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- Discussion Type
- discussion topicmodel structure on sSet-categories
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by Urs
- Last Active 2 days ago

started adding some genuine substance to model structure on sSet-categories (which used to be just a template).

- Discussion Type
- discussion topiccategory of V-enriched categories
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active 2 days ago

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.

- Discussion Type
- discussion topicenriched functor category
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topicrigidification of quasi-categories
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topicrelation between quasi-categories and simplicial categories
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active 2 days ago

one more remark at relation between quasi-categories and simplicial categories

(to be expanded...)

- Discussion Type
- discussion topicenriched category theory contents
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 2 days ago

started floating toc enriched category theory contents and added it to relevant entries

- Discussion Type
- discussion topicfull and faithful functor
- Category Latest Changes
- Started by Mike Shulman
- Comments 8
- Last comment by Todd_Trimble
- Last Active 2 days ago

I have added to full and faithful functor the fact that they are closed under pushouts in Cat, with references (thanks to MO twice).

- Discussion Type
- discussion topiccartesian closed enriched category
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topicdominant functor
- Category Latest Changes
- Started by varkor
- Comments 6
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topicadjoint functor
- Category Latest Changes
- Started by Urs
- Comments 72
- Last comment by Urs
- Last Active 2 days ago

added at adjoint functor

more details in the section In terms of universal arrows;

a bit in the section Examples

- Discussion Type
- discussion topicGrothendieck construction
- Category Latest Changes
- Started by Urs
- Comments 77
- Last comment by Urs
- Last Active 3 days ago

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.

- Discussion Type
- discussion topicsemidirect product group
- Category Latest Changes
- Started by nLab edit announcer
- Comments 8
- Last comment by Todd_Trimble
- Last Active 3 days ago

- Discussion Type
- discussion topicMorita equivalence
- Category Latest Changes
- Started by Thomas Holder
- Comments 21
- Last comment by Todd_Trimble
- Last Active 3 days ago

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.

- Discussion Type
- discussion topicmodel structure for quasi-categories
- Category Latest Changes
- Started by Urs
- Comments 19
- Last comment by Urs
- Last Active 3 days ago

added some basics to

*model structure for quasi-categories*at*general properties*

- Discussion Type
- discussion topicmultifunctor
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by maxsnew
- Last Active 3 days ago

- Discussion Type
- discussion topicFriedrich Ulmer
- Category Latest Changes
- Started by varkor
- Comments 2
- Last comment by Urs
- Last Active 3 days ago

- Discussion Type
- discussion topicbi-
- Category Latest Changes
- Started by varkor
- Comments 5
- Last comment by Urs
- Last Active 3 days ago

- Discussion Type
- discussion topicquasi-category
- Category Latest Changes
- Started by Urs
- Comments 19
- Last comment by Urs
- Last Active 3 days ago

added a second equivalent definition at quasi-category , one that may be easier to motivate

- Discussion Type
- discussion topiccomplete Segal space
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by Urs
- Last Active 3 days ago

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.)

- Discussion Type
- discussion topicAQFT on curved spacetimes
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topicPhilip W. Anderson
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topicquantum spin liquid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topicswampland cobordism conjecture -- references
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by David_Corfield
- Last Active 4 days ago

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).

- Discussion Type
- discussion topicTadasi Nakayama
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topicseparable algebra
- Category Latest Changes
- Started by zskoda
- Comments 7
- Last comment by Urs
- Last Active 4 days ago

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.

- Discussion Type
- discussion topicStiefel manifold
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Todd_Trimble
- Last Active 4 days ago

I have added some minimum content to

*Stiefel manifold*, also a little bit to*Grassmannian*

- Discussion Type
- discussion topiccompact object
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by Todd_Trimble
- Last Active 4 days ago

- Discussion Type
- discussion topicdiscrete category
- Category Latest Changes
- Started by varkor
- Comments 1
- Last comment by varkor
- Last Active 4 days ago

- Discussion Type
- discussion topicHopf monad
- Category Latest Changes
- Started by Mike Shulman
- Comments 6
- Last comment by varkor
- Last Active 4 days ago

Added to Hopf monad the Bruguières-Lack-Virelizier definition and some properties.

- Discussion Type
- discussion topicpushout
- Category Latest Changes
- Started by John Baez
- Comments 11
- Last comment by Urs
- Last Active 4 days ago

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.)

- Discussion Type
- discussion topicambidextrous adjunction
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by varkor
- Last Active 4 days ago

started a stub for

*ambidextrous adjunction*, but not much there yet

- Discussion Type
- discussion topic[[lawvere interval]]
- Category Latest Changes
- Started by Harry Gindi
- Comments 9
- Last comment by jonsterling
- Last Active 4 days ago

I stated this for presheaf categories, but I’m pretty sure that it carries over for any Grothendieck topos.

Check it out: lawvere interval

- Discussion Type
- discussion topicLawvere interval
- Category Latest Changes
- Started by jonsterling
- Comments 1
- Last comment by jonsterling
- Last Active 4 days ago

- Discussion Type
- discussion topicn-category
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 4 days ago

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.

- Discussion Type
- discussion topicDwyer-Kan fundamental simplicial groupoid
- Category Latest Changes
- Started by Urs
- Comments 37
- Last comment by Urs
- Last Active 5 days ago

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

$s = (d_1)^{n+1}$

$t = d_0(d_2)^n$

this must be wrong in itself: the “$d_2$” probably wants to be a “$d_1$”.

If anyone wants to fix this, feel invited. Otherwise I’ll change this to Dwyer & Kan’s definition.

- Discussion Type
- discussion topichigher-order logic
- Category Latest Changes
- Started by David_Corfield
- Comments 5
- Last comment by Hugo Herbelin
- Last Active 5 days ago

- Discussion Type
- discussion topiccompactly generated topological space
- Category Latest Changes
- Started by Todd_Trimble
- Comments 88
- Last comment by Todd_Trimble
- Last Active 5 days ago

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.

- Discussion Type
- discussion topicpower series
- Category Latest Changes
- Started by nLab edit announcer
- Comments 10
- Last comment by Todd_Trimble
- Last Active 5 days ago

- Discussion Type
- discussion topictraced monoidal category
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by J-B Vienney
- Last Active 5 days ago

created

*traced monoidal category*with a bare minimumI 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.

- Discussion Type
- discussion topicrelative monad
- Category Latest Changes
- Started by mattecapu
- Comments 14
- Last comment by BryceClarke
- Last Active 5 days ago

- Discussion Type
- discussion topicsemi-simplicial types in homotopy type theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 15
- Last comment by Hugo Herbelin
- Last Active 5 days ago

- Discussion Type
- discussion topicbimodule
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Todd_Trimble
- Last Active 5 days ago

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.

- Discussion Type
- discussion topiccyclic set
- Category Latest Changes
- Started by Urs
- Comments 29
- Last comment by zskoda
- Last Active 5 days ago

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

- Ieke Moerdijk,
*Cyclic sets as a classifying topos*, 1996 (pdf)

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.- Ieke Moerdijk,

- Discussion Type
- discussion topicmodule over a monoid
- Category Latest Changes
- Started by J-B Vienney
- Comments 1
- Last comment by J-B Vienney
- Last Active 6 days ago

- Discussion Type
- discussion topicIntroduction to Topology -- 1
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Guest
- Last Active 6 days ago

- Discussion Type
- discussion topicLawvere theory
- Category Latest Changes
- Started by Urs
- Comments 51
- Last comment by varkor
- Last Active 6 days ago

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.

- Discussion Type
- discussion topicreal quadratic function
- Category Latest Changes
- Started by nLab edit announcer
- Comments 8
- Last comment by Guest
- Last Active 6 days ago

- Discussion Type
- discussion topicleft division in modal type theory
- Category Latest Changes
- Started by Guest
- Comments 3
- Last comment by Todd_Trimble
- Last Active 6 days ago

- Discussion Type
- discussion topicinverse Galois problem
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 5
- Last comment by Urs
- Last Active 6 days ago

I created inverse Galois problem. However as it stand this stub does not meet nlab standards since it doesn’t refer to category theory.

- Discussion Type
- discussion topicquantum error correction
- Category Latest Changes
- Started by Urs
- Comments 43
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicAndreas Wallraff
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicMarkus Müller
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicAzumaya algebra
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by John Baez
- Last Active 7 days ago

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.

- Discussion Type
- discussion topicintegral
- Category Latest Changes
- Started by zskoda
- Comments 14
- Last comment by Guest
- Last Active 7 days ago

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 $n$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.

- Discussion Type
- discussion topicDarboux integration
- Category Latest Changes
- Started by Guest
- Comments 1
- Last comment by Guest
- Last Active 7 days ago

- Discussion Type
- discussion topicinvertible field theory
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicp-adic Teichmüller theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active 7 days ago

- Discussion Type
- discussion topiccategory of elements
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by hugopaquet
- Last Active 7 days ago

I added to category of elements an argument for why $El$ preserves colimits.

- Discussion Type
- discussion topicKurusch Ebrahimi-Fard
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 7 days ago

- Discussion Type
- discussion topicassociative dialgebra
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 7 days ago