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

• tried to polish a bit the matrial at Chern-Weil theory.

(not that there is much, yet, but still)

• Affinity in the context of D-modules, as defined by Alexander Beilinson is the subject of a new stub D-affinity. There is a categorical generalization in the MPI1996-53 preprint (pdf) of Lunts and Rosenberg in terms of differential monads. Many generalizations of Beilinson-Bernstein localization theorem have their intuitive explanation in a two-step reasoning. First the noncommutative algebra in question is understood as a noncommutative (or maybe categorical) resolution of singularities of a commutative object. Then the latter satisfies D-affinity and one can localize.

• New entry domain globalization of functors (zoranskoda) under development. The codomain globalization is more trivial. This are questions of extending the constructions related to Beck’s comonadicity from categories to functors. Our interest with Gabi Bohm are mainly for covers by localizations with some equivariance/compatibility with respect to additional (co)monad, which are a matter of ongoing work. This compatibility is like, or some dual of the one in the definition of morphisms of Q-categories and also the compatibility of differential monads and localization, studied by Lunts and Rosenberg. The latter is related to the classical fact that the assignment of ring of regular differential operators to a commutative ring $R\mapsto Diff(R)$ is compatible with exact localizations, in the sense that $S^{-1}R \mapsto S^{-1}Diff(R)$; and also to Beilinson’s notion of D-affinity.

• I have created stubs for the missing entries to complete this table:

The main actual content I added are, (at 2-type theory and 2-logic): pointers to Dan Licata’s thesis and to Mike’s personal wiki pages.

I’d hope that one outcome of the present $n$Café discussion is that eventually some of these entries get equipped with some useful content.

(P.S. I would have linked to material by Mike Stay, too, but I don’t know what to link to.)

• I am about to create an entry called locally algebra-ed topos in the spirit of the section for local algebras at classifying topos.

I tend to think this terminology is better than the undescriptive “structured topos”, but please let me know what you think.

I would like to amplify the following fact:

if we agree to say (which is reasonable) that

• an algebra is a model of some essentially algebraic theory, hence a lex functor out of a finite-limite category;

• a local algebra with respect to a coverage on the category is such a lex functor that preserves covers.

then the statement is:

• geometric theories are equivalently theories of local algebras.
• 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.

• 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