preserve structure). Certainly if you look at Joy of Cats, the prime classes of examples are those of topological and algebraic categories.

But, as we know from topos theory, it is not foundationally robust to treat topological spaces as "sets with structure", i.e. point-set topology. In general we have to work point-free, at least if we want to save important parts of topology from going down the drain.

If such an important source of examples, the point-set topological spaces, turned out to be misleading, then, in retrospect, any "precise meaning [of structure] on which the community of mathematicians agree", was probably misguided.

It's like looking for a definition of "fish", but on the understanding that it has to include whales.

All the best,

Steve.

> On 9 Feb 2017, at 16:38, Jean Benabou <jean.benabou@wanadoo.fr> wrote:
> 
> Dear Christopher,
> What I, personally, mean by structure is not the point. This word is used,  very often, in mathematical texts. Sometimes giving the impression that it has a precise meaning on which the community of mathematicians agree. And I was sure there was at least one definition on which the majority of users did  agree
> 
> Then I received 3 answers all referring to: The joy of Cats, but different:
> For Carsten Führman, only faithfulness is required, which obviously is not enough
> Jiri Adamek adds: an isomorphism in S is an identity if its image is. I agree with this; but again not enough.
> Thomas Streicher adds a third condition, with which I would probably agree  if was sure of the precise meaning of isofibration. Could you please, even at the risk of being pedantic say what you mean by that
> 
> Many thanks to all
> 
> 
>> â€ژHi Jean - I don't quite understand this question but  would like to. What do you mean by 'structure'? Thanks
>> 

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