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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory kan lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology natural nforum nlab nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory type type-theory universal variational-calculus

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- Discussion Type
- discussion topicJohn Stillwell
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 26th 2021

- Discussion Type
- discussion topicAnalysis Situs
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 26th 2021

- Discussion Type
- discussion topicidempotent monoid in a monoidal category
- Category Latest Changes
- Started by Théo de Oliveira S.
- Comments 9
- Last comment by Théo de Oliveira S.
- Last Active Sep 25th 2021

Page created. Idempotent monoids should be to monoids as idempotent monads are to monads.

I’ve added the examples of idempotent elements in (ordinary) monoids (1), idempotent morphisms in categories (2), solid rings (3), idempotent monads (4), idempotent $1$-morphisms in bicategories (5), and “solid ring spectra” (6) ―What are other examples?

Also, should idempotent monoids have a unit? The examples 1 and 2 I mentioned above don’t, but 3, 4, and 6 do, while whether 5 does or doesn’t seems to vary a bit among the literature (AFAIU).

- Discussion Type
- discussion topicSamson Abramsky
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Sep 25th 2021

Added writing here

- Maximilian Doré, Samson Abramsky,
*Towards Simplicial Complexes in Homotopy Type Theory*(pdf)

- Maximilian Doré, Samson Abramsky,

- Discussion Type
- discussion topicsimplicial complex
- Category Latest Changes
- Started by Todd_Trimble
- Comments 37
- Last comment by David_Corfield
- Last Active Sep 25th 2021

Made some edits and additions at simplicial complex.

- Discussion Type
- discussion topicStandard Conjectures on Algebraic Cycles
- Category Latest Changes
- Started by Mateo Carmona
- Comments 1
- Last comment by Mateo Carmona
- Last Active Sep 25th 2021

- Discussion Type
- discussion topicdomain specific embedded programming language
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by nLab edit announcer
- Last Active Sep 25th 2021

Roux Cody kindly alerted me of the fact that analog of synthetic mathematics under “computational trinitarianism” in programming theory is that of

*domain specific embedded programming languages*. To record this neat insight I have now created a minimum entry on the latter and cross-linked a bit.

- Discussion Type
- discussion topicSL(2,H)
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Sep 25th 2021

- Discussion Type
- discussion topicMisha Gavrilovich
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 24th 2021

added “Selected writings”:

Misha Gavrilovich,

*Point set topology as diagram chasing computations*, The De Morgan Gazette. 2014. Vol. 5. No. 4. P. 23-32. (arXiv:1408.6710, pdf)Misha Gavrilovich,

*The unreasonable power of the lifting property in elementary mathematics*, 2017 (arXiv:1707.06615, pdf)

- Discussion Type
- discussion topicspectral super-scheme
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Sep 24th 2021

I have added pointers to Mikhail Kapranov’s talks on the sphere spectrum in relation to super-algebra, and added some words at the beginning that this was the original motivation for the proposed definition of spectral supergeometry in the entry.

Also I fixed the link to the video recording of Krapranov’s 2013 talk. The previous link no longer worked but there is a YouTube copy of the video. Fixed this also at

*superalgebra*, see there at Kaprananov 13

- Discussion Type
- discussion topicperiodic table
- Category Latest Changes
- Started by John Baez
- Comments 4
- Last comment by Richard Williamson
- Last Active Sep 24th 2021

- Discussion Type
- discussion topicreal homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Sep 23rd 2021

created a minimum at

*real homotopy theory*

- Discussion Type
- discussion topicHandbook of Algebraic Topology
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Sep 23rd 2021

- Discussion Type
- discussion topicMP
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Sep 23rd 2021

- Discussion Type
- discussion topiclocally compact topological space
- Category Latest Changes
- Started by Urs
- Comments 64
- Last comment by Urs
- Last Active Sep 23rd 2021

I gave

*locally compact topological space*an Idea-section and added the other equivalent definition (here).

- Discussion Type
- discussion topiccore of a ring
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Sep 23rd 2021

gave

*core of a ring*some minimum content

- Discussion Type
- discussion topiceven cohomology theory
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by Urs
- Last Active Sep 23rd 2021

- Discussion Type
- discussion topicRobert H. Szczarba
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 23rd 2021

brief

`category:people`

-entry for hyperlinking references at*real homotopy theory*and*continuous cohomology*

- Discussion Type
- discussion topicstrong monad
- Category Latest Changes
- Started by Sam Staton
- Comments 21
- Last comment by mattecapu
- Last Active Sep 23rd 2021

- Discussion Type
- discussion topicDavid Mehrle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 23rd 2021

- Discussion Type
- discussion topicRenzo A. Piccinini
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 22nd 2021

- Discussion Type
- discussion topicK-orientation
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Sep 22nd 2021

- Discussion Type
- discussion topicBorel construction
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Sep 22nd 2021

started an entry on the

*Borel construction*, indicating its relation to the nerve of the action groupoid.

- Discussion Type
- discussion topicfive lemma
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Sep 22nd 2021

touched

*five lemma*

- Discussion Type
- discussion topichomological category
- Category Latest Changes
- Started by zskoda
- Comments 7
- Last comment by Urs
- Last Active Sep 22nd 2021

New entry homological category.

- Discussion Type
- discussion topicprotomodular category
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by Urs
- Last Active Sep 22nd 2021

I added some content in protomodular category. It’s mostly drawn from Bourn’s papers. It will need brushing up, cross-linking, etc.

There are many further related concepts. Don’t know how important they are, e.g., Bourn says

The dual of a topos is arithmetical.

Is that a standard concept? And ’affine categories’?

- Discussion Type
- discussion topicMalcev category
- Category Latest Changes
- Started by Keith Harbaugh
- Comments 6
- Last comment by Urs
- Last Active Sep 22nd 2021

- Discussion Type
- discussion topicTamar Janelidze
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 22nd 2021

- Discussion Type
- discussion topicMalcev, protomodular, homological and semi-abelian categories
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Sep 22nd 2021

- Discussion Type
- discussion topichomotopy theory
- Category Latest Changes
- Started by Tim_Porter
- Comments 46
- Last comment by Urs
- Last Active Sep 22nd 2021

I have deleted an old out of date query box from homotopy theory.

- Discussion Type
- discussion topicSabrina Pasterski
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 22nd 2021

brief

`category:people`

-entry for hyperlinking references at*celestial amplitudes*

- Discussion Type
- discussion topicflat space holography
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Sep 22nd 2021

- Discussion Type
- discussion topicDaniel Grumiller
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 22nd 2021

brief

`category:people`

-entry for hyperlinking references at*flat space holography*

- Discussion Type
- discussion topicequality
- Category Latest Changes
- Started by Urs
- Comments 43
- Last comment by Urs
- Last Active Sep 21st 2021

I was involved in some discussion about where the word “intensional” as in “intensional equality” comes from and how it really differs from “intenTional” and what the point is of having such a trap of terms.

Somebody dug out Martin-Löf’s lecture notes “Intuitionistic type theory” from 1980 to check. Having it in front of me and so before I forget, I have now briefly made a note on some aspects at

*equality*in the section*Different kinds of equalits*(below the first paragraph which was there before I arrived.)Anyway, on p. 31 Martin-Löf has

intensional (sameness of meaning)

I have to say that the difference between “sameness of meaning” and “sameness of intenTion”, if that really is the difference one wants to make, is at best subtle.

- 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,