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- Discussion Type
- discussion topicindiscrete category
- Category Latest Changes
- Started by varkor
- Comments 1
- Last comment by varkor
- Last Active Mar 2nd 2023

Added link to category enriched in a bicategory.

- Discussion Type
- discussion topicspecial unitary group
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Mar 2nd 2023

added some very basic facts on $SU(2)$ here to

*special unitary group*. Just so as to be able to link to them.

- Discussion Type
- discussion topicAnthony Voutas
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 2nd 2023

- Discussion Type
- discussion topicTim Campion
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 2nd 2023

- Discussion Type
- discussion topicabstract elementary class
- Category Latest Changes
- Started by Mike Shulman
- Comments 6
- Last comment by Urs
- Last Active Mar 2nd 2023

- Discussion Type
- discussion topicEilenberg-Moore category
- Category Latest Changes
- Started by FinnLawler
- Comments 35
- Last comment by anuyts
- Last Active Mar 2nd 2023

I’ve added to Eilenberg-Moore category an explicit definition of EM objects in a 2-category and some other universal properties of EM categories, including Linton’s construction of the EM category as a subcategory of the presheaves on the Kleisli category.

Question: can anyone tell me what Street–Walters mean when they say that this construction (and their generalised one, in a 2-category with a Yoneda structure) exhibits the EM category as the ‘category of sheaves for a certain generalised topology on’ the Kleisli category?

- Discussion Type
- discussion topicequivariant stable homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by David_Corfield
- Last Active Mar 2nd 2023

I am slowly creating a bunch of entries on basic concepts of equivariant stable homotopy theory, such as

- equivariant suspension spectrum, equivariant sphere spectrum, equivariant homotopy groups, RO(G)-grading, fixed point spectrum, tom Dieck splitting

At the moment I am mostly just indexing Stefan Schwede’s

- Discussion Type
- discussion topicuniversal algebra
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by anuyts
- Last Active Mar 2nd 2023

- Discussion Type
- discussion topicXTT
- Category Latest Changes
- Started by nLab edit announcer
- Comments 13
- Last comment by jonsterling
- Last Active Mar 2nd 2023

- Discussion Type
- discussion topicquantum circuits via dependent linear types
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by David_Corfield
- Last Active Mar 2nd 2023

Am starting a write-up (here) of how (programming languages for) quantum circuits “with classical control and/by measurement” have a rather natural and elegant formulation within the linear homotopy type theory of Riley 2022.

Aspects of this have a resemblance to some constructions considered in/with “Quipper”, but maybe it helps clarify some issues there, such as that of “dynamic lifting”.

The entry is currently written without TOC and without Idea-section etc, but rather as a single top-level section that could be

`!include`

-ed into relevant entries (such as at*quantum circuit*and at*dependent linear type theory*). But for the moment I haven’t included it anywhere yet, and maybe I’ll eventually change my mind about it.

- Discussion Type
- discussion topicfactorization homology
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Mar 2nd 2023

created a stub for John Francis’ notion of

*factorization homology*.

- Discussion Type
- discussion topiccircle type
- Category Latest Changes
- Started by nLab edit announcer
- Comments 9
- Last comment by Urs
- Last Active Mar 2nd 2023

- Discussion Type
- discussion topictwistor fibration
- Category Latest Changes
- Started by Urs
- Comments 30
- Last comment by David_Chester
- Last Active Mar 1st 2023

- Discussion Type
- discussion topicSchauenburg bialgebroid
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Mar 1st 2023

- Discussion Type
- discussion topicdistributive category
- Category Latest Changes
- Started by Todd_Trimble
- Comments 28
- Last comment by Sam Staton
- Last Active Mar 1st 2023

Spurred by an MO discussion, I added the observation that coproduct inclusions are monic in a distributive category.

- Discussion Type
- discussion topicFedor Bogomolov
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Urs
- Last Active Mar 1st 2023

Created:

A mathematician at NYU Courant and HSE University Moscow.

## Selected writings

- Fedor Bogomolov, Yuri Tschinkel,
*Monodromy of elliptic surfaces*(MonodromyOfEllipticSurfaces.pdf:file)

- Fedor Bogomolov, Yuri Tschinkel,
*Introduction to birational anabelian geometry*(pdf)

- Fedor Bogomolov, Yuri Tschinkel,

- Discussion Type
- discussion topicFrank Berkshire
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 1st 2023

- Discussion Type
- discussion topicTom W. B. Kibble
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 1st 2023

- Discussion Type
- discussion topicclassical mechanics
- Category Latest Changes
- Started by Urs
- Comments 55
- Last comment by Urs
- Last Active Mar 1st 2023

I’ll try to start add some actual content to the entries classical mechanics, quantum mechanics, etc. For the time being I added a simple but good definition to classical mechanics. Of course this must eventually go with more discussion to show any value. I hope to be able to use some nice lecture notes from Igir Khavkine for this eventually.

For the time being, notice there was this old discussion box, which I am herby mving to the forum here:

–

+–{.query} Edit: I changed the above text, incorporating a part of the discussion (Zoran).

Zoran: I disagree. Classical mechanics is classical mechanics of anything: point particles, rigid bodies (the latter I already included), infinite systems (mechanics of strings, membranes, springs, elastic media, classical fields). It includes statics, not only dynamics. The standard textbooks like Goldstein take it exactly in that generality.

One could even count the simplified beginning part of the specialized branches like aerodynamics and hydrodynamics (ideal liquids for example), which are usually studied in separate courses and which in full formulation are not just mechanical systems, as the thermodynamics also affects the dynamics. There are also mechanical models of dissipative systems, where the dissipative part is taken only phenomenologically, e.g. as friction terms. Hydrodynamics can also be considered as a part of rheology.

*Toby*: I take your point that ’dynamics’ was not the right word. But do you draw any distinction between ’classical mechanics’ and ’classical physics’? Conversely, what word*would*you use to restrict attention to particles instead of fields, if not ’mechanics’? (Incidentally, I would take point particles as possibly spinning, although I agree that I should not assume that the particle are points anyway.)*Zoran*: you see, in classical mechanics you express all you have by attaching mass, position, velocity etc. to the parfts of mechanical systems. Not all classical physics belongs to this kind of description. The thermodynamical quantities may influence the motion of the systemm, but their description is out of the frame of classical mechanics. If you study liquids you have to take into account both the classical mechanics of the liquid continuum but also variations of its temperature, entropy and so on, which are not expressable within the variables of mechanics. Formally speaking of course, the thermodynamics has very similar formal structure as mechanics, for example Gibbs and Helmholtz free energies and enthalpy are like Lagrangean, the quantities which are extremized when certain theremodynamical quantities are kept constant. To answer the terminological question, there is a classical mechanics of point particles and it is called classical mechanics of point particles, there is also cm of fields and cm of rigid bodies.*Toby*: So ’mechanics’ for you means ‹not taking into account thermal physics›? That's not the way that I learned it! But I admit that I do not have a slick phrase for that (any more than you have a slick phrase for ‹mechanics of point particles›), so I will try to ascertain how the term is usually used and defer to that. =–

- Discussion Type
- discussion topicwave function collapse
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Mar 1st 2023

I have added to

*wave function collapse*its relation to the expression for conditional expectation values in quantum probability: here (e.g. Kuperberg 05, section 1.2, Yuan 12)

- Discussion Type
- discussion topicadjoint modality
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 1st 2023

added pointer to p. 245 of

$U X \;\colon\; \array{ comodal X &\longrightarrow& X &\longrightarrow& modal X \\ opposite\;1 && unity && opposite\;2 }$*Sets for Mathematics*for the idea of

- Discussion Type
- discussion topicDorje C. Brody
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 1st 2023

- Discussion Type
- discussion topicLane P. Hughston
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 1st 2023

- Discussion Type
- discussion topicEfimov K-theory
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Feb 28th 2023

## Idea

A generalization of Waldhausen K-theory to dualizable dg-categories and dualizable stable ∞-categories.

For compactly generated inputs, recovers the Waldhausen K-theory of the full subcategory of compact objects.

The formalism is applicable to $\lambda$-presentable stable ∞-categories, where $\lambda$ can be uncountable (for example, various categories of sheaves, or categories occurring in functional analysis).

## References

Alexander Efimov,

*On the K-theory of large triangulated categories*, ICM 2022, https://www.youtube.com/watch?v=RUDeLo9JTroMarc Hoyois,

*K-theory of dualizable categories (after A. Efimov)*, https://hoyois.app.uni-regensburg.de/papers/efimov.pdf.Li He,

*Efimov K-theory and universal localizing invariant*, arXiv:2302.13052.

- Discussion Type
- discussion topiclax-idempotent 2-monad
- Category Latest Changes
- Started by John Baez
- Comments 3
- Last comment by anuyts
- Last Active Feb 28th 2023

I corrected an apparent typo:

A 2-monad $T$ as above is lax-idempotent if and only if for any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\theta_a \colon 1 \Rightarrow \eta \circ a$

to

A 2-monad $T$ as above is lax-idempotent if and only if for any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\theta_a \colon 1 \Rightarrow \eta_A \circ a$

It might be nice to say $\eta_A$ is the unit of the algebra….

- Discussion Type
- discussion topiclax-idempotent 2-adjunction
- Category Latest Changes
- Started by Mike Shulman
- Comments 5
- Last comment by anuyts
- Last Active Feb 28th 2023

I’ve wondered for a while whether there is a notion of lax-idempotent 2-adjunction, but for some reason until now I’d never thought to try the obvious route of simply generalizing the conditions defining an idempotent adjunction. Haven’t had time to cross-link it yet.

- Discussion Type
- discussion topicJ. Roger Hindley
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 28th 2023

- Discussion Type
- discussion topiccombinatory logic
- Category Latest Changes
- Started by Mike Shulman
- Comments 12
- Last comment by Urs
- Last Active Feb 28th 2023

I rescued combinatory logic from being a “my first slide” spam and gave it some content, mainly to record the fact (which I just learned) that under propositions as types, combinatory logic corresponds to a Hilbert system.

I feel like there should be something semantic to say here too, like $\lambda$-calculus corresponding to a “closed, unital, cartesian multicategory” (a cartesian multicategory that is “closed and unital” as in the second example here) and combinatory logic corresponding to a closed category that is also “cartesian” in some sense. Has anyone defined such a sense?

Relatedly, is there a notion of “linear combinatory logic” that would correspond to ordinary (symmetric) closed categories? My best guess is that instead of $S$ and $K$ you would have combinators with the following types:

$(B\to C) \to (A\to B) \to (A\to C)$ $(A \to (B\to C)) \to B \to A\to C$coming from the two ways to eliminate a dependency in $S$ to make it linear ($K$ is irreducibly nonlinear). These are of course the ways that you express composition and symmetry in a closed category.

- Discussion Type
- discussion topicsimple type theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by Urs
- Last Active Feb 28th 2023

- Discussion Type
- discussion topiclambda-calculus
- Category Latest Changes
- Started by DavidRoberts
- Comments 3
- Last comment by Urs
- Last Active Feb 28th 2023

- Discussion Type
- discussion topicNew Spaces for Mathematics and Physics
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by David_Corfield
- Last Active Feb 28th 2023

I have started a category:reference page

such as to be able to point to it for reference, e.g. from Kontsevich 15 etc.

- Discussion Type
- discussion topicCorina Keller
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 28th 2023

- Discussion Type
- discussion topicquantization of M2-brane to matrix model -- references
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Feb 28th 2023

- Discussion Type
- discussion topicAeysha Khalique
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 28th 2023

- Discussion Type
- discussion topicquantum teleportation
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Feb 28th 2023

- Discussion Type
- discussion topictwisted generalized cohomology in linear homotopy-type theory
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by David_Corfield
- Last Active Feb 27th 2023

I’d be trying to write out a more detailed exposition of how fiber integration in twisted generalized cohomology/twisted Umkehr maps are axiomaized in linear homotopy-type theory.

To start with I produced a dictionary table, for inclusion in relevant entries:

- Discussion Type
- discussion topicGiacomo Tendas
- Category Latest Changes
- Started by BryceClarke
- Comments 2
- Last comment by Urs
- Last Active Feb 27th 2023

- Discussion Type
- discussion topicBob Walters
- Category Latest Changes
- Started by John Baez
- Comments 5
- Last comment by Urs
- Last Active Feb 27th 2023

- Discussion Type
- discussion topicEmil Génetay-Johansen
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 27th 2023

- Discussion Type
- discussion topicTapio Simula
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 27th 2023

- Discussion Type
- discussion topicSolovay-Kitaev theorem
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Feb 27th 2023

- Discussion Type
- discussion topicsmooth Serre-Swan theorem
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active Feb 27th 2023

- Discussion Type
- discussion topicMorita equivalence
- Category Latest Changes
- Started by Thomas Holder
- Comments 18
- Last comment by Dmitri Pavlov
- Last Active Feb 27th 2023

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 topicNikita Kolganov
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 26th 2023

- Discussion Type
- discussion topicqudit
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Feb 26th 2023

- Discussion Type
- discussion topicDmitry Melnikov
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 26th 2023

- Discussion Type
- discussion topicAndrei Mironov
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 26th 2023

- Discussion Type
- discussion topicAlexei Morozov
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 26th 2023

- Discussion Type
- discussion topicAndrey Morozov
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 26th 2023

- Discussion Type
- discussion topicSergey Mironov
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 26th 2023

- Discussion Type
- discussion topicsu(2)-anyon
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Feb 26th 2023

- Discussion Type
- discussion topicconcrete sheaf
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Dmitri Pavlov
- Last Active Feb 26th 2023

expanded concrete sheaf: added the precise definition and some important properties.

- Discussion Type
- discussion topicfunctorial aspects of the GNS representation
- Category Latest Changes
- Started by nLab edit announcer
- Comments 12
- Last comment by Tom Mainiero
- Last Active Feb 26th 2023

- Discussion Type
- discussion topicbraid group
- Category Latest Changes
- Started by Urs
- Comments 35
- Last comment by Urs
- Last Active Feb 26th 2023

the entry

*braid group*said what a braid is, but forgot to say what the braid group is; I added in a sentence, right at the beginning (and fixed some other minor things).

- Discussion Type
- discussion topicshape via cohesive path ∞-groupoid
- Category Latest Changes
- Started by Urs
- Comments 57
- Last comment by Dmitri Pavlov
- Last Active Feb 25th 2023

I have added pointer to the arXiv copy to the item

- Daniel Berwick-Evans, Pedro Boavida de Brito, Dmitri Pavlov,
*Classifying spaces of infinity-sheaves*(arXiv:1912.10544)

- Daniel Berwick-Evans, Pedro Boavida de Brito, Dmitri Pavlov,

- Discussion Type
- discussion topichomotopy group
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Feb 25th 2023

I have added to homotopy group a very brief pointer to Mike’s HoTT formalization of $\pi_1(S^1)$.

Eventually I would like to have by default our $n$Lab entries be equipped with detailed pointers to which aspects have been formalized in HoTT (if they have), and in which .v-file precisely.

- Discussion Type
- discussion topichomotopy theory and algebraic topology -- references
- Category Latest Changes
- Started by Urs
- Comments 29
- Last comment by Urs
- Last Active Feb 25th 2023

following discussion here I am starting an entry with a bare list of references (sub-sectioned), to be

`!include`

-ed into the References sections of relevant entries (mainly at*homotopy theory*and at*algebraic topology*) for ease of updating and syncing these lists.The organization of the subsections and their items here needs work, this is just a start. Let’s work on it.

I’ll just check now that I have all items copied, and then I will

`!include`

this entry here into*homotopy theory*and*algebraic topology*. It may best be*viewed*withing these entries, because there – but not here – will there be a table of contents showing the subsections here.

- Discussion Type
- discussion topicLectures on Étale Cohomology
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Feb 25th 2023

added jstor:j.ctt1bpmbk1

- Discussion Type
- discussion topicbraid group representations -- references
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Feb 25th 2023

a bare list of references, to be

`!include`

-ed both at*braid group*and at*topological quantum computation*, for ease of updating

- Discussion Type
- discussion topico-minimal structure
- Category Latest Changes
- Started by Todd_Trimble
- Comments 10
- Last comment by perezl.alonso
- Last Active Feb 25th 2023

I started some short articles on o-minimal structure and structure (model theory).