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Added a table of contents to topos, a section on "special classes" and one on "higher toposes".
Oh, and I added a section with references.
I was – and still am – unhappy with the state of the entry topos. It did not really convey much of the grand idea of toposes.
In an attempt to remedy this, I now wrote an expanded Idea-section.
There are always two aspects of toposes. In what I wrote, I start the motivation with one of the two aspects, and then try to naturally lead over to the other. I wrote this in a way that I imagine I would have found useful reading back before I knew toposes.
But of course tastes may differ. So please have a look and see where you think this needs urgent improvement (that the entire entry generally needs improvement is clear…)
I wrote:
that the entire entry generally needs improvement is clear…)
Maybe I should clarify what I mean: not that there is anything wrong, as far as I can see, with the material that is there. What I mean is that there should be more material there Eventually this entry should be a more comprehensive discussion of the notion of topos.
There were once some blind men trying to describe an elephant…
Exactly. And now make them type what they have to say into a wiki entry, for the benefit of mankind.
I like it, thank you. I added some additional comments about the third “petit” perspective on toposes, as being themselves generalized spaces.
Steve Vickers wrote an article ’Toposes pour les nuls’, and following its distribution and some seminars he wrote a sequel ’pour les vraiment nuls’. I mention this to show that even the experts in the area, and who have worked with toposes for a long time, find them hard to present. I remember looking at ’pour les nuls’ but have only just noticed the sequel.
I like it, thank you.
Okay, thanks for letting me know!
I added some additional comments about the third “petit” perspective on toposes, as being themselves generalized spaces.
Very nice. Now I feel much better about this entry.
Seen this page: http://topos-physics.org
I had noticed this a long while back, but then never got back to it.
I think eventually this kind of undertaking needs to be soberified to take off. Eventually it would be nice to reduce the chat to a minimum and then just go Definition-Theorem-Proof-Definition-Theorem-Proof-so-there.
I am currently supervising a Bachelor thesis on something related. Should be finished later this summer. Then you can check if we are succeeding with following this suggestion ;-)
At topos it says
Since the definition of elementary topos is “algebraic,” there exist free toposes generated by various kinds of data.
What does the “algebraic” do there? Can one say what kinds of data can play the role of generators for free toposes?
What can be said in general about free generation? I see at free cartesian category, generators might form a set, a category or a signature.
It also says
any higher-order type theory (of the sort which can be interpreted in the internal logic of a topos) generates a free topos containing a model of that theory. Such toposes (for a consistent theory) are never Grothendieck’s.
Does that suggest a way to produce a non-Grothendieck elementary $(\infty, 1)$-topos?
I think algebraic refers to generalized algebraic theory. Yes, it does suggest that. That idea of a higher version of the free topos has been along for a long time. I guess at least since Mike’s work from 2012. However, I don’t think the details have been worked out.
Mike mentions it as an open problem in this video from the Voevodsky memorial.
Actually I believe “algebraic” there means “monadic”: elementary toposes (and logical functors) are 2-monadic over $Cat_g$, the 2-category of categories, functors, and natural isomorphisms (not all natural transformations, because some of the topos structure is contravariant in some arguments).
Yes, the syntactic category of HoTT ought to present a non-Grothendieck elementary $(\infty,1)$-topos, but I don’t think anyone has written out the details yet. To my knowledge the state of the art is Chris’s proof that it yields locally cartesian closed $(\infty,1)$-catetgories.
Thanks.
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