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I understood that the old terminology was ’projective system’, and ’projective limit’ refereed to the limit of a projective system. Can anyone confirm that? if I am right the present entry is slightly incorrect, but this needs checking first before changing it.
brief category:people-entry for hyperlinking references at L-complete module
Under definition 1 of salamander lemma, I fixed a mistake in the definition of where there was a direct sum of two submodules, where there needed to be a sum (i.e., join) instead.
Made a few additions to preimage. Added missing word; added a brief mention of the widely-known general reason for the good preservation-properties of this endofunctor.
The mention of these properties had already been there in preimage, but a reason was still missing. My parenthetical remark should perhaps be expanded and harmonized with existing relevant material on the nLab ( and are already well-documented on some pages), but this requires more care than I can apply to it today. Intend to return to the remark before long.
brief category:people-entry for hyperlinking references at equivariant K-theory
Adding redirect for Euler’s constant.
added references by Pronk-Scull and by Schwede, and wrote an Idea-section that tries to highlight the expected relation to global equivariant homotopy theory. Right now it reads like so:
On general grounds, since orbifolds are special cases of stacks, there is an evident definition of cohomology of orbifolds, given by forming (stable) homotopy groups of derived hom-spaces
into any desired coefficient ∞-stack (or sheaf of spectra) .
More specifically, often one is interested in viewing orbifold cohomology as a variant of Bredon equivariant cohomology, based on the idea that the cohomology of a global homotopy quotient orbifold
for a given -action on some manifold , should coincide with the -equivariant cohomology of . However, such an identification (1) is not unique: For any closed subgroup, we have
This means that if one is to regard orbifold cohomology as a variant of equivariant cohomology, then one needs to work “globally” in terms of global equivariant homotopy theory, where one considers equivariance with respect to “all compact Lie groups at once”, in a suitable sense.
Concretely, in global equivariant homotopy theory the plain orbit category of -equivariant Bredon cohomology is replaced by the global orbit category whose objects are the delooping stacks , and then any orbifold becomes an (∞,1)-presheaf over by the evident “external Yoneda embedding”
More generally, this makes sense for any orbispace. In fact, as a construction of an (∞,1)-presheaf on it makes sense for any ∞-stack, but supposedly precisely if is an orbispace among all ∞-stacks does the cohomology of in the sense of global equivariant homotopy theory coincide the cohomology of in the intended sense of ∞-stacks, in particular reproducing the intended sense of orbifold cohomology.
At least for topological orbifolds this is indicated in (Schwede 17, Introduction, Schwede 18, p. ix-x, see also Pronk-Scull 07)
Strangely, we don’t seem to have an nForum discussion for probability theory.
I added a reference there to
It replaces the category of measurable spaces, which isn’t cartesian closed, with the category of quasi-Borel spaces, which is. As they point out in section IX, what they’re doing is working with concrete sheaves on an established category of spaces, rather like the move to diffeological spaces.
[Given the interest in topology around these parts at the moment, we hear of ’C-spaces’ as generalized topological spaces arising from a similar sheaf construction in C. Xu and M. Escardo, “A constructive model of uniform continuity,” in Proc. TLCA, 2013.]
added publication data to:
touched five lemma
I have added to homotopy coherent diagram and to model structure on algebras over an operad the fact that one can describe homotopy coherent diagrams as algebras over the Boardman-Vogt resolution of some operad. Vogt’s rectification theorem is then a special case of the general Berger-Moerdijk result.
In the course of this I reorganized and expanded homtopy coherent diagram a bit. It still needs to be polished a bit.
touched the first paragraph, making hyperlinked pointers to compact Lie groups, positive energy representations and the Verlinde ring.
Then I replaced “twisted euqivariant K-theoyry” with twisted ad-equivariant K-theory and will give this its own entry now…
have split-off a stub positive energy representation from loop group
added complete referencing of FHT and cross-link with twisted ad-equivariant K-theory
I am filling in more details (definitions, properties) about the various toposes at Models for Smooth Infinitesimal Analysis
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.
Moved talk by McLarty to the History section of the refernces from functorial geometry
added DOI to:
@Todd. Thanks for correcting my atrocious English!
Does anyone have any ideas as to how we could provide a bit more for this entry?