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added pointer to:
Added to paracompact topological space in the Properties-section the statement that every bounded hypercover over a paracompact space is refined by the Čech nerve of a plain open cover… by appealing to a lemma in HTT. (Thanks to Danny Stevenson for pointing this out.)
created nerve theorem
linked to it from homotopy groups in an (infinity,1)-topos, where it had implicitly been mentioned before, but not made explicit.
Apart from stating the theorem, I wrote a section that explains what’s going on from the nPOV. As far as I can see, at least.
I added a bit to category of simplices, including the fact that the category of nondegenerate simplices is final and thus colimits can be computed using only that, and that the nerve of the category of simplices itself is colimit-preserving.
Added to group object the Yoneda-embedding-style definition and added supergroup to the list of examples.
brief category:people-entry for hyperlinking references at cosmic structure formation and cosmic reionization
a stub in cosmology, for the moment just so as to satisfy links at cosmic structure formation
I have created Locally Presentable and Accessible Categories, with redirect LPAC, since we seem to refer to it a lot. I also added it as a reference to compact object, where it was surprisingly absent.
added some more keywords to the “index” (came here to add action object, but added a few more while I was at it)
expanded the Idea-section and added a reference:
starting something, to be another item of internalization along with monoid object and group object.
Not done yet, but need to save.
(Our entry “action” does cover internal notions of actions, but it seems hard to bring that old entry into a good shape, and in any case splitting off an entry on the internal notion might just be the first step of bringing it into shape)
The evident question for compiling this entry is: What are the canonical original references?
Eckmann-Hilton in their articles originally introducing the notion of internal group objects (here) have all the ingredients in hand to say “action object”, but it seems they don’t (?).
I found the notion stated clearly and explicitly, albeit somewhat in passing, in:
but if anyone has other/better references, let’s add them.
I have edited at HQFT, touched the general formatting and structuring a good bit, trying to clean it up and beautify it a bit, and added a brief cross-pointer to the cobordisms hypothesis for cobordisms with maps into a base manifold.
am splitting this off from holographic QCD, since the latter is getting too crowded. Prompted by today’s
I added an explicit definition of cartesian model category to cartesian closed model category to highlight the convention that the terminal object is assumed cofibrant.
I noticed that there was no entry quotient stack, so I quickly started one, just to be able to point to it from elswhere.
added to Eckmann-Hilton argument the formal proposition formulated in any 2-category.
BTW, doesn’t anyone have a gif with the nice picture proof?
I pasted in something Mike wrote on sketches and accessible models to sketch. But now it needs tidying up, and I’m wondering if it might have been better placed at accessible category. Alternatively we start a new page on sketch-theoretic model theory. Ideas?
two-valued objects are topos-theoretic analogues of the set with two elements in , and the categorical semantics of the type of booleans. Different from subobject classifiers. Still to do: categorical semantics of binary coproduct types and binary product types using dependent sums and products and two-valued objects.
Anonymous
splitting off this definition from neighborhood retract, for ease of linking, and in order to record the characterization AR = ANR+contractible
just for completeness, to go alongside dimension of a cell complex and dimension of a manifold
Am finally giving this its own entry, alongside dimension of a cell complex
(this here used to just redirect to dimension)
have created covering dimension
felt like archiving a quote by Paul Taylor somewhere, it is now at folklore.
Besides being funny, it is actually a useful comment for the newbie, and so I linked to it from category theory.
I rephrased and restructured the Definition-section at join of simplicial sets a bit, in an attempt to make the exposition clearer. Please check if I succeeded.
I had started an entry “exponentiation” but then thought better of it and instead expanded the existing exponential object: added an examples-section specifically for and made some remarks on exponentiation of numbers.
brief category:people-entry for hyperlinking references at absolute neighbourhood retract
Added
For discussion of the hexagon in an equivariant setting, see
- Andreas Kübel, Andreas Thom, Equivariant Differential Cohomology, (arXiv:1510.06392)
added publication data to:
I have split off an entry classical model structure on simplicial sets from “model structure on simplicial set”. This entry should eventually contain detailed, self-contained and polished discussion of the definition, verification and key properties of the standard Kan-Quillen model structure.
So far I have inserted fair bit of background material regarding (minimal) fibrations and geometric realizations, essentially the material in chapter 1 of Goerss-Jardine. A bunch of little proofs are spelled out, but not yet the more laborious ones. Discussion of the verification of the axioms is not yet in the entry, but the key parts of the Quillen equivalence to are (modulo relying on previous lemmas that don’t have proofs spelled out yet).
The somewhat random list of properties of that used to be sitting at “model structure on simplicial sets” I have copied over to a section “Basic properties”, just for completenes, but this now needs re-organization to give decent logical flow.
For the moment I have to leave it at that, need to take care of something else now for a little bit.
Moved some content from polyhedron.
Several recent updates to literature at philosophy, the latest being
which is more into cognition and language problem, but still very relevant, and by a top mathematician. As these 2 are still manuscripts I put them under articles, though I should eventually classify those as books…
Added a table of contents to topos, a section on "special classes" and one on "higher toposes".
felt like adding a handful of basic properties to epimorphism
Added the references
Aurelio Carboni, Bob Walters, Cartesian Bicategories, I, article
Aurelio Carboni, Max Kelly, Bob Walters, Richard Wood, Cartesian Bicategories II, (arXiv:0708.1921)