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As an outcome of recent discussion at Math Overflow here, Mike Shulman suggested some nLab pages where comparisons of different definitions of compactness are rigorously established. I have created one such page: compactness and stable closure. (The importance and significance of the stable closure condition should be brought out better.)
copied over the homotopy-theoretic references from modal type theory to here.
brief category:people-page for hyperlinking references at flavour anomaly and at LHCb experiment
Added a reference
Corrected the spelling.
In English, “étale” in “étale space” is definitely an adjective, not a past participle (otherwise it would be “étaled space”).
The last “e” is always silent in the English pronunciation, so cannot be an “é”.
This also matches the usage in the literature.
I have removed the few stub lines that were in this entry before, and wrote out a more informative Idea-section. Currently it reads as follows:
In basic topology and differential geometry, by an atlas of/for a topological-, differentiable- or smooth manifold one means a collection of coordinate charts which form an open cover of .
If one considers here the disjoint union of all the choordinate charts, then the separate chart embeddings give rise to a single map (continuous/differentiable function)
and now the condition for an atlas is that this is a surjective étale map/local diffeomorphism.
If, next, one regards this morphism, under the Yoneda embedding, inside the topos of formal smooth sets, then these conditions on an atlas say that this morphism is
In this abstract form the concept of an atlas generalizes to any cohesive higher geometry (KS 17, Def. 3.3, Wellen 18, Def 4.13).
Next, for a geometric stack , an atlas is a smooth manifold (for differentiable stacks) or scheme (for algebraic stacks) or similar, equipped with a morphism
that is an effective epimorphism and formally étale morphism in the corresponding higher topos (for instance in that of formal smooth infinity-groupoids).
Here the terminology has a bifurcation:
In the general context of geometric stacks one typically drops the second condition and calls any effective epimorphism from a smooth manifold or scheme to a differentiable stack or algebraic stack, respectively, an atlas.
If in addition the condition is imposed that such an effective epimorphism exists which is also formally étale, then the geometric stack is called an orbifold or Deligne-Mumford stack (often with various further conditions imposed).
have added some further reference at gauged supergravity and related entries.
Edit to: GUT by Urs Schreiber at 2018-04-01 01:21:13 UTC.
Author comments:
added pointer to textbook account
added pointer to these two recent references, identifying further -algebra structure in Feynman amplitudes/S-matrices of perturbative quantum field theory:
Markus B. Fröb, Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories (arXiv:1803.10235)
Alex Arvanitakis, The -algebra of the S-matrix (arXiv:1903.05643)
this is a bare sub-section, to be !include-ed into the pertinent entries (at embedding tensor, at tensor hierarchy and at super Lie algebra) in order to avoid having to copy this stuff around and to facilitate updating and syncing it across these entries
brief category:people-entry for hyperlinking references at super Lie algebra and at Kantor-Koecher-Tits construction
Updated Jim’s (non)-affiliation. He was attached to UCR (clearly) and Macquarie. Anywhere else that should be added? I think more links to other pages could be added, and certainly the ones I’ve now added could have commentary around them. For instance, the list of neologisms that Jim had added deserve a section of their own, I think. But no time now.
added a few references and links to super Lie algebra