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- Discussion Type
- discussion topicalgebraic topology
- Category Latest Changes
- Started by Urs
- Comments 23
- Last comment by Urs
- Last Active Sep 21st 2021

I have tried to give

*algebraic topology*a better Idea-section.

- Discussion Type
- discussion topicCarlos Prieto
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 21st 2021

- Discussion Type
- discussion topicMarcelo Aguilar
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 21st 2021

- Discussion Type
- discussion topicJeffrey Strom
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 21st 2021

brief

`category:people`

-entry for hyperlinking references at*homotopy theory*

- Discussion Type
- discussion topicMonoidal Functors, Species and Hopf Algebras
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Sep 21st 2021

the book Monoidal Functors, Species and Hopf Algebras is very good, but still being written. Clearly the current link under which it is found on the web is not going to be the permanent link. So I thought it is a bad idea to link to it directly. Instead I created that page now which we can reference then from nLab entries. When the pdf link changes, we only need to adapt it at that single page.

- Discussion Type
- discussion topicinternal category in a monoidal category
- Category Latest Changes
- Started by FinnLawler
- Comments 6
- Last comment by Urs
- Last Active Sep 21st 2021

I came across the page internal category in a monoidal category, which was lacking even a definition, so I put one in.

- Discussion Type
- discussion topicAtiyah-Singer index theorem
- Category Latest Changes
- Started by David_Corfield
- Comments 8
- Last comment by David_Corfield
- Last Active Sep 21st 2021

Where the page has

The index theorem is supposed to have an interpretation in terms of the quantum field theory of the superparticle on the given space,

is the “is supposed to” necessary? Why not “has an interpretation”? Is it just the general issue of any translation from mathematics to physics?

- Discussion Type
- discussion topicmodel category theory - contents
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Sep 21st 2021

started model category theory - contents and added this as floating toc to relevant entries

- Discussion Type
- discussion topicmodel structure on Delta-generated topological spaces
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Sep 21st 2021

am giving this its own entry for ease of hyperlinking, split off from

*Delta-generated space*

- Discussion Type
- discussion topicsemigroup
- Category Latest Changes
- Started by nLab edit announcer
- Comments 8
- Last comment by RodMcGuire
- Last Active Sep 20th 2021

- Discussion Type
- discussion topicHahn-Banach theorem
- Category Latest Changes
- Started by DavidJaz
- Comments 1
- Last comment by DavidJaz
- Last Active Sep 20th 2021

- Discussion Type
- discussion topicHurewicz cofibration
- Category Latest Changes
- Started by DavidRoberts
- Comments 29
- Last comment by Urs
- Last Active Sep 20th 2021

Gave proper reference for (Kieboom 1987).

- Discussion Type
- discussion topicBlakers-Massey theorem
- Category Latest Changes
- Started by Urs
- Comments 31
- Last comment by Urs
- Last Active Sep 20th 2021

stub for

*Blakers-Massey theorem*. Need to add more references…

- Discussion Type
- discussion topiccompact-open topology
- Category Latest Changes
- Started by Urs
- Comments 30
- Last comment by Urs
- Last Active Sep 20th 2021

Did anyone ever write out on the $n$Lab the proof that for $X$ locally compact and Hausdorff, then $Map(X,Y)$ with the compact-open topology is an exponential object? (Many entries mention this, but I don’t find any that gets into details.)

I have tried to at least add a pointer in the entry to places where the proof is given. There is prop. 1.3.1 in

- Marcelo Aguilar, Samuel Gitler, Carlos Prieto, sections 1.2, 1.3 of
*Algebraic topology from a homotopical viewpoint*, Springer (2002) (toc pdf)

but of course there are more canonical references. I also added pointer to

- Eva Lowen-Colebunders, Günther Richter,
*An Elementary Approach to Exponential Spaces*, Applied Categorical Structures May 2001, Volume 9, Issue 3, pp 303-310 (publisher)

- Marcelo Aguilar, Samuel Gitler, Carlos Prieto, sections 1.2, 1.3 of

- Discussion Type
- discussion topicVera Serganova
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 20th 2021

- Discussion Type
- discussion topicMark Haiman
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 20th 2021

- Discussion Type
- discussion topicquantum decoherence
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Sep 20th 2021

added pointer to today’s

- Chris Nagele, Oliver Janssen, Matthew Kleban,
*Decoherence: A Numerical Study*(arXiv:2010.04803)

- Chris Nagele, Oliver Janssen, Matthew Kleban,

- Discussion Type
- discussion topicStrøm model structure
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by Urs
- Last Active Sep 20th 2021

Is the Strøm model category left proper? I know that pushout along cofibrations of homotopy equivalences of the form $A \to \ast$ are again homotopy equivalences. (e.g. Hatcher 0.17) Maybe the proof directly generalizes, haven’t checked.

- Discussion Type
- discussion topicA Concise Course in Algebraic Topology
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Sep 20th 2021

- Discussion Type
- discussion topicKuiper's theorem
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Sep 20th 2021

following a suggestion by Zoran, I have created a stub (nothing more) for Kuiper’s theorem

- Discussion Type
- discussion topicU(ℋ)
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Sep 20th 2021

- Discussion Type
- discussion topicHopf fibration
- Category Latest Changes
- Started by Urs
- Comments 36
- Last comment by DavidRoberts
- Last Active Sep 20th 2021

- Discussion Type
- discussion topicphysical unit
- Category Latest Changes
- Started by Urs
- Comments 19
- Last comment by Urs
- Last Active Sep 20th 2021

started some remarks at

*physical unit*. But I really need to stop with that now and do more urgent things…

- Discussion Type
- discussion topicunit
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Sep 20th 2021

The entry used to start out with the line “not to be confused with neutral element”. This was rather suboptimal. I have removed that sentence and instead expanded the Idea-section to read now as follows:

Considering a ring $R$, then by

*the unit element*one usually means the neutral element $1 \in R$ with respect to multiplication. This is the sense of “unit” in terms such as nonunital ring.But more generally

*a unit element*in a unital (!) ring is any element that has an inverse element under multiplication.This concept generalizes beyond rings, and this is what is discussed in the following.

- Discussion Type
- discussion topicRobert Loren Jaffe
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 20th 2021

- Discussion Type
- discussion topicParametrized Homotopy Theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 20th 2021

In view of discussion in another thread (here), I have added (here) the following warning:

Beware that section 4.4 claims a new proof of the Strøm model structure, but relying on a statement in

- Michael Cole,
*Many homotopy categories are homotopy categories*, Topology and its Applications 153 (2006) 1084–1099 (doi:10.1016/j.topol.2005.02.006)

which later was noticed to be false, by Richard Williamson, for details see p. 2 and Rem 5.12 and Sec. 6.1 in:

- Tobias Barthel, Emily Riehl,
*On the construction of functorial factorizations for model categories*, Algebr. Geom. Topol. 13 (2013) 1089-1124 (arXiv:1204.5427, doi:10.2140/agt.2013.13.1089, euclid:agt/1513715550)

It remains unclear, to me anyways, what this implies for the sliced generalization which is the core claim of May & Sigurdsson’s book.(?)

- Michael Cole,

- 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 topicPU(ℋ)
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Sep 19th 2021

for completeness, to go with

*U(ℋ)*, for the moment mainly in order to record references, such as:- David John Simms,
*Topological aspects of the projective unitary group*, Math. Proc. Camb. Phil. Soc.**68**1 (1970) 57-60 (doi:10.1017/S0305004100001043)

- David John Simms,

- 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 topicBorel model structure
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Sep 17th 2021

- 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.