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added to locale a section relation to toposes stating localic reflection
I have added some minimum content to Stiefel manifold, also a little bit to Grassmannian
weirdly, this page never linked to simplicial groupoid. I am adjusting a little…
Added to Hopf monad the Bruguières-Lack-Virelizier definition and some properties.
I made some minor improvements to the Properties section of pushout, making it match the similar section in pullback insofar as it can. (It’s a bit tiring to have to look at both these pages to get all the basic properties, so I fixed that, but for properties that hold both for pullbacks and dually for pushouts I’m happy to have all the proofs at pullback - that’s how it works now.)
I have touched the formatting at free groupoid. Then I added the statement that the fundamental groups of a free groupoid are free. Also added a pointer to a writeup of the proof.
started adding some genuine substance to model structure on sSet-categories (which used to be just a template).
added some basics to model structure for quasi-categories at general properties
As there had been a change to the entry for Ross Street I gave it a glance. Is there a reason that the second reference is to a paper without Ross as an author?I hesitate to delete it as there may be a hidden reason. (I have edited this discussion entry to remedy the point that Todd and Urs have made below. I also edited the title of this discussion!)
started a stub for ambidextrous adjunction, but not much there yet
I stated this for presheaf categories, but I’m pretty sure that it carries over for any Grothendieck topos.
Check it out: lawvere interval
When pointing somebody to it, I noticed that the entry n-category is in a rather sad state and in particular it used to start out in a rather unhelpful fashion. I have now tried to briefly fix at least the latter problem by expanding and editing the first two sentences a bit. Notably I made sure that a pointer to (∞,n)-category appears early on, for that is a place with more robust information, currently.
I have adjusted and expanded wording and formatting in this entry.
Notice that the definition of the source and target maps that was (and still is) given here differs from that in Dwyer & Kan (1984):
where Dwyer & Kan’s §3.1(ii) “discards vertices from the right”, the definition that was (and still is) given in the entry seems to want to switch to the convention where vertices are discarded “from the left”.
With due care this can probably be made an equivalent definition, but as currently stated
this must be wrong in itself: the “” probably wants to be a “”.
If anyone wants to fix this, feel invited. Otherwise I’ll change this to Dwyer & Kan’s definition.
one more remark at relation between quasi-categories and simplicial categories
(to be expanded...)
I left a counter-query underneath Zoran’s query at compactly generated space. It may be time for a clean-up of this article; the query boxes have been left dangling and unanswered for quite some time. Either proofs or references to detailed proofs would be welcome.
created traced monoidal category with a bare minimum
I would have sworn that we already had an entry on that, but it seems we didn’t. If I somehow missed it , let me know and we need to fix things then.
I noticed only now that the entry bimodule is in bad shape and needs some attention. For the moment I have added here a mentioning of the 2-category of algebras, bimodules and intertwiners and a pointer to the Eilenberg-Watts theorem.
Have added to cyclic set a pointer to notes from 1996 by Ieke Moerdijk where the theory classified by the topos of cyclic sets is identified (abstract circles).
This is an unpublished note, but on request I have now uploaded it to the nLab
I have also added a corresponding brief section to classifying topos.
By the way, there is an old query box with an exchange between Mike and Zoran at cyclic set. It seems to me that this has been resolved and the query box could be removed (to make the entry read more smoothly). Maybe Mike and/or Zoran could briefly look into this.
started a Properties-section at Lawvere theory with some basic propositions.
Would be thankful if some experts looked over this.
Also added the example of the theory of sets. (A longer list of examples would be good!) And added the canonical reference.
I created inverse Galois problem. However as it stand this stub does not meet nlab standards since it doesn’t refer to category theory.
added looping/delooping statements to model structure on simplicial groupoids and model structure on presheaves of simplicial groupoids.
Question: Suppose we take the catgeory of genuine simplicial groupoids (no restriction on the simplicial set of objects). Does send degreewise fibrations to fibrations?
earlier today I had added at Azumaya algebra a new section In terms of (derived) étale cohomology with some notes that I took during a talk by David Gepner. He has a quite beautiful picture. Maybe I find the time to expand a bit on this entry and related entries.
Urs has added Euler integration prompted by Tom’s post at nCafe; I wanted to do that and will contribute soon. I noticed there is no entry integral in Lab, but it redirects to integration. I personally think that integral as a mathematical object is a slightly more canonical name for a mathematical entry than integration, if the two are not kept separated. Second, the entry is written as an (incomplete) disambiguation entry and with a subdivision into measure approach versus few odd entries. I was taught long time ago by a couple of experts in probability and measure theory that a complete subordination to the concept of integral to a concept of measure is pedagogically harmful, and lacks some important insights. This has also to do with the choice of the title: integration points to a process, and the underlying process may involve measure. Integral is about an object which is usually some sort of functional, or operator, on distributions which are to be acted upon.
Thus I would like to rename the entry into integral (or to create a separate entry from integration) and make it into a real entry, the list of variants being just a section, unlike in the disambiguation only version. What do you think. Then I would add some real ideas about it.
I added to category of elements an argument for why preserves colimits.
brief category:people
-entry for satisfying links now requested at p-adic Teichmüller theory
New entry noncommutative differential calculus with redirect Batalin-Vilkovisky module.
Add example of Eilenberg–Moore objects.
Some comments at BLG model in the section “3-algebras” on a confusion of terms.
considerably expanded the entry strict 2-group.
Apart from adding an introductory discussion, and expanding the list of examples, in particular by adding that of automorphism 2-groups ...
... I in particular give the detailed translation prescription for how to encode a 2-group by a crossed module at In terms of crossed modules
This is to eventually serve as a supplement to the discussion at nonabelian group cohomology. So I spent some energy on disentangling the four different (though isomorphic) ways a crossed module gives rise to a 2-group (following my article with David Roberts).
At crossed module it seems we are missing what i think should be the prototypical example: the relative second homotopy group together with the bundary map and the -action on . As someone confirms this example is correct I’ll add it to crossed module.
I have briefly recorded the equivalence of FinSet with finite Booplean algebras at FinSet – Properties – Opposite category. Then I linked to this from various related entries, such as finite set, power set, Stone duality, opposite category.
(I thought we long had that information on the Lab, but it seems we didn’t)
Noted that this follows from Day’s reflection theorem.
I’ve been adding a considerable amount of detail to relational beta-module. Not quite finished, but a lot of former ellipses have been filled in, and technical results proved from scratch.
If anyone knows this material, I’d enjoy hearing of any proposed simplifications. My local university library does not have the Springer Lecture Notes where Barr first wrote his article on relational beta-modules, and it’s hard finding detailed material at this ground level on the internet, so this represents pretty much what I’ve been able to work out by myself. (And it might read that way.)
I tried to prettify the entry topological space a bit more:
made an attempt at adding an Idea-section (feel free to work on that, it’s just a quick idea motivated more from the desire to have such a section at all than from an attempt to do it any justice).
collected the three Definition-sections to subsections of a single Definition-section
polished and expanded the Standard definition section.