Re: Grothendieck toposes

[majordomo@mlist.mta.ca]

Dear Marta,

Mathematics and science are very often regarded
as the pure product of human rationality.
I can agree with the importance of rationality, except that humanity
is as much the product of nature as it is of rational choices.
You will agree that the natural world is only partially explained by science.
The rest is a big mystery. Not that the mystery is absolutly impenetrable.
I feel compelled to recognize the presence of mysteries even
in mathematics. The history of complex numbers, from the
discovery by Cardano to their applications in quantum physics is bewildering.
They belong to this universe as much as the electron and the human mind.
The fact that we human can understand  complex numbers may
have a metaphysical meaning.
What is it?

Best,
André


________________________________
From: Marta Bunge [martabunge@hotmail.com]
Sent: Wednesday, November 02, 2016 7:18 AM
To: categories@mta.ca
Cc: Steve Vickers; Patrik Eklund; Joyal, André
Subject: Re: categories: Re: Grothendieck toposes


Dear all,



> It is marvelous that the two notions should be so related.
> But it is be better to keep them appart before uniting them.
> Otherwise the miracle disappear in confusion.



The above is a quotation from a recent posting by Andre Joyal.  To the risk  of boring everyone I offer the following comment on it here. There is no need to talk about miracles in mathematics, not even as some sort of analogy. Why not instead give credit to the very important insight of an elementary topos as embodying both the logic and the geometry? There are two notions  of morphism between elementary toposes, not a preferred one - the geometric and the logical. One structure - to wit that of an elementary topos, can be seen in two different ways depending on what the mathematical uses one wants to give it. There is no confusion here  - just richness. Let me be more specific.


Thinking of an elementary topos S as the chosen "set theory", a Grothendieck topos (including any category of the form Sh(X) for X a locale in S, but more generally as a category of sheaves on a site in S) can be recovered as  a pair (E, e) where E is another elementary topos and e: E -> S a bounded geometric morphism. Thinking of elementary toposes from the logical point of view, and so of logical morphisms between them, there are other ideas and  constructions that profit from this point of view - for instance a formulation and proof of realizability by means of Artin-Wraith glueing.


Both the geometric and the logical are sides of the same coin. The notion of an elementary topos (or "topos" for short) is simple yet powerful and until now it has served most of the mathematical purposes for which it was intended and more.


Best wishes,

Marta




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