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- Discussion Type
- discussion topicThomason-type model structure
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 5
- Last comment by Dmitri Pavlov
- Last Active Sep 20th 2021

Initial version:

## Idea

Thomason-type model categories provide simple 1-categorical models for (∞,1)-categorical objects.

The provide a particularly convenient setting for results like Quillen’s Theorem A and Theorem B.

## Examples

|(∞,1)-categorical structure|1-categorical structure|model structure| |∞-groupoid|category|Thomason model structure| |∞-groupoid|poset|model structure on posets| |(∞,1)-category|relative category|Barwick–Kan model structure| |connective spectra|symmetric monoidal groupoid|Fuentes-Keuthan model structure|

## Related concepts

## References

[…]

- Discussion Type
- discussion topicidempotent complete (infinity,1)-category
- Category Latest Changes
- Started by Mike Shulman
- Comments 18
- Last comment by Théo de Oliveira S.
- Last Active Sep 19th 2021

At first Zoran's reply to my query at structured (infinity,1)-topos sounded as though he were saying "being idempotent-complete" were a structure on an (oo,1)-category rather than just a property of it. That had me worried for a while. It looks, though, like what he meant is that "being idempotent" is structure rather than a property, and that makes perfect sense. So I created idempotent complete (infinity,1)-category.

- Discussion Type
- discussion topicmonoidal functor
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Sep 19th 2021

created (finally) lax monoidal functor (redirecting monoidal functor to that) and strong monoidal functor.

Hope I got the relation to 2-functors right. I remember there was some subtlety to be aware of, but I forget which one. I could look it up, but I guess you can easily tell me.

- Discussion Type
- discussion topicgeometric realization of simplicial topological spaces
- Category Latest Changes
- Started by Urs
- Comments 97
- Last comment by Urs
- Last Active Sep 19th 2021

started an entry geometric realization of simplicial topological spaces.

I decided this is a topic big enough to justify splitting it off from geometric realization (of simplicial sets).

But not much there yet. I just wanted to record for the moment that this realization too, does preserve pullbacks.

- Discussion Type
- discussion topicBachir Bekka
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 19th 2021

- Discussion Type
- discussion topiccoset space coprojection admitting local sections
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Sep 19th 2021

am giving this its own entry, in order to record sufficient conditions on topological subgroup inclusions $H \subset G$ for the coset space coprodoction $G \to G/H$ to admit local sections.

So far I have two original references here (Gleason 50, Mostert 53). One should add some textbook account, too.

Am also referencing this at

*closed subgroup*, at*coset space*and maybe elsewhere.

- Discussion Type
- discussion topicenriched category
- Category Latest Changes
- Started by Mike Shulman
- Comments 16
- Last comment by varkor
- Last Active Sep 19th 2021

I added two recent examples of enriched categories: tangent bundle categories and Lawvere theories.

- Discussion Type
- discussion topicFabian Hebestreit
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 19th 2021

brief

`category:people`

-entry for hyperlinking references at*twisted K-theory*

- Discussion Type
- discussion topicwell-pointed topological space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 19th 2021

- Discussion Type
- discussion topicquantum circuit diagram
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Sep 19th 2021

- Discussion Type
- discussion topicJoachim Hilgert
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 19th 2021

- Discussion Type
- discussion topicBanach manifold
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 19th 2021

- Discussion Type
- discussion topicabsolute retract
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Sep 19th 2021

splitting off this definition from

*neighborhood retract*, for ease of linking, and in order to record the characterization AR = ANR+contractible

- Discussion Type
- discussion topicLeibniz algebra
- Category Latest Changes
- Started by zskoda
- Comments 18
- Last comment by Urs
- Last Active Sep 18th 2021

Many additions and changes to Leibniz algebra. The purpose is to outline that the (co)homology and abelian and even nonabelian extensions of Leibniz algebras follow the same pattern as Lie algebras. One of the historical motivations was that the Lie algebra homology of matrices which lead Tsygan to the discovery of the (the parallel discovery by Connes was just a stroke of genius without an apparent calculational need) cyclic homology. Now, if one does the Leibniz homology instead then one is supposedly lead the same way toward the Leibniz homology (for me there are other motivations for Leibniz algebras, including the business of double derivations relevant for the study of integrable systems).

Matija and I have a proposal how to proceed toward candidates for Leibniz groups, that is an integration theory. But the proposal is going indirectly through an algebraic geometry of Lie algebras in Loday-Pirashvili category. Maybe Urs will come up with another path if it drags his interest.

- Discussion Type
- discussion topictopology
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Sep 18th 2021

added pointer to

- Michael Müger,
*Topology for the working mathematician*, Nijmegen 2018 (pdf)

- Michael Müger,

- Discussion Type
- discussion topichomotopy coend
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Dmitri Pavlov
- Last Active Sep 18th 2021

added pointer to:

- Sergey Arkhipov, Sebastian Ørsted,
*Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories*(arXiv:1807.03266)

- Sergey Arkhipov, Sebastian Ørsted,

- Discussion Type
- discussion topicBernard Bolzano
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Sep 18th 2021

- Discussion Type
- discussion topicKan complex
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Urs
- Last Active Sep 18th 2021

added a sentence to the Idea-section at Kan complex

- Discussion Type
- discussion topicaxiom of full comprehension
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by DavidRoberts
- Last Active Sep 18th 2021

- Discussion Type
- discussion topichomotopy weighted colimit
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 7
- Last comment by Tim_Porter
- Last Active Sep 18th 2021

- Discussion Type
- discussion topicSergey Arkhipov
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 18th 2021

- Discussion Type
- discussion topicSebastian Ørsted
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 18th 2021

- Discussion Type
- discussion topicFrederic Fitch
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 18th 2021

brief

`category:people`

-entry for hyperlinking references at*Curry’s pardox*and*axiom of full comprehension*

- Discussion Type
- discussion topicHaag–Łopuszański–Sohnius theorem
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Urs
- Last Active Sep 18th 2021

- Discussion Type
- discussion topicMartin Sohnius
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 18th 2021

brief

`category:people`

-entry for hyperliunking references at*Haag-Łopuszański-Sohnius theorem*

- Discussion Type
- discussion topicenhanced triangulated category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Sep 17th 2021

there were no references here. I have added one:

- Alexei Bondal, Mikhail Kapranov,
*Enhanced Triangulated Categories*, Sbornik: Mathematics, Volume 70, Issue 1, pp. 93-107 (1991). doi:10.1070/SM1991v070n01ABEH001253

- Alexei Bondal, Mikhail Kapranov,

- Discussion Type
- discussion topicstable (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Sep 17th 2021

- I edited stable (infinity,1)-category a bit:

* rephrased the intro part, trying to make it more forcefully to the point (not claiming to have found the optimum, though)

* added a dedicated section <a href="http://ncatlab.org/nlab/show/stable+(infinity%2C1)-category#the_homotopy_category_of_a_stable_category_triangulated_categories_7">The homotopy cat of a stable (oo,1)-cat: traingulated categories</a> to highlight the important statement here, which was previously a bit hidden in the main text.

- Discussion Type
- discussion topicStein manifold
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Sep 17th 2021

have added a tad more content to

*Stein manifold*and cross-linked a bit more

- Discussion Type
- discussion topicsimplicial topological group
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Sep 16th 2021

have started simplicial topological group

- Discussion Type
- discussion topicopen problem of confinement -- references
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Sep 16th 2021

a bare sub-section with a list of references – to be

`!included`

into relevant entries – mainly at*confinement*and at*mass gap problem*(where this list already used to live)

- Discussion Type
- discussion topicmonadicity theorem
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Sep 16th 2021

at monadicity theorem in the second formulation of the theorem, item 3, it said

$C$ has

I think it must be

$D$ has

and have changed it accordingly. But have a look.

- Discussion Type
- discussion topicstate on a star-algebra
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Sep 16th 2021

I have expanded the Idea section at

*state on a star-algebra*and added a bunch of references.The entry used to be called “state on an operator algebra”, but I renamed it (keeping the redirect) because part of the whole point of the definition is that it makes sense without necessarily having represented the “abstract” star-algebra as a C*-algebra of linear operators.

- Discussion Type
- discussion topicdouble functor
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by mattecapu
- Last Active Sep 16th 2021

- Discussion Type
- discussion topicMoritz Groth
- Category Latest Changes
- Started by Marc
- Comments 1
- Last comment by Marc
- Last Active Sep 16th 2021

- Discussion Type
- discussion topic(infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 23
- Last comment by Marc
- Last Active Sep 16th 2021

I worked on brushing up (infinity,1)-category a little

mostly I added in a section on homotopical categories, using some paragraphs from Andre Joyal's message to the CatTheory mailing list.

in this context I also rearranged the order of the subsections

I removed in the introduction the link to the page "Why (oo,1)-categories" and instead expanded the Idea section a bit.

added a paragraph to the beginning of the subsection on model categories

added the new Dugger/Spivak references on the relation between quasi-cats and SSet-cats (added that also to quasi-category and to relation between quasi-categories and simplicial categories)

- Discussion Type
- discussion topicHilbert space
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Sep 16th 2021

there had been no references at

*Hilbert space*, I have added the following, focusing on the origin and application in quantum mechanics:John von Neumann,

*Mathematische Grundlagen der Quantenmechanik*. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.George Mackey,

*The Mathematical Foundations of Quamtum Mechanics*A Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963E. Prugovecki,

*Quantum mechanics in Hilbert Space*. Academic Press, 1971.

- Discussion Type
- discussion topicHilb
- Category Latest Changes
- Started by David_Corfield
- Comments 3
- Last comment by Urs
- Last Active Sep 16th 2021

There’s a paper out characterising the category of continuous linear functions between Hilbert spaces

- Chris Heunen, Andre Kornell,
*Axioms for the category of Hilbert spaces*(arXiv:2109.07418)

But Hilb concerns short linear maps between Hilbert spaces. Should we have a page for the former category?

- Chris Heunen, Andre Kornell,

- Discussion Type
- discussion topictopological string
- Category Latest Changes
- Started by zskoda
- Comments 8
- Last comment by Urs
- Last Active Sep 15th 2021

Stub for topological string with redirect topological string theory.

- Discussion Type
- discussion topicDavid John Simms
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Tim_Porter
- Last Active Sep 15th 2021

- Discussion Type
- discussion topichomotopy extension property
- Category Latest Changes
- Started by zskoda
- Comments 8
- Last comment by Urs
- Last Active Sep 15th 2021

I created homotopy extension property and homotopy lifting property. If somebody wonders why I made identical copy of one of them on my personal nlab part is because there I want to keep conservative page for students and here in the main nlab I expect more vigorous extensions by others. On the other hand, I would like to have under homotopy lifting property mention of various variants like "soft map" homotopy lifting property, the homotopical variant of Dold etc. all in one place.

- Discussion Type
- discussion topicnearby homomorphisms from compact Lie groups are conjugate
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Sep 15th 2021

- Discussion Type
- discussion topicGudrun Hiller
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 15th 2021

brief

`category:people`

-entry for hyperlinking references at*flavour anomaly*

- Discussion Type
- discussion topictoo simple to be simple
- Category Latest Changes
- Started by Oscar_Cunningham
- Comments 40
- Last comment by TobyBartels
- Last Active Sep 14th 2021

(Hi, I’m new)

I added some examples relating too simple to be simple to the idea of unbiased definitions. The point is that we often define things to be simple whenever they are not a non-trivial (co)product of two objects, and we can extend this definition to cover the “to simple to be simple case” by removing the word “two”. The trivial object is often the

*empty*(co)product. If we had been using an unbiased definition we would have automatically covered this case from the beginning.I also noticed that the page about the empty space referred to the naive definition of connectedness as being

“a space is connected if it cannot be partitioned into disjoint nonempty open subsets”

but this misses out the word “two” and so is accidentally giving the sophisticated definition! I’ve now corrected it to make it wrong (as it were).

- Discussion Type
- discussion topicquasi-category
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by DavidRoberts
- Last Active Sep 14th 2021

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

- Discussion Type
- discussion topicfine model structure on topological G-spaces
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Sep 14th 2021

am finally giving this its own entry (this used to be treated within the entry on

*Elmendorf’s theorem*)but just a stub for the moment

- Discussion Type
- discussion topicRiemann hypothesis and physics
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Sep 14th 2021

started to collect some references at

*Riemann hypothesis and physics*. But just a puny start so far, have to quit now.

- Discussion Type
- discussion topic1-category equipped with relations
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Sep 13th 2021

- Discussion Type
- discussion topiclinear logic
- Category Latest Changes
- Started by Mike Shulman
- Comments 22
- Last comment by nLab edit announcer
- Last Active Sep 13th 2021

The cut rule for linear logic used to be stated as

If $\Gamma \vdash A$ and $A \vdash \Delta$, then $\Gamma \vdash \Delta$.

I don’t think this is general enough, so I corrected it to

If $\Gamma \vdash A, \Phi$ and $\Psi,A \vdash \Delta$, then $\Psi,\Gamma \vdash \Delta,\Phi$.

- Discussion Type
- discussion topicG-CW complex
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Sep 13th 2021

started

*G-CW complex*.

- Discussion Type
- discussion topicEL-∞ algebra
- Category Latest Changes
- Started by Luigi
- Comments 1
- Last comment by Luigi
- Last Active Sep 13th 2021

- Discussion Type
- discussion topicπ-Kan condition
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 13th 2021

- Discussion Type
- discussion topicIntroduction to Topology -- 2
- Category Latest Changes
- Started by nLab edit announcer
- Comments 6
- Last comment by nLab edit announcer
- Last Active Sep 13th 2021

- Discussion Type
- discussion topicPawel Sobocinski
- Category Latest Changes
- Started by RodMcGuire
- Comments 1
- Last comment by RodMcGuire
- Last Active Sep 12th 2021

updated 2012 webpage from

to

- Discussion Type
- discussion topicsheaf of meromorphic functions
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Sep 12th 2021

Adeel Khan created

*sheaf of meromorphic functions*.(He currently has problems logging into here, that’s why I am posting this for the moment.)

- Discussion Type
- discussion topicMaster course on algebraic stacks
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active Sep 12th 2021

- Discussion Type
- discussion topiccovariant derivative
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Dmitri Pavlov
- Last Active Sep 11th 2021

wrote a definition and short discussion of covariant derivative in the spirit of oo-Chern-Weil theory

- Discussion Type
- discussion topicadhesive category
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by nLab edit announcer
- Last Active Sep 11th 2021

- Discussion Type
- discussion topicclassifying topos
- Category Latest Changes
- Started by Urs
- Comments 56
- Last comment by JonasFrey
- Last Active Sep 11th 2021

edited classifying topos and added three bits to it. They are each marked with a comment "check the following".

This is in reaction to a discussion Mike and I are having with Richard Williamson by email.

- Discussion Type
- discussion topicJoachim Kock
- Category Latest Changes
- Started by David_Corfield
- Comments 2
- Last comment by Tim_Porter
- Last Active Sep 11th 2021

Added a couple more papers.

- Imma Gálvez-Carrillo, Joachim Kock, and Andrew Tonks.
*Homotopy linear algebra*. Proc. Roy. Soc. Edinburgh Sect. A, 148(2):293–325, 2018 (arXiv:1602.05082).

looks interesting as a way to take linear algebra in a slightly alternative way to groupoidification.

- Imma Gálvez-Carrillo, Joachim Kock, and Andrew Tonks.

- Discussion Type
- discussion topicAdjointness for 2-Categories
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by David_Corfield
- Last Active Sep 11th 2021

Slighly adjusted the lead-in of this page: Added a line on what the book is actually about, and moved the line advertizing how great this is to after that.

Added the line:

The approach is echoed in Riehl & Verity 13 with Cat enhanced to the homotopy 2-category of (∞,1)-categories.

Also touched some wording further below (“is very difficult to read” $\mapsto$ “may be difficult to read”)

and added the missing cross-link with

*John Gray*(1)Finally, I made “formal category theory” a redirect to this page (this would deserve to point to a dedicated page, but as long as that doesn’t exit, it’s good to have it redirect here)