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I worked on synthetic differential geometry:
I rearranged slightly and then expanded the "Idea" section, trying to give a more comprehensive discussion and more links to related entries. Also added more (and briefly commented) references. Much more about references can probably be said, I have only a vague idea of the "prehistory" of the subject, before it became enshrined in the textbooks by Kock, Lavendhomme and Moerdijk-Reyes.
Also, does anyone have an electronic copy of that famous 1967 lecture by Lawvere on "categorical dynamics"? It would be nice to have an entry on that, as it seems to be a most visionary and influential text. If I understand right it gave birth to topos theory, to synthetic differential geometry and all that just as a spin-off of a more ambitious program to formalize physics. If I am not mistaken, we are currently at a point where finally also that last bit is finding a full implmenetation as a research program.
Made some some small improvements (ordering of sections, note on how the definition defaults to the usual definition of adjoints, fixing broken link in the references, etc) in relative adjoint functor.
New page: double category of algebras.
At the old entry cohomotopy used to be a section on how it may be thought of as a special case of non-abelian cohomology. While I (still) think this is an excellent point to highlight, re-reading this old paragraph now made me feel that it was rather clumsily expressed. Therefore I have rewritten (and shortened) it, now the third paragraph of the Idea-section.
(We had had long discussion about this entry back in the days, but it must have been before we switched to nForum discussion, because on the nForum there seems to be no trace of it.)
am starting fibrations of quasi-categories
Is there a particular reference where these (or rather, their super analogs) are computed for super Riemann surfaces?
felt like adding a handful of basic properties to epimorphism
pointer
added pointer to:
In locally cartesian closed category, I wrote out an explicit proof that pullback functors between slices preserve exponentials (so that Frobenius reciprocity is satisfied).
added the full definition to factorization algebra
We already have main references at Richard Askey, so why not ?
brief category:people-entry for hyperlinking references at elliptic genus
Not to confuse with Macdonald conjecture on plane partitions.
started a minimum at functor with smash products (the realization of ring spectra in terms of lax monoidal functors)
In the end this is entirely a story about monoids with respect to Day convolution tensor products. I suppose there is room to say this yet a bit more general abstractly than MMSS00 did.
added at dendroidal set
a section on the relation to simplicial sets
a section on the symmetric monoidal structure on the cat of dendroidal sets
(also added a stubby "overview" section to model structure on dendroidal sets)
I have expanded Lawvere-Tierney topology, also reorganized it in the process
some bare minimum on the free coproduct cocompletion.
The term used to redirect to the entry free cartesian category, where however the simple idea of free coproduct completion wasn’t really brought out.
Person entry.
Warning: we have a webpage for another algebraist, the group theorist Robert A. Wilson, and elsewhere in the Lab Robert Wilson shortcut is carelessly used for the latter. People tend to shorten links in Lab and the confusion and illegal links might occur in this case. Maybe we should not use version without middle initial for these two guys in links.
In statu nascendi.
A method for calculating determinants. It is related to cluster algebras and a special case of Sylvester identity.
I have been adding some material to matroid. I haven’t gotten around to defining oriented matroid yet (and of course there’s much besides to add).
Added more material to Boolean algebra, particularly the principle of duality and the connection to Boolean rings, and a wee bit of material on Stone duality.
Stone duality deserves greater expansion, bringing out the dualities via ambimorphic (ahem, schizophrenic) structures on the 2-element set, and mentioning the connection to Chu spaces. Another day, another dollar.
I should say – for those watching the logs and wondering – that I started editing the entry global equivariant homotopy theory such as to reflect Charles Rezk’s account in a coherent way.
But I am not done yet. The entry has now some of the key basics, but is still missing the general statement in its relation to orbispaces. Also some harmonizing of the whole entry may be necessary now, as I moved around some stuff.
So better don’t look at it yet. I hope to bring it into shape tomorrow or so.
(In the process I have split off global orbit category now.)
created a minimum at Penrose-Hawking singularity theorem
I have expanded the Idea-section at 3d quantum gravity and reorganized the remaining material slightly.
I feel unsure about the pointer to “group field theory” in the References. Can anyone list results that have come out of group field theory that are relevant here?
I find the following noteworthy, and I am not sure if this is widely appreciated:
the original discussion of the quantization of 3d gravity by Witten in 1988 happens work out to be precisely along the lines that “loop quantum gravity” once set out to get to work in higher dimensions: one realizes
that the configuration space is equivalently a space of connections;
that these can be characterized by their parallel transport along paths in base space;
that therefore observables of the theory are given by evaluating on choices of paths (an idea that goes by the unfortunate name “spin network”).
All this is in Witten’s 1988 article. Of course the point there is that in the case of 3d this can actually be made to work. The reason is that in this case it is sufficient to restrict to flat connections and for these everything drastically simplifies: their parallel transport depends not on the actual paths but just on their homotopy class, rel boundary. Accordingly the “spin networks” reduce to evaluations on generators of the fundamental group, etc.
Notice that in 4d the analog of this step that Witten easily performs in 3d was never carried out: instead, because it seemed to hard, the LQG literature always passes to a different system, where smooth connections are replaced by parallel transport that is required to be neigher smooth nor in fact continuous. These are called “generalized connections” in the LQG literature. Of course these have nothing much to do with Einstein-gravity: because there the configuration space does not contain such “generalized” fields.
For these reasons I feel a bit uneasy when the entry refers to LQG or spin foams as “other approaches” to discuss 3d quantum gravity. First of all, the existing good discussion by Witten did realize the LQG idea already in that dimension, and it did it correctly. So in which sense are there “other approaches”?
Which insights on 3d quantum gravity do “spin foam”s or does “group field theory”add? If anyone could list some results with concrete pointers to the literature, I’d be most grateful.