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started adding some genuine substance to model structure on sSet-categories (which used to be just a template).
I want to be able to point to category of V-enriched categories, so I created an entry, so far just with a brief Idea-paragraph.
one more remark at relation between quasi-categories and simplicial categories
(to be expanded...)
started floating toc enriched category theory contents and added it to relevant entries
I have added to full and faithful functor the fact that they are closed under pushouts in Cat, with references (thanks to MO twice).
added at adjoint functor
more details in the section In terms of universal arrows;
a bit in the section Examples
added to Grothendieck construction a section Adjoints to the Grothendieck construction
There I talk about the left adjoint to the Grothendieck construction the way it is traditionally written in the literature, and then make a remark on how one can look at this from a slightly different perspective, which then is the perspective that seamlessly leads over to Lurie's realization of the (oo,1)-Grothendieck construction.
There is a CLAIM there which is maybe not entirely obvious, but straightforward to check. I'll provide the proof later.
This is intended to continue the issues discussed in the Lafforgue thread!
I have added an idea section to Morita equivalence where I sketch what I perceive to be the overarching pattern stressing in particular the two completion processes involved. I worked with ’hyphens’ there but judging from a look in Street’s quantum group book the pattern can be spelled out exactly at a bicategorical level.
I might occasionally add further material on the Morita theory for algebraic theories where especially the book by Adamek-Rosicky-Vitale (pdf-draft) contains a general 2-categorical theorem for algebraic theories.
Another thing that always intrigued me is the connection with shape theory where there is a result from Betti that the endomorphism module involved in ring Morita theory occurs as the shape category of a ring morphism in the sense of Bourn-Cordier. Another thing worth mentioning on the page is that the Cauchy completion of a ring in the enriched sense is actually its cat of modules (this is in Borceux-Dejean) - this brings out the parallel between Morita for cats and rings.
added some basics to model structure for quasi-categories at general properties
added a second equivalent definition at quasi-category , one that may be easier to motivate
added to complete Segal space a discussion of what an ordinary category looks like when regarded as a complete Segal space.
(This is meant to be pedagogical, therefore the recollection of all the basics at the beginning.)
This is a bare list of references, to be !include-ed into relevant entries, such as at swampland and 24 branes transverse to K3, for ease of cross-linking and updating.
I am taking the liberty of including a pointer to our upcoming M/F-Theory as Mf-Theory which has some details on a precise version of the conjecture and a proof (from Hypothesis H).
I wrote an entry (short for now) separable algebra. It is a sort of support for the current Galois theory/Tannakian reconstruction/covering space/monodromy interest of Urs.
I have added some minimum content to Stiefel manifold, also a little bit to Grassmannian
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.)
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.
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.
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.