Re: Grothendieck toposes

[Marta Bunge <martabunge@hotmail.com>]

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



************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics
McGill University
Montreal, QC, Canada H3A 2K6
Home: (514) 935-3618
marta.bunge@mcgill.ca
http://www.math.mcgill.ca/people/bunge
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________________________________
From: Patrik Eklund <peklund@cs.umu.se>
Sent: November 2, 2016 6:13:30 AM
To: Joyal, André
Cc: Marta Bunge; categories@mta.ca; Steve Vickers
Subject: Re: categories: Re: Grothendieck toposes

On 2016-11-01 17:16, Joyal wrote:

> 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 "miracle disappears in confusion" is an important observation, as is
the need to "keep apart before uniting".

Syntax and semantics is like that, or meta and object language.
Foundations of mathematics without categorical consideration is
basically then over  Set, naively speaking. Logic is similar. Fons et
origo logic from late 19th century and decades after is confused about
being before set theory or after. Topos internalizes logic but is
different from the Goguen-Meseguer approach to institutions and
entailment systems. The apples and pears of logic should not be seen as
a fruit salad.

I've sometimes thought (and written some pieces about, e.g. to be found
under www.glioc.com<http://www.glioc.com>) what if Gödel's Incompleteness  Theorem wasn't a
Theorem but a Paradox. After all, Gödel basically transforms the Liar
Paradox to a Liar Theorem, and logicians at that time (except maybe
Hilbert, but he was too old to quarrel) found it to be very smart.
However, if we use underlying categories and functors to start from
signatures, then create terms, then sentences, then entailments, then
models, then proof strategies, and so on, it means we close doors behind
us, so that we disable ourselves to mix truth and provability as being
"of the same kind or type", which Gödel did. Categorically, terms come
from monads, because they enable substitution, but sentences just from
functors, because otherwise everything is 'term'. The functorial
description and generality of entailment and model is of course more
tricky, in particular if the underlying category is something more
elaborate (like monoidal closed categories) than just Set.

In this (heretic?) view, Gödel's Theorem/Paradox is actually an example
where that miracle appears because of the unintended (?) confusion, so
this is why I sometimes think what if it was ween as a Paradox.

Best,

Patrik

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