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

• I am trying to write up an elementary exposition for how the Hochschild chain complex for a commutative associate algebra is the normalized chains/Moore complex of the simplicial algebra that one gets by tensoring the algebra $A$ with the simplicial set $\Delta[1]/\partial \Delta[1]$:

$C_\bullet(A,A) = N_\bullet( (\Delta[1]/\partial \Delta[1]) \cdot A ) \,.$

I would like to get feedback on whether or not my exposition is in fact understandable in an elementary way.

The section that contains this material is the section

The simplicial circle algebra

at the entry Hochschild cohomology. Just this one section. It’s not long.

It describes first the simplicial set $\Delta[1]/\partial \Delta[1]$, then discusses how the coproduct in $CAlg_k$ is given by the tensor product over $k$, and deduces from that what the simplicial algebra $(\Delta[1]/\partial \Delta[1])$ is like.

After taking the normalized chains of that, the result is Pirashvili’s construction of a chain complex from a simplicial set and a commutative algebra. I just think it is important to amplify that this construction of Pirashvili’s is a categorical tensoring=copower operation. Because that connects the construction to general abstract constructions. That’s what the beginning of the above entry is about. But for the moment I would just like to make the elementary exposition of the tensoring operation itself pretty and understandable.

• tried to bring the old neglected entry sSet-category roughly into some kind of stubby shape. Added Porter-Cordier and LurieA.3 as references. The former was my motivation for doing this. Eventually it would be good to have here a detailed discussion of $sSet$-category models for $(\infty,1)$-category theory. See the discussion with Tim over in the other thread on the $(\infty,1)$-Yoneda lemma.

(I don’t have time for this now. I am saying all this in the hope that somebody else has.)

• I’ve cleaned up diffeological space a little. In particular:

1. I’ve removed all references to Chen spaces. There is a relationship, but not what was implied on that page.
2. I’ve tried to clean up the distinction between the definition in the literature (which uses all open subsets of Euclidean spaces) and the preferred nLab definition (which uses CartSp).
3. Other minor cleaning.
• I have been advising Herman Stel on his master thesis, which is due out in a few days. I thought it would be nice to have an nLab entry on the topic of the thesis, and so I started one: function algebras on infinity-stacks.

For $T$ any abelian Lawvere theory, we establish a simplicial Quillen adjunction between model category structures on cosimplicial $T$-algebras and on simplicial presheaves over duals of $T$-algebras. We find mild general conditions under which this descends to the local model structure that models $\infty$-stacks over duals of $T$-algebras. In these cases the Quillen adjunction models small objects relative to a choice of a small full subcategory $C \subset T Alg^{op}$ of the localization

$\mathbf{L} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} \mathbf{H} = Sh_{(\infty,1)}(C )$

of the $(\infty,1)$-topos of $(\infty,1)$-sheaves over duals of $T$-algebras at those morphisms that induce isomorphisms in cohomology with coefficients the canonical $T$-line object. In as far as objects of $\mathbf{H}$ have the interpretation of ∞-Lie groupoids the objects of $\mathbf{L}$ have the interpretatin of ∞-Lie algebroids.

For the special case where $T$ is the theory of ordinary commutative algebras this reproduces the situation of (Toën) and many statements are straightforward generalizations from that situation. For the case that $T$ is the theory of smooth algebras ($C^\infty$-rings) we obtain a refinement of this to the context of synthetic differential geometry.

As an application, we show how Anders Kock’s simplicial model for synthetic combinatorial differential forms finds a natural interpretation as the differentiable $\infty$-stack of infinitesimal paths of a manifold. This construction is an $\infty$-categorical and synthetic differential resolution of the de Rham space functor introduced by Grothendieck for the cohomological description of flat connections. We observe that also the construction of the $\infty$-stack of modules lifts to the synthetic differential setup and thus obtain a notion of synthetic $\infty$-vector bundles with flat connection.

The entry is of course as yet incomplete, as you will see.

• I created split coequalizer and absolute coequalizer, the latter including a characterization of all absolute coequalizers via an “$n$-ary splitting.” While I was doing this, I noticed that monadic adjunction included a statement of the monadicity theorem without a link to the corresponding page, so I added one. (The discussion at the bottom of monadic adjunction should probably be merged into the page somehow.) Then I noticed that while we had a page preserved limit, we didn’t have reflected limit or created limit, so I created them. They could use some examples, however.

I would also like to include an example of how to actually use the monadicity theorem to prove that a functor is monadic. Something simpler than the classic example in CWM about compact Hausdorff spaces; maybe monadicity of categories over quivers? Probably not something that you would need the monadicity theorem for in practice, so that it can be simple and easy to understand.

• Under the Definitions at topos, I gave definitions of Grothendieck toposes and W-toposes, since these are two very important kinds of toposes that some authors (at least) often call simply ‘toposes’. (Also it gave me a place to redirect W-topos and its synonym topos with NNO.)

• On Tim’s suggestion, I have been in contact with Ronnie Brown on some of the history behind “convenient categories”, and received a wealth of information from him. I have made an initial attempt to summarize what I have learned in the Historical Remarks section of convenient category of topological spaces, but it might be somewhat garbled still. Hopefully Ronnie and/or Tim will have a look. I will be adding more references by and by.

I also added in the follow-up discussion with Callot under Counterexamples.

• Francois Métayer already has a lab-entry. (NB the acute accent is missing on the new one! That was the cause of the error.)

• I added some more details to the section on ultrafilter monad at ultrafilter. Incidentally, it seems to me that the bit on Barr’s observation (“topological spaces = relational $\beta$-modules”) is too terse. There is a lot of generalized topology via abstract nonsense that deserves more explanation.

I also added some elementary material to cartesian closed category, mainly to indicate to the novice how exponentials deserve to be thought of as function spaces, how internal composition works, and so forth. I left the job somewhat unfinished.

• Krzysztof Worytkiewicz kindly informs me that an old question to Francois Metayer has now been answered: the folk model structure on strict $\omega$-categories does restrict to the Brown-Golasinski model structure on strict $\omega$-groupoids. (The latter is indeed the transferred model structure along the forgetful functor to the former).

This is now written up in

Ara, Métayer, The Brown-Golasinki model structure on oo-groupoids revisited (pdf)

As my connection allows, I will insert this into the nLab entry now…

• created 1-sentence entry conical limit (so satisfy requested links from four other entries)

• New stub hom-connection. I should figure it out once. While tensor product is involved in many constructions in algebra, some are dual with Hom instead, for example there are contramodules in addition to comodules over a coring. In similar vain hom-connections were devised, but there are some really intriguing examples (including superconnections, right connections of Manin etc.) and there are relations to examples of noncommutative integration of various kind.

• I have added some discussion to the page on orientals (in the sense of Ross Street), regarding the link to the convex geometry of cyclic polytopes (as discussed by Kapranov and Voevodsky).

My selfish motive for doing so is that I am curious if my recent work with Steffen Oppermann which includes a new description of the triangulations of (even-dimensional) cyclic polytopes, has any relevance to the study of orientals, or higher category theory more broadly. (In particular, if there are explicit questions about the internal structure of orientals which are of interest, I would like to hear about them.)

A particularly speculative version of my question, would be whether there is a natural connection between orientals and the representation theory which we are studying in that paper (which necessitated a detour into convex geometry). We biject triangulations of an even-dimensional cyclic polytope to (a nice class of) tilting objects for a certain algebra. The simplest version of this (which was already known) is that triangulations of an $n$-gon correspond to tilting objects for the path algebra of the quiver consisting of a directed path with $n-2$ vertices. (These tilting objects then give derived equivalences between the derived category of this path algebra, and the derived category of the endomorphism ring of this tilting object.)

Questions, speculations, or suggestions would be very welcome.

Hugh

• to the functional analysis crew of the $n$Lab: where should operator spectrum point to? Do we have any suitable entry?

• New stubs Oka principle, Oka manifold (with redirect Oka map) and Franc Forstnerič. Jardine has shown that one can use the Toen-Vezzosi like engineering with his intermediate model structure on the category of simplicial presheaves on a simplicial version of the Stein site. The $(\infty,1)$-stacks/fibrants will be Oka maps; those cofibrants which are represented by complex manifolds are in fact Stein manifolds.

• I expanded some entries related to the Café-discussion:

• at over-(infinity,1)-topos I expanded the Idea-section, added a few remarks on proofs and polished a bit,

and added the equivalence $\infty Grpd/X \simeq PSh_{\infty}(X)$ to the Examples-section

• at base change geometric morphism I restructured the entry a little and then included the proof of the existence of the base change geometric morphism

• added to adjunct the description in terms of units and counits.

• created (infinity,1)-algebraic theory.

I tried to adapt Rosicky’s and Lurie’s terminology such as to match that at algebraic theory, but Mike, Toby, Todd and whoever else feels expert should please check if I did it right.

• New reference entry FAC aka Faisceaux algébriques cohérents. And few improvements to coherent sheaf including historical note.

$Top/X \simeq Sh_{(\infty,1)}(X)$

here

• Kevin Walker was so kind to add a bit of material to blob homology. Notably he added a link to a set of notes now available that has more details.

• I added to loop space a reference to Jim’s classic article, which was only linked to from H-space and put pointers indicating that his delooping result in $Top$ is a special case of a general statement in any $\infty$-topos.

By the way: it seems we have slight collision of terminology convention here: at “loop space” it says that H-spaces are homotopy associative, but at “H-space” only a homotopy-unital binary composition is required, no associativity. I think this is the standard use. I’d think we need to modify the wording at loop space a little.

• I reworked A-infinity algebra so as to apply to algebras over any $A_\infty$-operad in any ambient category. So I created subsections “In chain complexes”, “In topological spaces”.

I think if we speak generally of “algebra over an operad” then we should also speak generally of “$A_\infty$-algebra” even if the enriching category is not chain complexes. Otherwise it will become a mess. But I did link to A-infinity space.

(by the way, should we not rather call these “pedestrian definition” or so instead of “rough”? The latter seems to suggest that there is something not quite working yet with these definitions, while in fact they are perfectly fine, just not as high-brow as other definitions.)

• 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 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.