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I have half an idea to choose this as a topic for a contribution to a book on category theory and philosophy. It was once a very interesting subject, one to which Kant, Riemann, Helmholtz, Lie, Poincare, and Hilbert contributed, but fell into disuse most likely with the rise of set theory as putative foundation. Perhaps now is the moment to revive it with cohesive $\infty$-toposes grounding physics!
Looking at higher geometry, how should one think about the comparison between big and little structured $\infty$-toposes? I mean would there be anything against claiming that the ’big’ approach is the far more important one? Here we can tell a nice story about adding extra structure, right up to differential cohesion and its importance for physics, tying all this in to homotopy type theory, etc. In sum, Urs’s book.
But what kind of story can be told about the importance of structured little $\infty$-toposes?
For one, every object in a differentially cohesive infinity-topos canonically induces its petit structured inginity-topos see the entry on differential cohesion
Sorry for the telegraphic and typo-ridden message above, was posted in a hurry from my phone.
The canonical construction of an $\mathbf{H}$-structured “petit” $\infty$-topos $Sh_{\mathbf{H}}(X)$ for any object $X$ of a differentially cohesive $\infty$-topos is discussed here.
Thanks, so that’s Def. 10 there. But still, what’s the big deal? It seems that the petit topos is the minor concept – just one small consequence of the gros topos juggernaut.
Yes, but that is useful. Effectively, I think of this as being the/a synthetic way of defining
internal sheaves
étale stacks
locally ringed petit toposes
internal to (differentially cohesive) homotopy type theory.
This is a big deal because all three of these are constructions that one needs in applications all along. So the fact that it comes out elegantly synthetically from the axioms of differential cohesion here makes this more interesting, not less.
Think of the other concepts that come out in a simple fashion from cohesion, such as the whole theory of differential cohomology. Traditionally this is a thorny subject. Now it all flows out just like this from the axioms of cohesion. That is not to make us say “So what’s the big deal about differential cohomology?”. On the contrary, we know that differential cohomology is a big deal, and so this makes us see that the axioms of cohesion are more powerful than their simplicity might superficially make one think.
That makes sense. Thanks.
I should come back to the original question at the end of #1 above:
I think that’s generally a great question to ask, as it aims at some pretty fundamental stuff, essentially it aims at what Grothendieck was aiming for, as best extracted in Monique Hakim’s thesis.
Effectively, there was Monique Hakim, then there was nothing for decades, and then there was DAG V.
I guess if we (the nForum regulars) join forces here, we can write a good entry that explains the fundamental relevance of locally ringed toposes and structured $\infty$-toposes. I won’t try it tonight, but let’s keep this in mind and come back to it.
Perhaps it would be good to look back at this. In general, I think the picture we came up with of the big/little relationship is that little toposes are the general “higher notion of space”, whereas if we restrict ourselves to a particular (more or less arbitrary) class of well-behaved higher spaces determined, then they can be assembled into the objects of a big topos. This should be equally true in the structured case.
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