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

• Began entry with that name.
• I wrote Poincare group as an entree to the project of carrying on in nLab the blog discussion on unitary representations of the Poincare group. I'm not a specialist of course, so I ask the experts to please examine for accuracy.

• I expanded and polished the discussion of the abstract definition of of G-principal oo-bundles in an arbitrary (oo,1)-topos at principal infinity-bundle.

Parts of this could/should eventually be moved/copied to action and action groupoid, but I won't do that now.

I'd be interested in comments. One would expect that for the case that the ambient (oo,1)-topos is Top this style of definition should be well known in the literature, but I am not sure if it is. In fact, the examples listed further below in the entry, (the construction by Quillen and the Stasheff-Wirth construction) seems to indicate that this very simple very general nonsense picture has not been conceived as such before. Could that be true?

• I've removed the request for help link from the main contents. It didn't get used much (though I got answers to my questions there!). Since we have yet to actually delete a page, rather than just blank the request for help page I've put a pointer to where one can ask questions (pretty similar to that on the FAQ).

• I created a page for S-Sch as a notation for S-schemes to refer to in another post. Zoran pointed out that the notation is nonstandard (I do not know why I thought it was normal) and changed the title to [[Sch/S]]. I thus changed the first sentence to read Sch/S instead.

• I added a description of the degenerate affine Hecke algebra to the Hecke algebra page as one of the many variants.

I added the categorical generalization of Schur's lemma to that page.

I wrote a short stub on the additive envelope of a category, which Mike Shulman has expanded.

I mentioned the generalization of the Morse lemma to Hilbert manifolds.

I added the generalization of Hilbert's basis theorem to the case of where the ground ring is noetherian (not necessarily a field).

I wrote a short page on the Eilenberg swindle.
• I see that Akil Mathew has worked on a bunch of entries. Great! We should try to contact him and ask hom to record his changes here.

• I added Alex's recent lecture notes to cobordism hypothesis and in that process polished some typesetting there slightly.

Then I was pleased to note that Noah Snyder joined us and worked on fusion category. I created a page for him.

• I don't think that the (non-full) essential image of an arbitrary functor is well-defined.

• I added a fairly long (but still immensely incomplete) examples section to smooth topos.

I mention the "well adapted models" and say a few words about the point of it. Then I have a sectoin on how and in which sense algebraic geometry over a field takes place in a smooth topos. here the model is described easily, but I spend some lines on how to think of this. In the last example sections I have some remarks on non-preservation of limits in included subcategories of tame objects, but all that deserves further expansion of course.

• I continued working my way through the lower realms of the Whitehead tower of the orthogonal group by creating special orthogonal group and, yes, orthogonal group.

For the time being the material present there just keeps repeating the Whitehead-tower story.

But I want more there, eventually: I have a query box at orthogonal group. The most general sensible-nonsense context to talk about the orthogonal group should be any lined topos.

I am wondering if there is anything interesting to be said, from that perspective. Incidentally, I was prepared in this context to also have to create general linear group, only to find to my pleasant surprise that Zoran had already created that some time back. And in fact, Zoran discusses there an algebro-geometric perspective on GL(n) which, I think, is actually usefully thought of as the perspective of GL(n) in the lined topos of, at least, presheaves on $CRing^op$.

Presently I feel that I want eventually a discussion of all those seemingly boring old friends such as $\mathbb{Z}$ and $\mathbb{R} / \mathbb{Z}$ and $GL(n)$ etc. in lined toposes and smooth toposes. Inspired not the least by the wealth of cool structure that even just $\mathbb{Z}$ carries in cases such as the $\mathbb{B}$-topos in Models for Smooth Infinitesimal Analysis.

• created a page for Haynes Miller, since I just mentioned his name at string group as the one who coined that term.

not much on the page so far. Curiously, I found only a German Wikipedia page for him

• I've started listing differences between iTeX and LaTeX in the FAQ. That seemed the most logical place (I don't think we want a proliferation of places where users should look to find simple information) so either here or the HowTo seemed best. I chose the FAQ because the most likely time someone is going to look for this is when they notice something didn't look right.

The issue is that whilst iTeX is meant to be close to LaTeX they are never going to be the same so it's worth listing known differences with their work-arounds.

So far I've noted operator names, whitespace in \text, and some oddities on number handling.

• a bit rough for the time being.

• Vishal Lama joined the Lab!

on his page he promises to create Lab pages on some books on category theory and topos theory. Great, I am looking forward to it

• I have typed into infinitesimal interval object a detailed description of the simplicial object inuced on a microlinear space from the infinitesimal interval in immediate analogy to the construction of the finite path simplicial object induced from an interval object (as discussed there).

I also give the inclusion of the infinitesimal simplicial object into the finite one.

All the proofs here are straightforward checking, which I think I have done rather carefully on paper, but not typed up. What I would appreciate, though, is if somebody gave me a sanity check on the definition of the infinitesimal simplicial object (which is typed in detail).

In the very last section, which is the one that is still just a sketch, I am hoping to describe an isomorphism from my simplicial infinitesimal object to that considered by Anders Kock, which is currently described at infinitesimal singular simplicial complex in the case that the space X satisfies Kock's assumptions (it must be a "formal manifold").

The construction I discuss at infinitesimal interval object is supposed to generalize Kock's construction to all microlinear spaces and motivated by having that canonical obvious inclusion into the finite version at interval object.

The isomorphism should be evident: my construction evidently yields in degree k k-tuples of pairwise first oder neighbours if the space X admits that notion. But I want to sleep over this statement one more night...