Steve Vickers raises the issue of when a simplification is an over-simplification. One might think to view that as a function of the intended audience. For an audience as comfortable with the notion of a topological space as with the notions of sets and functions, I would agree that I'd oversimplified my account. Perhaps I was a little optimistic about mentioning graphs, but at least "graph" has no more syllables than "set", and these days half the student body is exposed to the concept thanks to Computer Science having recently passed Human Sexuality in enrollments. I had originally intended to mention the notion of an abelian category as being related to the notion of a topos. (Peter Freyd and I had a few exchanges on that topic in this forum a few decades ago that helped me a lot.) However I figured that abelian groups or vector spaces as typical of objects one might find in an abelian category were above the level of sets and functions, and so decided not to mention abelian categories. But we're all sufficiently fond of categories here to appreciate the value of being able to take the opposite of a category without feeling as ill as MSRI Director Bill Thurston claimed to be in his welcoming (and welcome) address to the Universal Algebra and Category Theory meeting at MSRI in July, 1993. What made Bill's remark memorable for me was the sharp intake of breath heard round the room when Bill made it. (And note how context can disambiguate a word like "welcome".) Does "function of the intended audience" have an opposite? Interestingly, it does. It is called the Gunning Fog Index for an article. It is calculated as 0.4 times the sum of the average sentence length and the percentage of three-syllable words or longer (with certain rules about counting syllables). The resulting number is the age group the article is optimum for. Unfortunately this does not take into account the level and type of education needed to know what a particular word means, and "topos" can only decrease the Fog Index. Likewise for philtrum, though acnestis will increase it, as words more likely to be encountered in Human Sexuality than Computer Science. But at least it's a start. A dictionary of obscure words with a score for each could be a blessing. And so one can just write at whatever level you feel is appropriate, and calculate its fog index when done. You then know what audience your exegesis is most suitable for. Vaughan Pratt On Mon, Sep 2, 2024 at 6:45 AM Steven Vickers <s.j.vickers.1@bham.ac.uk<mailto:s.j.vickers.1@bham.ac.uk>> 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<mailto:pratt@cs.stanford.edu>> Sent: Monday, September 2, 2024 6:32 AM To: Wesley Phoa <doctorwes@gmail.com<mailto:doctorwes@gmail.com>> Cc: categories@mq.edu.au<mailto:categories@mq.edu.au> <categories@mq.edu.au<mailto: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 You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>