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

• I started an idea section at transgression, but it could probably use some going over by an expert. I hope I didn’t mess things up too badly. I was reading Urs’ note on “integration without integration” on the train ride home and fooled myself into thinking I understood something.

By the way, this reminded me of a discussion we had a while back

Integrals: Loops space vs target space

• added very briefly the monoidal model structure on $G$-objects in a monoidal model category to monoidal model category (deserves expansion)

• there is a span of concepts

higher geometry $\leftarrow$ Isbell duality $\to$ higher algebra

which is a pretty fundamental thing about math, I think (well, this observation is at least to Lawvere, of course).

I put this span of links at the top of these three entries. I am enjoying that, but let me know if it is once again a silly idea of mine.

(maybe it should also be higher Isbell duality )

• I created Lawvere distribution. I decided to formulate it directly in the $(\infty,1)$-topos setup, because

1. there the analogy with distributions makes (even) better sense, as we can invoke $\infty$-groupoid cardinality to think of (tame) $(\infty,1)$-sheaves as $\mathbb{R}$-valued functions;

2. it reproduces then a special case of the discussion at Pr(∞,1)Cat and harmonizes with the interpretation in terms of $\infty$-vector spaces as described there.

• I just noticed that aparently last week Adam created indexed functor and has a question there

• Someone should improve this article so that it gives a definition of ‘algebraic theory’ before considering special cases such as ‘commutative algebraic theory’.

Thus is the current end to the entry on algebraic theory and I agree. Further I needed FP theory or FP sketch for something so looked at sketch. That looks as if it needs a bit of TLC as well, well not this afternoon as I have some other things that need doing. I did add the link to Barr and Wells, to sketch, however as this is now freely available as a TAC reprint.

• copy-and-pasted from MO some properties of homotopy groups of simplicial rings into simplicial ring (since Harry will probably forget to do it himself ;-)

• I made sure the following list of entries exists and interlinked everything. Some entries are still stubs that need to be filled with content. Am working on it.

• I’ve started cleaning up and adding stuff at fibration in a 2-category, but it’s bedtime now, so I’ll finish it tomorrow.

• I got annoyed with the fact that these links did not exist, and so I created now stubs for them:

To the latter entry I moved the references on $(\infty,1)$-topos theory that had been lsited at higher topos theory.

• I have added to universal covering space a discussion of the “fiber of $X\to \Pi_1(X)$” definition in terms of little toposes rather than big ones.

I find this definition of the universal cover extremely appealing. It seems that this sort of thing must have been on the tip of Grothendieck’s tongue, and likewise of all the other people who have studied fundamental groups and groupoids of a topos, but it all becomes so much clearer (I think) when you state it in the language of higher toposes. In this case, merely (2,1)-toposes are enough, so no one can argue that the categorical technology wasn’t there – so why didn’t people see this way of stating it until recently? Or did they?

• I made the following obvious fact more manifest in the respective $n$Lab entries:

a pregeometry (for structured (infinity,1)-toposes) $\mathcal{T}$ is a special case of a (multi-sorted) (infinity,1)-algbraic theory.

A structure $\infty$-sheaf

$\mathcal{O} : \mathcal{T} \to \mathcal{X}$

on $\mathcal{X}$ is an $\infty$-algebra over this $\infty$-algebraic theory in $\mathcal{X}$. The extra conditions on it ensure that it indeed looks like a sheaf of function algebras .

(I added a respective remark to the discussion of pre-geometries and added an Example-sectoin with this to the entry of oo-alghebraic theories.)

• I have created a stub for primary homotopy operation. At present it just refers to Whitehead products and composition operations and redirects attention to those entries and to Pi-algebras, which will be next on my list to be created. I do not have access to G. W. Whitehead’s book on homotopy theory so have not given a precise definition nor a discussion of what these are, although the entry on $\Pi$-algebras will to some extent cure that. If anyone knows the definition well or has Whitehead’s book, can they provide the details…. otherwise it will remain a stub. :-(

• The page join of simplicial sets is requesting a page titled “Jack Duskin”. We do have a page titled John Duskin. It that supposed to coincide?

In any case, if anyone who created that unsatisfied link to “Jack Duskin” at join of simplicial sets (also one to van Osdol) could do something such as to satisfy the links, that would be nice.

• started stub for operadic Dold-Kan correspondence (for simplicial- vs dg-algebras over operads)

with Birgit Richer’s article we’d also have a notion of “monadic DK correspondence” (for simplicial vs dg-algebras over monads)

does anyone know any direct considerations of “T-algebraic DK-correspondence” (for simplicial vs dg-algebras over a Lawvere theory)?

of course this is to some extent implied by the previous versions. But it would be good to have a direct description.

• started entries

But details still need to be filled in.

• I am in the process of reproducing the proof of the main theorem in Schwede-Shipley’s “Equivalence of monoidal model categories” at monoidal Quillen adjunction (see the references and pointers given there).

I find that there are some intermediate steps that need to be filled in and which require a tad more thinking than just copying what they write.

This mainly concerns some pure category-theoretic arguments about adjunctions, which is entirely independent of the model category theoretic argument that is later built on it. I am saying this in case you are an expert eager to help on some pure category theory issues but maybe not so much into model category theory.

I think I can figure things out myself eventually, but since I am a bit time pressured and since working toghether is fun anyway, I thought I’d just highlight here what I am doing and where there is still things remaining to be done.

So I am working on the section Lift to Quillen adjunction on monoids. This breaks up the Schwede-Shipley argument into a bunch of small lemmas and propositions and aims to write out the proofs. Partly this is spelled out. Whenever there is a gap in the argument that still needs to be written up or even figured out, I put ellipses

  (...)


for the moment. I’ll be working now on filling these ellipses with content, so where exactly you see them may change over time. But if you feel you can easily help fill some of them, you are kindly invited to do so!

• the book Monoidal Functors, Species and Hopf Algebras is very good, but still being written. Clearly the current link under which it is found on the web is not going to be the permanent link. So I thought it is a bad idea to link to it directly. Instead I created that page now which we can reference then from nLab entries. When the pdf link changes, we only need to adapt it at that single page.

• added to oplax monoidal functor the statement how an oplax monoidal structure is induced on a functor from a lax monoidal structure on a right adjoint.

• After getting myself confused about the distinction between the various notions of basis in infinite dimensions, I wrote up my attempt to disentangle myself at basis in functional analysis (also redirects from Hamel basis, topological basis, and Schauder basis. Hmm, now I think about it, maybe “topological basis” is too close to “basis of a topology”). I may still be confused about stuff, of course.