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- Discussion Type
- discussion topicexposition of Hochschild chains from simplical tensoring
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Dec 2nd 2010

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

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.

- Discussion Type
- discussion topicbicategory of maps
- Category Latest Changes
- Started by FinnLawler
- Comments 5
- Last comment by FinnLawler
- Last Active Dec 1st 2010

New page: bicategory of maps.

- Discussion Type
- discussion topicsSet-category
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Nov 30th 2010

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.)

- Discussion Type
- discussion topicDiffeological spaces
- Category Latest Changes
- Started by Andrew Stacey
- Comments 1
- Last comment by Andrew Stacey
- Last Active Nov 29th 2010

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

- I’ve removed all references to Chen spaces. There is a relationship, but not what was implied on that page.
- 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).
- Other minor cleaning.

- Discussion Type
- discussion topicfunction algebras on oo-stacks
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Nov 29th 2010

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.

- Discussion Type
- discussion topiclimits and monadicity
- Category Latest Changes
- Started by Mike Shulman
- Comments 12
- Last comment by Yaron
- Last Active Nov 29th 2010

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.

- Discussion Type
- discussion topicTypes of toposes
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Nov 29th 2010

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.)

- Discussion Type
- discussion topicconvenient category of topological spaces
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active Nov 28th 2010

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.

- Discussion Type
- discussion topicFrancois Métayer
- Category Latest Changes
- Started by Tim_Porter
- Comments 5
- Last comment by TobyBartels
- Last Active Nov 28th 2010

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

- Discussion Type
- discussion topicultrafilter monad, cartesian closed category
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active Nov 27th 2010

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.

- Discussion Type
- discussion topicmodel structure on strict omega-categories
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 26th 2010

split off model structure on strict omega-categories from the page canonical model structure (that page could do with some cleaning-up)

- Discussion Type
- discussion topicmodel structure on strict omega-groupoids from that on strict omega-categories
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Nov 26th 2010

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…

- Discussion Type
- discussion topicframed little n-disk operad
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 26th 2010

created framed little n-disk operad and added to BV-algebra its relation to higher BV-algebras

- Discussion Type
- discussion topicPoisson algebra
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by TobyBartels
- Last Active Nov 26th 2010

stub for Poisson algebra

- Discussion Type
- discussion topicmultivector field
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by TobyBartels
- Last Active Nov 26th 2010

stub for multivector field

- Discussion Type
- discussion topic(oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Nov 25th 2010

started adding to (infinity,1)-topos a section on the (oo,1)-category of (oo,1)-toposes.

- Discussion Type
- discussion topicconical limit
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 25th 2010

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

- Discussion Type
- discussion topichom-connection
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Nov 25th 2010

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.

- Discussion Type
- discussion topichomotopy equivalence of toposes
- Category Latest Changes
- Started by Mike Shulman
- Comments 62
- Last comment by Tim_Porter
- Last Active Nov 25th 2010

Created homotopy equivalence of toposes.

- Discussion Type
- discussion topicGerstenhaber algebra
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Nov 25th 2010

created stub for Gerstenhaber algebra

- Discussion Type
- discussion topicEn-algebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 25th 2010

stub for En-algebra

- Discussion Type
- discussion topicBV-algebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 25th 2010

stub for BV-algebra

- Discussion Type
- discussion topicspectral theory
- Category Latest Changes
- Started by zskoda
- Comments 11
- Last comment by zskoda
- Last Active Nov 24th 2010

I created a stub for spectral theory. It is related to the deficiency of the entries in functional analysis.

- Discussion Type
- discussion topicorientals, cyclic polytopes, representation theory
- Category Latest Changes
- Started by Hugh_Thomas
- Comments 4
- Last comment by Urs
- Last Active Nov 23rd 2010

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

- Discussion Type
- discussion topicoperator spectrum
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Nov 23rd 2010

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

- Discussion Type
- discussion topichyper-derived functor
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 23rd 2010

created hyper-derived functor

- Discussion Type
- discussion topicacyclic object
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 23rd 2010

a little acyclic object

- Discussion Type
- discussion topicOka
- Category Latest Changes
- Started by zskoda
- Comments 11
- Last comment by zskoda
- Last Active Nov 22nd 2010

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.

- Discussion Type
- discussion topicfuture cone
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by Tim_Porter
- Last Active Nov 22nd 2010

created a stub future cone. This was in the context of directed homotopy theory, but clearly could be developed and linked to other topics.

- Discussion Type
- discussion topicChoice functions
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Nov 19th 2010

New: choice function

- Discussion Type
- discussion topicVopenka's principle
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Mike Shulman
- Last Active Nov 19th 2010

expanded Vopenka’s principle

- Discussion Type
- discussion topicmodel structure on monoids in a monoidal model category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Nov 18th 2010

stub for model structure on monoids in a monoidal model category, for the moment only to host Schwede-Shipley’s lemma on homotopy pushouts along free monoid morphisms

- Discussion Type
- discussion topicmonoid axiom in a monoidal model category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 18th 2010

- Discussion Type
- discussion topictensor power
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 18th 2010

tried to brush-up tensor power a little

- Discussion Type
- discussion topicmodel structure on algebras over a monad
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 18th 2010

- Discussion Type
- discussion topicmodel structure on algebras over an operad
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Nov 17th 2010

am working on model structure on algebras over an operad.

- Discussion Type
- discussion topicover-topos and base change
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Nov 17th 2010

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

- Discussion Type
- discussion topiclimits in over-(oo,1)-categories
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 17th 2010

wrote out the proof of the expected statement at limits in over-(oo,1)-categories

- Discussion Type
- discussion topicadjunct
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 17th 2010

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

- Discussion Type
- discussion topic(oo,1)-algebraic theory
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Nov 17th 2010

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.

- Discussion Type
- discussion topicFAC
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Nov 15th 2010

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

- Discussion Type
- discussion topicoo-sheaves over paracompact spaces
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 15th 2010

- Discussion Type
- discussion topicPr(oo,1)Cat
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 15th 2010

added some propositions to Pr(infinity,1)Cat that support the analogy to linear algebra, as described there and at integral transforms on sheaves

- Discussion Type
- discussion topicmodel structure on operator algebras
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by Tim_Porter
- Last Active Nov 15th 2010

- Discussion Type
- discussion topicNew: ordered field
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Nov 15th 2010

I satisfied a few requests for ordered field. Pretty basic.

- Discussion Type
- discussion topicblob homology
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Tim_Porter
- Last Active Nov 12th 2010

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 formatting and some hyperlinks.

- Discussion Type
- discussion topicBoardman-Vogt resolution
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Nov 12th 2010

stub for Boardman-Vogt resolution

- Discussion Type
- discussion topicE-oo algebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 11th 2010

edited E-infinity algebra a little. Still all very stubby

- Discussion Type
- discussion topicloop spaces and H-spaces
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Nov 11th 2010

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.

- Discussion Type
- discussion topicA-oo algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Nov 11th 2010

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.

- Discussion Type
- discussion topiccoloured operad
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Nov 10th 2010

added the definition of “coloured operad” to operad in the section “Rough definition”

(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.)

- Discussion Type
- discussion topicmodel structure on modules over an algebra over an operad
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 10th 2010

created stub for model structure on modules over an algebra over an operad

- Discussion Type
- discussion topicmodule over an algebra over an operad
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 10th 2010

created stub for module over an algebra over an operad

- Discussion Type
- discussion topicresolutions of operads
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 10th 2010

started a section on cofibrant resolutions at model structure on operads. But incomplete for the time being.

- Discussion Type
- discussion topicmonoidal model structure on G-objects
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 10th 2010

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

- Discussion Type
- discussion topicendomorphism ring
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 10th 2010

stub for endomorphism ring

- Discussion Type
- discussion topichigher geometry <- Isbell duality -> higher algebra
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Nov 9th 2010

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*)

- Discussion Type
- discussion topicindexed functor
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by FinnLawler
- Last Active Nov 9th 2010

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

- Discussion Type
- discussion topicoo-algebra of an (oo,1)-operad
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Nov 8th 2010

in preparation of the next session of our Seminar on derived differential geometry I am starting

- Discussion Type
- discussion topicAlgebraic theories, sketches etc.
- Category Latest Changes
- Started by Tim_Porter
- Comments 5
- Last comment by Tim_Porter
- Last Active Nov 8th 2010

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.