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

• I started an important entry differential monad. According to Lunts-Rosenberg MPI 1996-53 pdf differential calculus on schemes and noncommutative schemes can be derived from the yoga of coreflective topologizing subcategories in the abelian category of quasicoherent sheaves on the scheme, like the $\mathbb{T}$-filtration, and $\mathbb{T}$-part, in the case when the topologizing subcategory is the diagonal in the sense of the smallest subcategory of the category of additive endofunctors having right adjoint which contains the identity functor – in that case we say differential filtration and differential part. The regular differential operators are the elements of the differential part of the bimodule of endomorphisms. Similarly, one can define the conormal bundle etc.

• have added to monoidal (infinity,1)-category the definition of $\mathcal{O}$-monoidal $(\infty,1)$-category, for $\mathcal{O}$ an $\infty$-operad

(though maybe this definition either deserves its own entry or ought to be included instead at symmetric monoidal (infnity,1)-category)

• A point of information. These constructions are due to Charles Wells in this particular setting and to Jonathan Leech, (H-coextensions of monoids, vol. 1, Mem. Amer. Math. Soc, no. 157, American Mathematical Society, 1975) in the single object case, and McLane introduces the category of factorisations I think. Charlie Wells even pushes things a bit further than Baues. Hans does not seem to have known of that work. (Charles Wells, Extension theories for categories (preliminary report), (available from http://www.cwru.edu/artsci/math/wells/pub/pdf/catext.pdf), 1979. ) I have been meaning to have a go at this entry as I have written up a modern version of Wells especially in the non-Abelian case. There is a very nice interpretation of Natural System as a lax functor. (I will do this some time…. but I can make the notes available to anyone interested.)

• Urs created Frechet manifold, so I created Frechet space. (We violated the naming conventions too, but I guess it's OK since we have the redirects in.)

• I am trying to begin to coherently add some of the topics of part D of the Elephant into the Lab.

Currently I am creating lots of stub entries, splitting them off from existing entries if necessary, cross-link them appropriately, and then eventually add content to them.

so far I have for instance created new (mostly stub) entries for things like

I have created

and made it a disambiguation page.

(or rather I will have in a few minutes. All my save-windows are currently stalled. Will have to restart the server.)

• created standard site (maybe not a great term, but since I am $n$Labifying the Elephant). Added the theorem that every sheaf topos has a standard site of definition to site

• started adding rules to sequent calculus, but have to interrup now and hunt some food

• I have created the following web of entries

Most of them stubs. Partly just material split off from other entries. But all with the relavent pointers to the Elephant or other literature. To be expanded.

• I have created degeneration conjecture required at Dmitri Kaledin. In my memory, I never heard ofthis degeneration conjecture by precisely that name and I do not like it (there are so many degeneration conjectures in other fields, some of which I heard under that name). It is usually said the degeneration of Hodge to de Rham spectral sequence (conjecture). It has a classical analogue. I put redirect degeneration of Hodge to de Rham spectral sequence.

• In differential cohomology in an (∞,1)-topos – survey, I can’t guess what ’nothing’ should be here:

The curvature characteristic forms / Chern characters in the traditional formulation of differential cohomology take values in abelian $\infty$-Lie algebras and are therefore effectively nothing differential forms with values in a complex of vector spaces

• touched string structure. Added some formal discussion, also polsihed layout and added references. But didn’t change the previous informal discussion.

• This is an excerpt I wrote at logical functor:

As far as cartesian morphism there are two different universal properties in the literature, which are equivalent for Grothendieck fibered categories but not in general. In what Urs calls the “traditional definition” (but is in fact a later one) one has for every $x'$, for every $h$, for every $g$ such that … there exist a unique da da da. This way it is spelled in Vistoli’s lectures. This is in fact the strongly cartesian property, stronger than one in Gabriel-Grothendieck SGA I.6. The usual, Grothendieck, or weak property takes for $g$ the identity, and the unique lift is of the identity at $p(x_1)$. Then a Grothendieck fibered category is the one which has cartesian lifts for all morphisms below and all targets, and cartesian morphisms are closed under composition. With the strong cartesian property one does not need to require the closedness under composition. Now a theorem says that in a Grothendieck fibered category, a morphism is strongly cartesian iff it is cartesian.

Now I have made some changes to cartesian morphism, so that the entry is aware of the two variants of the universal property, which are not equivalent in general but are equivalent for Grothendieck fibered categories.

There was also a statement there

In words: for all commuting triangles in Y and all lifts through p of its 2-horn to X, there is a unique refinement to a lift of the entire commuting triangle.

which is too vague and I am not happy with, as it does not involve the essential parameter: the morphism for which we test cartesianess. I made a hack to it, and still it is not something I happy with (I like the idea of horn mentioned, however not the lack of appropriate quantifiers/conditions etc.). It is cumbersome to talk horn. (Maybe we could skip the whole statement in this imprecise form, and just mention please note the filling of the horn in $X$ with prescribed projection in $Y$ or alike). Here is the temporary hack:

In imprecise words: for all commuting triangles in $Y$ (involving $p(f)$ as above) and all lifts through $p$ of its 2-horn to $X$ (involving $f$ as above), there is a unique refinement to a lift of the entire commuting triangle.