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

• 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.

• for the purposes of having direct links to it, I gave a side-remark at stable Dold-Kan correspondence its own page: rational stable homotopy theory, recording the equivalence

$(H \mathbb{Q}) ModSpectra \;\simeq\; Ch_\bullet(\mathbb{Q})$

I 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.

• 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?

• 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

• almost missed that meanwhile we have an entry pullback-power. So I added more redirects and expanded a little.

• 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.
• 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.

• 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?

• 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.

• 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.

(??)

• 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
• 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.

• 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!

• Included Lie integration of finite-dimensional real Lie algebras as an example of a coreflective subcategory. The coreflector is Lie differentiation.
• I added linear logic and type theory (homotopy type theory was already there) to true, which I renamed to truth to make it a noun (although something like true proposition, which I made a redirect, could also work). I then edited false (now falsehood) to include everything in truth.

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