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

• I have added a bit of history to the entry on Baues-Wirsching cohomology. Whilst looking for something else I found a paper by Charlie Wells from 1979, extending the earlier ideas of Leech cohomology for semigroups to small categories. He defines various types of extension and classifies them using the same methods as B and W used a few years later.

• added to exact functor a new subsection “Between abelian categories” and listed there (briefly) the standard characterizations of left/right exact functors in terms of preservation of left/right exact sequences.

Also added a reference by Michael Barr on the relation between exactness and respect for homology in very general contexts.

• added to injective object propositions and examples for injective modules and injective abelian groups

P.S. I am checking if I am missing something: Toën on page 48/49 here behaves as if it were clear that there is a model structure on positive cochain complexes of R-modules for all R in which the fibrations are the epis. But from the statements that i am aware of at model structure on chain complexes, in general the fibrations may be taken to be those epis that have injective kernels. For $R$ a field this is an empty condition and we are in business and find the familiar model structures. But for $R$ not a field? Notably simply $R = \mathbb{Z}$ What am I missing?

• I fixed the definition at over quasi-category so it makes the adjointness relationship clearer between overcategories and joins. In particular, Lurie’s notation and definition makes it very hard to see this. It’s much easier to see what’s going on when we look at things as follows: The join with $K$ fixed in the first coordinate, $S\mapsto i_K^{K\star S}: K\to K\star S$, where $i_K^{K\star S}$ is the canonical inclusion, is a functor $SSet\to (K\downarrow SSet)$. Then the undercategory construction gives the adjoint to this functor sending $(K\downarrow SSet) \to SSet$. This makes it substantially clearer to understand what’s going on, since $Hom_{SSet}(S, X_{F/}):= Hom_{K\downarrow SSet}(i_K^{K\star S}, F)$ is the set of those maps $f:K\star S\to X$ such that $f|K=f\circ i_K^{K\star S}=F$.

Lurie’s notation $Hom_{SSet}(S, X_{F/}):= Hom_F(K\star S, X)$ is nonstandard and inferior, since it obscures the obvious adjointness property.

The definition for overcategories is “dual” (by looking at the join of $K$ on the right).

• Aleks Kissinger has given us sifted colimit. Although I don’t quite understand the definition.

• I revisited some old discussion with Mike at sequence. Are you happy now, Mike?

• started oo-vector bundle on my personal web, following my latest remarks in the thread here on deformation theory.

• New page: indecomposable object, following (what I think is) Johnstone's definition. I also found it in some online topos theory lecture notes by Ieke Moerdijk and Jaap van Oosten.

Lambek and Scott give a different definition in Intro. to Higher-order Cat. Log., p. 168. I'm not sure how it relates to Johnstone's.

I've also given a proof that indecomposable <=> connected in an extensive category. I'd be interested to know whether this hypothesis is the weakest possible, if anyone has any ideas (or just likely-looking references).

• I could have sworn that we had something for thin category, at least a redirect, but we don’t. Or didn’t. Now we do.

Not much to it, just a note of terminology, like inductive limit or (0,1)-topos.

There’s also a diagram that I can’t to get to work there, if anybody wants to help.

• This is mathematically much simpler than the classical Gleason’s Theorem, but I added it to Gleason’s theorem anyway.

• In gluing categories from localizations (zoranskoda) the main section

From a family of localizations to a comonad

is fully rewritten in improved notation. In other way, it is explained better how to get a comonad from a cover of a category by not necessarily compatible flat localizations. This generalizes the Sweedler's coring to relative situations. Now from such data one can make a two category, which I will explain in few days.

This is a preliminary to something I am writing at the moment namely to explain in such terms actions of comonads and monoidal categories on such descent categories. This part will be analogous to description of equivariant maps among G-manifolds in pairs of local charts, but because of the distributive laws with coherences, the thing complicates.

• I started quantifier, but I ran out of time to say all that I wanted. I’ll probably get back to it in a couple of hours.

• I redid everything that includes contents using the new click-based menu system. This includes HomePage; there didn’t seem to be a need anymore to have two columns, so I put them back in one column. However, those are separate issues; we could put them back in two columns again and still do the click thingy.

• I added some stuff about states in statistical physics to state.