Added a reference to double profunctor

]]>Well, not always terminology happens to develop systematically. The meaning of “co-” here is certainly not to be understood as an operation that is somehow involutive up to some notion of equivalence. De Groots understanding of “co-X” seems to have been “belonging to X”, “something that goes/comes *with* X”.

Thanks, Todd.

]]>I would like to modify the “default” definition of topological concrete category to include the “evil” condition that $U(T)=X$, corresponding to the “usual” notion of Grothendieck fibration rather than the weaker one of Street fibration. I think this is usually included in definitions in the literature, and satisfied by most examples, and I don’t think there is a good reason to change it; the weaker notion can be called “weakly topological” or something. The proper way to formulate this sort of condition without equality of objects is probably using displayed categories rather than weakening the equality.

But if anyone has an objection, please raise it!

]]>I assumed $p$ was a prime, so I put that in. Also linked p-derivation from Fermat quotient.

]]>Why is it called cotopology?

]]>I created article cotopology including a redirect from cocompact space.

]]>I separated p-derivation from Fermat quotient.

]]>Thanks Max! It’s good to see this spelled-out.

]]>Hm, I didn’t say that well: That the base is not a topos need not be that bad. But what matters is that the two functors from the base are fully faithful, so that the three (co-)monads that are induced are indeed idempotent.

]]>have restated with reference to proof as suggested in #27

]]>IIRC, EKMM spectra use a different loophole than diagram spectra do. It has something to do with the unit of the EKMM monoidal structure not being cofibrant.

]]>Thanks for the alert. That’s really interesting.

]]>Re #79 and the superalgebra approach to the de Rham complex, I see that Buium uses this approach in arithmetic differential geometry. If you can see page 33 of his book, he points out that there is no de Rham calculus available, but he can then use the Lie superalgebra formalism.

]]>On the assumption that this was a mistake (or left over from an initial attempt), I renamed it.

]]>Why is the title of this page “uniform space draft” and not “uniform space”?

]]>The factorization through pointed spaces I took as evident, I can add details on that.

The case for orthogonal spectra happens to be the one for which a fully self-contained proof is spelled out on the $n$Lab. It also works for symmetric spectra, but due to the annyoing issue with the class of stable weak equivalences there not coinciding with the stable weak homotopy equivalences, I didn’t spell that out completely on the $n$Lab. Of course it is in *Model categories of diagram spectra*.

Once I knew which loophole allowed highly structured spectra to evade Lewis’ conclusion, but now I forget what it was. If we don’t already, then we should discuss this at *symmetric monoidal smash product of spectra*.

Thanks. You stated this only for orthogonal spectra and unpointed spaces. Presumably it is true for some other models of spectra and for based spaces as well? Also I don’t remember how this statement figures into Lewis’s no-go theorem about good categories of spectra and how it is avoided by the various models?

]]>One possibility of course could be to ’embrace the pro-setting’,

Yes, one can do that, but then one is no longer in an $\infty$-topos.

Same is true for the usual $\mathbb{A}^1$-local homotopy category: while it is of course a “shape-like” loclization, by construction, it is not an $\infty$-topos.

A cohesive version of algebraic geometry, in the usual $\infty$-topos theoretic sense should have some relation with $\mathbb{A}^1$-homotopy theory, but must be a little different.

There are maybe two different goals to be distinguished: The first is to put cohesive $\infty$-toposes to any usage in a context of algebraic geometry, and possibly interpret the result in terms of the general idea of motivic geometry, and the second is to explore relations specifically to $\mathbb{A}^1$-homotopy theory. I feel the former might be more interesting.

Unless I am really missing something, one is certainly getting all schemes over $k$ (with some finiteness condition that I cannot remember the name of) by glueing the $\mathbb{A}^{n}_{k}$’s using ordinary colimits when using the Zariski topology,

Could you dig out the precise statement that you are thinking of here?

]]>I added a reference for the generalisation of jet spaces to a finite set of primes.

]]>Re #30, so now we should have a version of:

The Eilenberg-Moore category of coalgebras over the Jet comonad has the interpretation of the category of partial differential equations

These coalgebras are just the images of base change along $X \to \Im(X)$, as Urs points out.

]]>Thanks, Todd. I have edited the formatting at *uniformly continuous map* in order to make it easier to spot what the entry offers.

Thanks!

I have taken the liberty or re-ordering the Idea-section, keeping the simple description at the beginning and your universal characterization afterwards.

In fact the universal characterization deserves to be (re-)stated in the Properties-section of the entry with some indication as to its proof, or at least with a reference.

]]>I have added statement of the example of stabilization at “monoidal adjunction” here and also at “monoidal Quillen adjunction” here

(I had thought that this example had long been stated there, but no)

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