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In “Axiomatic cohesion”, Lawvere claims that the the axiom “pieces of powers are powers of pieces” (which he calls “continuity”, although I don’t understand why) “holds if the contrast with S is determined as in IV below by an infinitely divisible interval in E”. Looking at his very sketchy IV, it seems that what he means is that the discrete objects are the ones $Y$ for which $Y\to Y^T$ is an isomorphism for some object $T$, and I guess $T$ is his “infinitely divisible interval”. I don’t know exactly what he means by “infinitely divisible”, but it seems that $\mathbb{R}$ ought to be “infinitely divisible”, so if his claim is true, then maybe this axiom holds automatically in real-cohesion, which would be nice. (“Pieces have points” also holds automatically in real-cohesion, essentially because $\mathbb{R}$ has a point.)
But is there a proof of this claim written out anywhere? Or is it easy to see?
This is the subject of a recent paper of Menni – see §8.
Ah, thanks!
Menni’s paper is indeed the most outspoken source on what Lawvere is after with this axiom. I think the only use of it in the ’axiomatic cohesion’ is in the construction of the canonical homotopy quality type in theorem 1. So it is a kind of way of throwing Sset in the bin and as Menni shows of ruling out presheaf toposes in general with the idea that what remains are the cats of continuous ’non-combinatorial’ spaces. I must confess though that it escapes me why the finite set valued reflexive graph toposes which satisfy the axiom would be more continuous and less combinatorial than Sset.
Concerning, the ’interval’ I think the claim here is that the construction of IV when applied to infinitesimal $T$ is supposed to satisfy the axiom though I am not 100% sure. You may want to check what he says on pp.10-12 of the Como lectures 2008 on the ’reals’. Otherwise the best source for his ideas on T, the algebra of time, is Lawvere’s paper ’Euler’s continuum functorially vindicated’ in the Bell festschrift (2011), for the construction of $X^T$ a look in the 2002 JPAA ’Categorical algebra for continuum micro physics’ might be worthwhile too. Well, my quick guess here is that , ’infinitely divisible’ refers to the construction of his abstract time algebra $T$ using an infinitesimal object (this should get clearest by having a look at ’Euler’s continuum’). [a less quick guess, is that the ’infinity’ involved is actually the pointedness of the object, so that Lawvere’s claim is as you say that the internal foundations on the T-discrete $X$ satisfy the axiom]
[There is also a construction in SDG in Lawvere’s ’laws of motions’ with an independent proof by Kock-Reyes in a late 90s tac paper which constructs a topos of laws of motion that might be related to this idea to generate a smooth topos infinitesimally. - this is probably less relevant]
Menni pursued some of Lawvere’s ideas on these abstract reals at CT15 but for the time being there is only his abstract available at the conference page.
As far as I can see right now, it seems that Menni’s paper only applies to intervals in a site, rather than abstract interval objects in a topos; is that right?
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