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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• At locally contractible space David Roberts had written a conjecture relating local contractibility to local triviality of constant n-stacks. I have added to that a converse proposition.

• Added a minor alternative rendition of the definition of natural transformation. I'm not sure that it's particularly useful for anything, but it is at least true, with some aesthetically pleasing qualities.
• I thought I'd amuse myself with creating a succinct list of all the useful structures that we have canonically in an (oo,1)-topos without any intervention by hand:

• principal oo-bundles, covering oo-bundles, oo-vector-bundles, fundamental groupoid, flat cohomology, deRham cohomology, Chern character, differential cohomology.

I started typing that at structures in a gros (oo,1)-topos on my personal web.

I think this gives a quite remarkable story of pure abstract nonsense. None of this is created "by man" in a way. It all just exists.

Certainly my list needs lots of improvements. But I am too tired now. I thought I'd share this anyway now. Comments are welcome.

Main point missing in the list currently is the free loop space object, Hochschild cohomology and Domenico's proposal to define the Chern character along that route. I am still puzzled by how exactly the derived loop space should interact with $\Pi^{inf}(X)$ and $\Pi(X)$.

• I am re-reading Simpson/Teleman's "de Rham theorem for oo-stacks" and realize what I missed on first reading:

they have essentually the statement that I produced recently, that oo-stacks on a site with "geometrically contractible" objects form a localy contractible (infinity,1)-topos in that the terminal geometric morphism is essential.

They say this (slightly implicitly) at the level of homotopy cateories on page 5.

I briefly recorded this in a somewhat stubby way now at locally contracible (oo,1)-topos - Examples

• I have created a brief entry for Phil Higgins. Clearly more could be added for instance his work on using groupoids to prove group theoretic results.

• I have started an entry on the decalage functor.

• I expanded the beginning of physicscontents and edited a bit to bring out more structure.

• Created solid functor with an SVG graphic. The SVG editor is awesome! Even if it is still a little buggy.

• added to derivation at the very end in the exampls section a discussion of derivations on smooth functions (and how they are vector fields) and f derivations on continuous functions (and how they are trivial).

• Jim Stasheff edited a bit at A-infinity category -- but did not yet work in the comments that he posted to the blog

• as part of a reply in my blog discussion with Mike here I created a section

• I have been editing the entry homotopy equivalence to include a brief discussion of strong homotopy equivalences and Vogt's lemma. In so doing, I have followed my nose and found various other entries to edit. For instance that for Hans Baues, that for cylinder functor, etc. I am thinking that the general area of Henry Whitehead's idea of algebraic homotopy, may be a useful intermediate one between the infinity category ideas (which could be seen as just a 'souped up' version of Kan complexes), (I am not saying they are just that a cynic might make them out to be!) and the algebraic topologists desire to perform calculations. Note the quotes at algebraic homotopy. Of course, they d not say what 'compute' means in this context. (Note we do not have an entry on Whitehead as yet.)

• I worked on improving (hopefully) and further expanding (a bit) the discussion of geometric homotopy groups at

I give at

• Idea now just the brief reminder of the simple situation for 1-toposes as described at locally connected topos;

• and then at Defintion I state the very obvious and simple generalization of this to homotopy oo-groupoids of objects in a locally contractible (oo,1)-topos.

Then I say something like: while this definition is very obvious and simple, it seems it has not quite been stated in the literature (except possibly in the thesis by Richad Williamson), but that there are old well-known results in the literature that essentially, with only slight modification of language, already do say precisely this.

Then I go through this claim in detail. I list three subsections with three different methods of how to construct that left adjoint $\Pi(-)$ to the constant oo-stack functor, and then discuss in some detail how old and new references do already -- if slightly implicitly -- discuss precisely this. The three sections are

You'll notice that I also link to the discussioon of the absgtract oo-adjunction on my personal web. Currently I am thinking of the entry on my personal web as talking about the abstract notion of a path oo-groupoid, and of this page here on the nLab as providing all the "well-known" aspects of it (in that these are in the literature).

Please complain if somehow this doesn't look like the right thing to do. I am currently a bit undecided as to what bits of this discussion should be on my personal web and which on the nLab.

• At category with weak equivalences we say that it is unclear whether every (oo,1)-category arises as the simplicial localization of a cat with weak equivalences, but that it seems plausible.

At (infinity,1)-category we say that indeed every (oo,1)-category arises as the simplicial localization of a homotopical category.

I had put in the paragraph that says this based on a message that Andre Joyal recently posted to the CatTheory mailing list.

It would be good to harmonize this with the discussion at category with weak equivalences and maybe to add some references.

• I think I understand locally contractible (oo,1)-toposes now, with their left-adjoint to the constant oo-sheaf functor.

The (simple) observation is here on my personal web.

• (edit: typo in the headline: meant is "bare" path oo-groupoid)

I think I have the proof that when the structured path oo-groupoid of an oo-stack oo-topos exists, as I use on my pages for differential nonabelian cohomology, then its global sections/evaluation on the point yields the bare path oo-groupoid functor, left adjoint to the formation of constant oo-stacks.

A sketch of the proof is now here.

Recall that this goes along with the discussion at locally constant infinity-stack and homotopy group of an infinity-stack.

P.S.

Am in a rush, will get back to the other discussion here that are waiting for my replies a little later. Just wanted to et this here out of the way

• For the record, all I did at geometric morphism#sheaftopoi was to add a paragraph at the beginning of the example, substitute ‘sober’ for ‘Hausdorff’ in appropriate places, and add to the query box there. I mention this because the diff thinks that I did much more than that, and I don't want anybody to waste time looking for such changes!

I still to make the proof apply directly to sober spaces; the part that used that the space was Hausdorff is still in those terms.

• I fiddled a bit with direct image, but maybe didn't end up doing anything of real value...

• I have just uploaded a new 10 chapter version of the menagerie notes. It can be got at via my n-lab page then to my personal n-lab page and follow the link.