Dear Steven, I very much agree with all in your posting, and I would like to add some comments: "A topos is simply one of many possible generalization of sets and their functions ... " is very misleading in fact as a topos Sets is a point. The underlying category of a topos shares the exactness properties of the category of sets, and only secondarily it is an elementary topos Morphisms of elementary topoi are the logical ones, and they sit in the other side of the duality, morphism of topoi (or their inverse images which go in the same direction than morphisms of elementary topoi) do no agree with the elementary topos essential structure of the underlying category. As Steven say, "The generalized topological spaces are at the heart of Grothendieck's motivation" I remember I was a student at Chicago when Grothendieck visited, and after his talk, around a table at a bar (he would drink only water) he said that he was very upset that Lawvere had called his concept a topos. I imagine he could have called "generalized set" for example. Much confusion would have being avoided, and today elementary topi would be called "Lawvere Sets", an even strong recognition of the importance of Lawvere and his creation. Eduardo. On 02/09/2024 10:45 AM, Steven Vickers wrote:
Dear Vaughan,
It's easy to make the summary "A topos is simply one of many possible generalization of sets and their functions ...", but that's definitely an over-simplification. As Mac Lane and Moerdijk say, toposes have two facets: as generalized universes of sets (which is what you said), and as generalized topological spaces (which is what I was alluding to in my own Guardian comment). In fact Johnstone's Elephant explicitly tries to bring out even more facets.
The generalized topological spaces are at the heart of Grothendieck's motivation. On the one hand, that is in the sense of algebraic topology, in that topological invariants such as cohomologies can be calculated for them - and can be exploited in algebraic geometry. On the other hand, it is also in the sense of general topology, in that a topos can be fruitfully be viewed as a space whose points are the models of a geometric theory that the topos classifies.
The trouble is, the generalized topological space is easy to lose sight of, even easier if you move to elementary toposes (which are not classifying toposes in their own right, but only relative to other toposes), so many mathematicians just see the generalized universes of sets.
Actually, the "essential properties that make sets so valuable in mathematics" can be an obstruction to seeing the generalized topological spaces. The issue is that some of the "essential properties", such as cartesian closedness and subobject classifiers, do not interact successfully with geometric morphisms, the generalization of continuous maps. (They are not preserved by inverse image functors.) For an unobstructed view of the generalized topological spaces, at least in the sense of general topology, it is best to reject those non-geometric constructions.
To summarize: if you view toposes as "simply" the generalized universes of sets, then you risk overlooking the generalized topological spaces.
Steve.
------------------------------------------------------------------------ *From:* Vaughan Pratt <pratt@cs.stanford.edu> *Sent:* Monday, September 2, 2024 6:32 AM *To:* Wesley Phoa <doctorwes@gmail.com> *Cc:* categories@mq.edu.au <categories@mq.edu.au> *Subject:* Re: Grothendieck in the Guardian *CAUTION:* This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe.
"I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical."
Why would anyone object to journalists doing exactly those things? Do we want to keep mathematics a dark secret, or what?
A topos is simply one of many possible generalization of sets and their functions that allows many other mathematical objects besides sets to be imbued with some of the essential properties that make sets so valuable in mathematics.
For example graphs and their maps form a topos with very similar properties to sets and their functions, such as having the notion of a power set. But not all properties, for example the law of the excluded middle, which holds for sets but not graphs.
Vaughan Pratt
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