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- Discussion Type
- discussion topicMaterial set theory
- Category Latest Changes
- Started by TobyBartels
- Comments 24
- Last comment by Richard Williamson
- Last Active 20 minutes ago

I have reorganised set theory and spun off material set theory.

- Discussion Type
- discussion topicmodel structure on dg-Lie algebras
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 1 hour ago

stub for model structure on dg-Lie algebras

- Discussion Type
- discussion topicmodel stucture on simplicial Lie algebras
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 3 hours ago

The stub entry

*model structure on simplicial Lie algebras*used to point to*model structure on simplicial algebras*. But is it really a special case of the discussion there?Quillen 69 leaves the definition of the model structure to the reader. Is it with weak equivalences and fibrations those on the underlying simplicial sets? Is this a simplicially enriched model category?

- Discussion Type
- discussion topicLawvere theory
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by Urs
- Last Active 4 hours ago

started a Properties-section at Lawvere theory with some basic propositions.

Would be thankful if some experts looked over this.

Also added the example of the theory of sets. (A longer list of examples would be good!) And added the canonical reference.

- Discussion Type
- discussion topicrational stable homopy theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 5 hours ago

for the purposes of having direct links to it, I gave a side-remark at

$(H \mathbb{Q}) ModSpectra \;\simeq\; Ch_\bullet(\mathbb{Q})$*stable Dold-Kan correspondence*its own page: rational stable homotopy theory, recording the equivalenceI also added the claim that under this identification and that of classical rational homotopy theory then the derived version of the free-forgetful adjunction

$(dgcAlg^{\geq 2}_{\mathbb{Q}})_{/\mathbb{Q}[0]} \underoverset {\underset{U \circ ker(\epsilon_{(-)})}{\longrightarrow}} {\overset{Sym \circ cn}{\longleftarrow}} {\bot} Ch^{\bullet}(\mathbb{Q})$models the stabilization adjunction $(\Sigma^\infty \dashv \Omega^\infty)$. But I haven’t type the proof into the entry yet.

- Discussion Type
- discussion topicmodel structure on simplicial T-algebras
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active 6 hours ago

- Discussion Type
- discussion topicrational parameterized spectra
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by Urs
- Last Active 8 hours ago

Has anyone developed models for the homotopy theory of $H \mathbb{Q}$.module spectra over rational topological spaces a bit?

I expect there should be a model on the opposite category of dg-modules over rational dg-algebras. Restricted to the trivial modules it should reduce to the standard Sullivan/Quillen model of rational homotopy theory. Restricted to the dg-modules over $\mathbb{Q}$ it should reduce to the standard model for the homotopy theory of rational chain complexes, hence equivalently that of $H \mathbb{Q}$-module spectra.

Is there any work on this?

- Discussion Type
- discussion topicsimplicial Lawvere theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 8 hours ago

I gave

*simplicial Lawvere theory*an entry, stating Reedy’s result on the existence of the simplicial model structure of simplicial algebras over a simplicial Lawvere theory

- Discussion Type
- discussion topicpullback-power
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Dmitri Pavlov
- Last Active 11 hours ago

almost missed that meanwhile we have an entry

*pullback-power*. So I added more redirects and expanded a little.

- Discussion Type
- discussion topicLocally ringed topological space - added example
- Category Latest Changes
- Started by Bartek
- Comments 4
- Last comment by Bartek
- Last Active 12 hours ago

Added the example of smooth manifolds, which have a canonical fully faithful embedding into locally ringed spaces, citing Lucas Braune’s nice proof on stackexchange.

- Discussion Type
- discussion topicmodel structure on dg-coalgebras
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active 12 hours ago

am starting model structure on dg-coalgebras.

In the process I

created a stub for dg-coalgebra

and linked to it from L-infinity algebra

- Discussion Type
- discussion topicInfinity-category: added Emily Riehl's minicourse videos
- Category Latest Changes
- Started by Bartek
- Comments 1
- Last comment by Bartek
- Last Active 15 hours ago

- In the references on [Infinity-category](https://ncatlab.org/nlab/show/infinity-category), I added Emily Riehl's [lecture videos](http://hessbellwald-lab.epfl.ch/ytm2015) on infinity categories from the Young Topologists' Meeting 2015.

- Discussion Type
- discussion topiccontinuous map
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by IngoBlechschmidt
- Last Active 23 hours ago

I gave

*continuous map*a little bit of substance by giving it an actual Idea-paragraph and by writing out the epsilontic definition for the case of metric spaces, together with its equivalence to the “abstract” definition in terms of opens.

- Discussion Type
- discussion topicmodel structure on dg-algebras
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active 1 day ago

Fixed the comments in the reference list at model structure on dg-algebras: Gelfand-Manin just discuss the commutative case. The noncommutative case seems to be due to the Jardine reference. Or does anyone know an earlier one?

- Discussion Type
- discussion topicminimal fibration
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

The entry

*minimal fibration*used to be just a link-list for disambiguating the various versions. I have now given it some text in an Idea-section and a pointer to Roig 93 where the concept is considered in generality.

- Discussion Type
- discussion topicproper model category
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 2 days ago

I expanded proper model category a bit.

In particular I added statement and (simple) proof that in a left proper model category pushouts along cofibrations out of cofibrants are homotopy pushouts. This is at Proper model category -- properties

On page 9 here Clark Barwick supposedly proves the stronger statement that pushouts along all cofibrations in a left proper model category are homotopy pushouts, but for the time being I am failing to follow his proof.

(??)

- Discussion Type
- discussion topicDoctrines of algebraic geometry - links to lectures no longer work
- Category Latest Changes
- Started by Bartek
- Comments 5
- Last comment by DavidRoberts
- Last Active 2 days ago

- James Dolan gave a series of talks on algebraic geometry for category theorists at John Baez's seminar, but it seems that the links on the nLab page no longer work. Does anyone know if the videos have been uploaded elsewhere?

https://ncatlab.org/jamesdolan/published/Algebraic+Geometry

- Discussion Type
- discussion topicPr(infinity,1)Cat
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Mike Shulman
- Last Active 2 days ago

Somebody kindly pointed out by email to me two mistakes on the page Pr(infinity,1)Cat. I have fixed these now (I think).

The serious one was in the section Embedding into Cat where it said that $Pr(\infty,1)Cat \to (\infty,1)Cat$ preserves limits and colimits. But it only preserves limits. This is HTT, prop. 5.5.3.13. The wrong statement was induced from a stupid misreading of HTT, theorem. 5.5.3.18. Sorry.

The other mistake was that it said “full subcategory”. But of course by the very definition of $Pr(\infty,1)Cat$ if is not full in $(\infty,1)Cat$. I have fixed that, too, now.

- Discussion Type
- discussion topicsemisimple category
- Category Latest Changes
- Started by John Baez
- Comments 4
- Last comment by Charles Rezk
- Last Active 3 days ago

I was dissatisfied with the discussion at semisimple category because it only defined a semisimple

*monoidal Vect-enriched*category, completely ignoring the more common notion of semsimple abelian category.So, I stuck in the definition of semisimple abelian category.

However, I still think there is a lot that could be improved here: when is a semisimple abelian category which is also monoidal a semsimple monoidal category in some sense like that espoused here???

I think this article is currently a bit under the sway of Bruce Bartlett’s desire to avoid abelian categories. This could be good in some contexts, but not necessarily in all!

- Discussion Type
- discussion topicsound doctrine
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active 3 days ago

Created sound doctrine as a stub to record relevant references.

- Discussion Type
- discussion topiccoreflective subcategory - examples
- Category Latest Changes
- Started by Bartek
- Comments 2
- Last comment by Todd_Trimble
- Last Active 3 days ago

- Included Lie integration of finite-dimensional real Lie algebras as an example of a coreflective subcategory. The coreflector is Lie differentiation.

- Discussion Type
- discussion topicIsabelle
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by Daniel Luckhardt
- Last Active 5 days ago

- Discussion Type
- discussion topiclocally presentable categories - table
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active 6 days ago

I came to think that the pattern of interrelations of notions in the context of locally presentable categories deserves to be drawn out explicitly. So I started:

Currently it contains the following table, to be further fine-tuned. Comments are welcome.

| | | inclusion of left exaxt localizations | generated under colimits from small objects | | localization of free cocompletion | | generated under filtered colimits from small objects | |–|–|–|–|–|—-|–|–| |

**(0,1)-category theory**| (0,1)-toposes | $\hookrightarrow$ | algebraic lattices | $\simeq$ Porst’s theorem | subobject lattices in accessible reflective subcategories of presheaf categories | | | |**category theory**| toposes | $\hookrightarrow$ | locally presentable categories | $\simeq$ Adámek-Rosický’s theorem | accessible reflective subcategories of presheaf categories | $\hookrightarrow$ | accessible categories | |**model category theory**| model toposes | $\hookrightarrow$ | combinatorial model categories | $\simeq$ Dugger’s theorem | left Bousfield localization of global model structures on simplicial presheaves | | | |**(∞,1)-topos theory**| (∞,1)-toposes |$\hookrightarrow$ | locally presentable (∞,1)-categories | $\simeq$ <br/> Simpson’s theorem | accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories | $\hookrightarrow$ |accessible (∞,1)-categories |

- Discussion Type
- discussion topicTruth
- Category Latest Changes
- Started by TobyBartels
- Comments 12
- Last comment by TobyBartels
- Last Active 6 days ago

- Discussion Type
- discussion topicBorel's Theorem
- Category Latest Changes
- Started by TobyBartels
- Comments 12
- Last comment by TobyBartels
- Last Active 6 days ago

I wanted to understand Borel's Theorem better, so I wrote out a fairly explicit proof of the one-dimensional case.

- Discussion Type
- discussion topicrational homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active 7 days ago

motivated by the blog discussion I added to rational homotopy theory a section Differential forms on topological spaces

- Discussion Type
- discussion topicSullivan model
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 7 days ago

created Sullivan model

- Discussion Type
- discussion topicdifferential forms on simplices
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

I have expanded, streamlined and re-organized a little at

*differential forms on simplices*.

- Discussion Type
- discussion topiccurved L-infinity algebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

made

*curved L-infinity algebra*explicit

- Discussion Type
- discussion topicdouble dimensional reduction
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Feb 14th 2017

am in the process of adding some notes on how the D=5 super Yang-Mills theory on the worldvolume of the D4-brane is the double dimensional reduction of the 6d (2,0)-superconformal QFT in the M5-brane.

started a stubby

*double dimensional reduction*in this context and added some first further pointers and references to*M5-brane*, to*D=5 super Yang-Mills theory*and maybe elsewhere.But this still needs more details to be satisfactory, clearly.