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

• 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.

(??)

• Added some criteria for spatiality.

• starting something – not done yet

• Many additions and changes to Leibniz algebra. The purpose is to outline that the (co)homology and abelian and even nonabelian extensions of Leibniz algebras follow the same pattern as Lie algebras. One of the historical motivations was that the Lie algebra homology of matrices which lead Tsygan to the discovery of the (the parallel discovery by Connes was just a stroke of genius without an apparent calculational need) cyclic homology. Now, if one does the Leibniz homology instead then one is supposedly lead the same way toward the Leibniz homology (for me there are other motivations for Leibniz algebras, including the business of double derivations relevant for the study of integrable systems).

Matija and I have a proposal how to proceed toward candidates for Leibniz groups, that is an integration theory. But the proposal is going indirectly through an algebraic geometry of Lie algebras in Loday-Pirashvili category. Maybe Urs will come up with another path if it drags his interest.

• brief category:people-entry for hyperlinking references at Leibniz algebra

• brief category:people-entry fro hyperlinking references

• brief category:people-page for hyperlinking references

• am finally giving this event/proceedings its own category:reference-page, for ease of equipping the respective contributions with pointer to publication data

not done yet, but need to save

• brief category:people-entry for hyperlinking references

• Fixed a LaTeX typo in Remark 3.12. I’m not an expert so could someone double-check?

• Added some remarks on the theory of empty objects.

• following up on our discussion in the thread <a href="http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=593&page=1#Item_12">oo-vector bundle (forum)</a> here on the forum, I have now spent a bit of time on expanding the entry quasicoherent sheaf

- I fixed the formulas in the section <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf#AsSheaves">As sheaves on Aff/X</a>. They were a bit rough and typo-ridden in the first version (which likely was my fault, not anyone else's). Since there are 50 variants in the literature to state this, I also pointed to page and verse in a lecture by Goerss where the statement is given explicitly the way it now appears there in the entry

- then I added a new section <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf#AsSheaves">As homs into the stack of modules</a> where I aim to describe in great detail how this definition is equivalent to the even simpler one, where we just say that the category of quasicoherent sheaves on a sheaf $X$ is the Hom-category $QC(X) = Hom(X,QC)$ for $QC : Ring \to Cat$ the stack of modules, classifying the canonical bifibration $Mod \to Rings$. This is the statement that my discussion at <a href="http://ncatlab.org/schreiber/show/%E2%88%9E-vector+bundle">oo-vector bundle (schreiber)</a> was secretly based on, which I promised to make more explicit.

- then I added a section <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf#Higher">Higher/derived quasicoherent sheaves</a>, where I indicate the now obvious oo-categorification discussed in more detail at <a href="http://ncatlab.org/schreiber/show/%E2%88%9E-vector+bundle">oo-vector bundle (schreiber)</a> and point out how this gives the derived QC sheaves used by Ben-Zvi et al as discussed at geometric infinity-function theory

- finally I wrote a fairly detailed <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf#Idea">Idea</a> section for quasicoherent sheaves, that previews the content of all these sections.
• am splitting this off from Lie algebra, for ease of cross-linking.

• brief category:people-entry for hyperlinking references at Leibniz algebra

• just for completeness

• Somebody named Adam left a comment box a while ago at premonoidal category saying that naturality of the associator requires three naturality squares. I believe that this is true when phrased explicitly in terms of one-variable functors, but the slick approach using the “funny tensor product” allows us to rephrase it as a single natural transformation between functors $C\otimes C\otimes C\to C$. I’ve edited the page accordingly. I also added the motivating example (the Kleisli category of a strong monad) and a link to sesquicategory.

There is a comment on the page that “It may be possible to weaken the above make $(Cat,\otimes)$ a symmetric monoidal 2-category, in which a monoid object is precisely a premonoidal category”. However, the Power-Robinson paper says that “We remark that $(C \otimes -) : Cat \to Cat$ is not a 2-functor,” which seems to throw some cold water on the obvious approach to that idea. Was the thought to define a different 2-categorical structure on $Cat$ than the usual one, e.g. using unnatural transformations? It seems that at least one would still have to explicitly require centrality of the coherence isomorphisms.

• I started an article about Martin-Löf dependent type theory. I hope there aren't any major mistakes!

One minor point: I overloaded $\mathrm{cases}$ by using it for both finite sum types and dependent sum types. Can anyone think of a better name for the operation for finite sum types?

• Build upon the „(possibly zero [length])“ remark and mention what happens with these paths.

• I started writing up the actual theorem from Day’s paper “On closed categories of functors”, regarding an extension of the “usual” Day convolution. He identifies an equivalence of categories between biclosed monoidal structures on the presheaf category $V^{A^{op}}$ and what are called pro-monoidal structures on A (with appropriate notions of morphisms between them) (“pro-monoidal” structures were originally called “pre-monoidal”, but in the second paper in the series, he changed the name to “pro-monoidal” (probably because they are equivalent to monoidal structures on the category of “pro-objects”, that is to say, presheaves)).

This is quite a bit stronger than the version that was up on the lab, and it is very powerful. For instance, it allows us to seamlessly extend the Crans-Gray tensor product from strict ω-categories to cellular sets (such that the reflector and Θ-nerve functors are strong monoidal). This is the key ingredient to defining lax constructions for ω-quasicategories, and in particular, it’s an important step towards the higher Grothendieck construction, which makes use of lax cones constructed using the Crans-Gray tensor product.

• started an Examples-section at natural numbers object

btw, the natural numbers objects of a topos is unique, up to isomorphism, right? if so, we should say that

• Simply a typo; the constant functor is of type $C \rightarrow \text{Func}(K, C)$.

• I noticed that there was a neglected stub entry universe that failed to link to the fairly detailed (though left in an unpolished state full of forgotten discussions) Grothendieck universe.

I renamed the former to universe > history and made “universe” redirect to “Grothendieck universe”

• I created biactegory following my 2006 work and being prompted by overlapping work of a student of Nikshych which appeared on the arXiv today.
• I have split off from holographic principle and then expanded a good bit a few paragraphs on