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

• 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 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!

• 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.

• I have separated Eilenberg-Watts theorem from abelian category and added the references and MR links. One of the queries from the abelian category is moved here with backpointer there. I cleaned up some typoi.

The following discussion is about to which extent abelian categories are a general context for homological algebra.

Zoran: I strongly disagree with the first sentence, particularly with THE (it is THE general context for linear algebra and homological algebra). MacLane was (according to Janelidze) looking whole life for what is the general context for homological algebra, and the current answer of expert are semi-abelian categories of Borceux and Janelidze, and homological categories…Linear algebra as well makes sense in many other contexts. This “idea’ is to me very misleading. MacLane in 1950 was lead by the idea to axiomatize the categories which behave like abelian groups. Grothendieck wanted to unify on the obsrervation that the categories of abelian sheaves and categories of R-modules have the same setup for homological algebra as in Tohoku.

There is much linear algebra you can do with cokernels, for example, as well as much linear algebra which you can not do if you are not over a field for example. So, saying that abelian categories are distinguished is only among categories which have closest properties to abelian sheaves and R-modules, not among principles for homlogical algebra and linear algebra that uniquely (although the strong motivation was ever there).

Mike: I changed it to “a” general context; is that satisfactory? Once we have pages about those other notions, there can be links from here to there.

Toby: I've made the phrasing even weaker. Abelian categories are pretty cool, but (if you don't already have the examples that make it so useful) the definition is a fairly arbitrary place to draw the line.

Tim : I note that sometimes we (collectively) take parts of a discussion and turn it into part of an entry, because of that I would like to note two points here. The first is that the accepted first definition of semi-abelian category is in the Janelidze, Marki, and Tholen (JPAA, but we have a link on the semi-abelian entry.)

The other point is that Tim Van der Linden’s thesis does a lot of stuff that could be useful. It is available online http://arxiv.org/abs/math/0607100

• I tried to brush-up Warsaw circle and in the process created a bunch of simple stubs:

• At flat functor there is a statement that a functor $F : C \to \mathcal{E}$ on a complete category to a cocomplete category is left exact precisely if its Yoneda extension is.

I know this for $\mathcal{E} = Set$. There must be some extra conditions on $\mathcal{E}$ that have to be mentioned here.

• I have started a page on knot groups. So far I have outlined how to get the Dehn presentation.

• I have started a page for epistemic logic and one on $K_{(m)}$, the most basic form of epistemic logic.

• I tried to brush-up the entry fundamental infinity-groupoid which had been in a rather sorry state. Some things I did:

• stated the definition (!) $Sing X$

• removed Ronnie’s remark that there is a problem with this definition due to lack of chosen fillers and instead added Thomas Nikolaus’s theorem that when you choose fillers to get an algebraic Kan complex $\Pi X$ there is (still) a direct proof of the homotopy hypothesis

• made the statement that $Sing X$ is equivalently computed by the abstract $\infty$-topos-theoretic definition of fundamental $\infty$-groupoid a formal proposition.

• Created progroup, with remarks about the equivalence between surjective progroups and prodiscrete localic groups.

Why do we have separate pages profinite space and Stone space which do nothing but point to each other? Is there any reason not to merge them?

• I started at cohesive (infinity,1)-topos a section van Kampen theorem

In the cohesive $\infty$-topos itself the theorem holds trivially. The interesting part is, I think, to which extent it restricts to the concrete cohesive objects under the embedding $Conc(\mathbf{H}) \hookrightarrow \mathbf{H}$.

• I have added an entry on Yde Venema who is active in Coalgebras etc. in Modal Logic.

He has looked at arrow logics that I would be interested in others views on as they may be useful. They seem to be related to a from of category in which composition is a relation. (but I have not read his crash course on them in detail yet.)

• I’ve had a first pass at some (mostly minor) tidying up of unbounded operator: some reformatting, some editing of the English. More to come. There is a a lot of useful material in that article and it would be great to have some more dedicated articles on spectral theory. (Note to self, perhaps.)

• I added to Hochschild homology in the section Function algebra on derived loop space a statement and proof of the theorem that “the function complex $C(\mathcal{L}X)$ on the derived loop space $C(\mathcal{L}C)$ is the Hochschild homology complex of $C(X)$”.

There is a curious aspect to this: we are to compute the corresponding pushout in $\infty$-algebras. But in the literature on Hochschild homology, the pushout is of course taken not in algebras, but in modules

$HH_k(C(X)) := Tor_k(C(X), C(X))_{C(X) \otimes C(X)} \,.$

So how is that $HH$-complex actually an derived algebra?

The solution of this little conundrum is remarkably trivial using Badzioch-Berger-Lurie’s result on homotopy T-algebras .This tells us that we may model the $\infty$-algebras as simplicial copresheaves on our syntactic category $T$, using the left Bousfield localizatoin of the injective model structure at maps that enforce the algebra property.

But since we are computing a pushout and since the traditional bar complex provides a cofibrant resolution of our pushout diagram already in the unlocalized structure, and since left Bousfield localization does not affect the cofibrations, due to all these reasons we may (or actually: have to) compute the pushout of $\infty$-algebras as just a pushout in simplicial copresheaves.

In particular it follows that the pushout of our product-preserving coproseheaves is not actually product-presrving itself. Instread, it is (the simplicial set underlying) the standard Hochschild complex. So everything comes together. We know that if we wanted to find the actual $\infty$-algebra structure on this, we’d have to form the fibrant replacement in the localized model structure. That would make a bit of machinery kick in and actually produce the $\infty$-algebra structure on the Hochschild complex for us.

But if we don’t feel like doing that, we don’t have to. The homotopy groups of our simplicial copresheaf won’t change by that replacement.

• I split off ∞-connected (∞,1)-topos from locally ∞-connected (∞,1)-topos and added a proof that a locally ∞-connected (∞,1)-topos is ∞-connected iff the left adjoint $\Pi$ preserves the terminal object, just as in the 1-categorical case. I also added a related remark to shape of an (∞,1)-topos saying that when H is locally ∞-connected, its shape is represented by $\Pi(*)$.

I hope that these are correct, but it would be helpful if someone with a little more $\infty$-categorical confidence could make sure I’m not assuming something that doesn’t carry over from the 1-categorical world.

• Mike,

I have expanded your discussion of the sheaf topos on a locally connected site at locally connected site. Please check if you can live with what I did.

• added to cohesive site an example in the section Examples – families of sets. It is intentionally simplistic. And depending on which axioms we settle on, it is a counter-example. But maybe still of some use.

• I presume that

Definition

Let $C$ be a category with pullbacks. Then the tangent category $T_C$ of $C$ is the category whose

• objects are pairs $(A,\mathcal{A})$ with $A \in Ob(C)$ and with $\mathcal{A}$ an abelian group object in the overcategory $C/B$;

contained a minor typo, so I replaced $C/B$ with $C/A$.

• added a stubby section on free operads (free on a "collection") to operad, but a bit example-less at the moment. Have to run...

• Added another characterisation at connected category, this time in terms of the localisation at all arrows.

• Please check the statement of Reidemeister’s theorem at Reidemeister moves, I was not that happy with the precise wording of the previous version as it made everything look as if it was happening in the plane, rather than indicating that what was happening in the plane mirrored what was happening in 3-dimensions. (Note that there was a discussion on MO, [here], on the proof.)

• I am being asked for a list of references on the little disks operad, their action on higher categories, their higher traces, higher centers, etc.

So I went and improved the entry little k-cubes operad a little. Copied over some theorems, and then created/expanded the list of references.

If you have a favorite reference not yet listed there, this would be a good chance to list it, as I wil now point the people who asked me to this list

• This area is linked to cubical sets and I just came on a recent paper by Glynn Winskel and Sam Staton, that may be of interest as it links several of the models for concurrency with presheaves. The paper is here.

(Edit: I have also linked to another paper by Winskel, Events, causality, and symmetry, (online version), from 2009. This may be useful for various aspects of the Physics-Theoretical Computer Science/Logic interface. It is well written and reasonably chatty.)

• in the course of last night’s events, I created a handful of stubs for some basic concepts: