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- Discussion Type
- discussion topic(infinity,1)-category of (infinity,1)-functors
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active May 15th 2023

- Discussion Type
- discussion topicYves Diers
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active May 15th 2023

brief

`category:people`

-entry for hyperlinking references at*spectrum (geometry)*and at*multi-adjoint*

- Discussion Type
- discussion topicpseudoform
- Category Latest Changes
- Started by gregprice
- Comments 1
- Last comment by gregprice
- Last Active May 15th 2023

Created this page partly based on material found within differential+form, and partly on related other pages and the linked Usenet thread. The main reason I wanted a separate page is to have a good home for some of the examples and other material in that thread that hadn’t already made it onto the nLab.

It’d be good to add discussion of pseudoforms’ behavior under wedge product and differentiation, and more details on the examples: in particular explaining the notion of absolute value, and why the volume is a pseudoform and not an untwisted form.

- Discussion Type
- discussion topicHomotopical Algebra
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 7
- Last comment by Dmitri Pavlov
- Last Active May 14th 2023

- Discussion Type
- discussion topicmodel structure on functors
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active May 14th 2023

expanded model structure on functors by adding a long list of properties

- Discussion Type
- discussion topicfunctorial factorization
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active May 14th 2023

I gave

*functorial factorization*its own little entry, for ease of pointing to the precise definition.This is for the moment just copied over from the corresponding paragraph at

*weak factorization system*(where I have re-organized the sectzion outline slightly, for clarity). Also I added cross-links with some relevant entries.

- Discussion Type
- discussion topichomotopy category of a model category
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active May 14th 2023

I am splitting off

*homotopy category of a model category*from*model category*. Have spelled out statement and proof of the localization construction there.

- Discussion Type
- discussion topicbifibrant object
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 14th 2023

- Discussion Type
- discussion topicMartin Speirs
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 14th 2023

- Discussion Type
- discussion topicpseudoorientation
- Category Latest Changes
- Started by gregprice
- Comments 1
- Last comment by gregprice
- Last Active May 14th 2023

This gives the idea and definition of a pseudoorientation, and the key properties related to integration. The definition is taken from integration of differential forms, and the rest is largely from my reading of several of Toby’s messages in the linked Usenet thread.

It’d be good to add some examples, particularly with n=2 and n=3, including in the electromagnetism context where one traditionally uses a right-hand rule to conflate orientations with pseudoorientations (and 2- with 1- and pseudo- with untwisted forms). A couple of the messages in that thread had some nice examples, but I don’t have the specific messages in front of me at the moment. I may return to this in the next few days to try to track those down.

- Discussion Type
- discussion topicsimplicial group
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by Urs
- Last Active May 14th 2023

at simplicial group I added/expanded the section delooping and simplicial principal bundles

I discuss this in revisionistic terms meant to exhibit the simple general underlying structure, and then try to spell out how it corresponds to vaarious explicit constructions in the literature, trying to point out page and verse in May’s “Simplicial objects in algebraic topology” and discuss how that yields what I am discussing.

By the way, did anyone ever find the time to make a sanity check of my query-box claim at decalage that forming decalage in sSet is nothing but forming the standard based path space object?

- Discussion Type
- discussion topicmanifold-diagrammatic n-category
- Category Latest Changes
- Started by Christoph Dorn
- Comments 33
- Last comment by Christoph Dorn
- Last Active May 13th 2023

- Discussion Type
- discussion topichomotopy coequalizer
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 3
- Last comment by Urs
- Last Active May 13th 2023

- Discussion Type
- discussion topicdirected topological space
- Category Latest Changes
- Started by nLab edit announcer
- Comments 11
- Last comment by Guest
- Last Active May 13th 2023

- Discussion Type
- discussion topicmanifold diagram
- Category Latest Changes
- Started by Christoph Dorn
- Comments 7
- Last comment by Christoph Dorn
- Last Active May 12th 2023

- Discussion Type
- discussion topicn-truss
- Category Latest Changes
- Started by Christoph Dorn
- Comments 3
- Last comment by Christoph Dorn
- Last Active May 12th 2023

- Discussion Type
- discussion topicn-mesh
- Category Latest Changes
- Started by Christoph Dorn
- Comments 5
- Last comment by Christoph Dorn
- Last Active May 12th 2023

- Discussion Type
- discussion topicFederico Campanini
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 12th 2023

- Discussion Type
- discussion topicinternal profunctor
- Category Latest Changes
- Started by FinnLawler
- Comments 16
- Last comment by ncfavier
- Last Active May 12th 2023

Created internal profunctor, which also describes an idea I saw somewhere about internal diagrams in fibrations over the base category. I added what I think are two examples, and asked a generic 'Help!' question. It might be better off on a page of its own, though.

- Discussion Type
- discussion topiccoextension of scalars
- Category Latest Changes
- Started by John Baez
- Comments 1
- Last comment by John Baez
- Last Active May 12th 2023

- Discussion Type
- discussion topiccontext lock
- Category Latest Changes
- Started by Guest
- Comments 2
- Last comment by Urs
- Last Active May 12th 2023

- Discussion Type
- discussion topicrestriction of scalars
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by John Baez
- Last Active May 12th 2023

- Discussion Type
- discussion topiclax-idempotent 2-adjunction
- Category Latest Changes
- Started by Mike Shulman
- Comments 6
- Last comment by varkor
- Last Active May 12th 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 topicsound doctrine of limits
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by Urs
- Last Active May 12th 2023

Created sound doctrine as a stub to record relevant references.

- Discussion Type
- discussion topictopological quantum computation with anyons -- references
- Category Latest Changes
- Started by Urs
- Comments 35
- Last comment by Urs
- Last Active May 12th 2023

I am giving this bare list of references its own entry, so that it may be

`!include`

-ed into related entries (such as*topological quantum computation*,*anyon*and*Chern-Simons theory*but maybe also elsewhere) for ease of updating and synchronizing

- Discussion Type
- discussion topicIntroduction to Topology -- 2
- Category Latest Changes
- Started by nLab edit announcer
- Comments 10
- Last comment by Urs
- Last Active May 12th 2023

- Discussion Type
- discussion topicskeleton
- Category Latest Changes
- Started by Alec Rhea
- Comments 19
- Last comment by Urs
- Last Active May 12th 2023

- Discussion Type
- discussion topicconservative functor
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by John Baez
- Last Active May 11th 2023

added to

*conservative functor*the proposition saying that pullback along strong epis is a conservative functor (if strong epis pull back).How about the $\infty$-version?

- Discussion Type
- discussion topicSteven V Sam
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 11th 2023

- Discussion Type
- discussion topicrepresentation stability
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active May 11th 2023

- Discussion Type
- discussion topicstratified space
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Christoph Dorn
- Last Active May 11th 2023

I have created

*stratified space*in order to collect some references

- Discussion Type
- discussion topicdirected univalence axiom
- Category Latest Changes
- Started by Guest
- Comments 1
- Last comment by Guest
- Last Active May 10th 2023

- Discussion Type
- discussion topictwo-variable adjunction
- Category Latest Changes
- Started by Mike Shulman
- Comments 6
- Last comment by Urs
- Last Active May 10th 2023

I have added a reference to Cheng-Gurski-Riehl to two-variable adjunction, and some comments about the cyclic action.

- Discussion Type
- discussion topictensor category
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active May 10th 2023

added this pointer:

- Damien Calaque, Pavel Etingof,
*Lectures on tensor categories*, IRMA Lectures in Mathematics and Theoretical Physics 12, 1-38 (2008) (arXiv:math/0401246)

- Damien Calaque, Pavel Etingof,

- Discussion Type
- discussion topiccopower
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active May 10th 2023

expanded copower:

added an Idea-section, an Example-section, and a paragraph on copowers in higher category theory.

- Discussion Type
- discussion topicalgebraic model category
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by Urs
- Last Active May 10th 2023

created algebraic model category

- Discussion Type
- discussion topicweak complicial set
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active May 10th 2023

added pointer to

- Viktoriya Ozornova, Martina Rovelli,
*A Quillen adjunction between globular and complicial approaches to $(\infty,n)$-categories*, Advances in Mathematics**421**(2023) 108980 [doi:10.1016/j.aim.2023.108980]

and publication data for this item:

- Viktoriya Ozornova, Martina Rovelli,
*Model structures for (∞,n)-categories on (pre)stratified simplicial sets and prestratified simplicial spaces*, Algebr. Geom. Topol.**20**(2020) 1543-1600 [arxiv:1809.10621, doi:10.2140/agt.2020.20.1543]

- Viktoriya Ozornova, Martina Rovelli,

- Discussion Type
- discussion topicJohn Elias Roberts
- Category Latest Changes
- Started by Urs
- Comments 29
- Last comment by Urs
- Last Active May 10th 2023

since we were talking about rigged Hilbert spaces, I figured it was time to create an entry on John Roberts

- Discussion Type
- discussion topicmeasurable function
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active May 10th 2023

- Discussion Type
- discussion topicRohit Nagpal
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 9th 2023

- Discussion Type
- discussion topicJordan S. Ellenberg
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 9th 2023

- Discussion Type
- discussion topicFI-representation
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active May 9th 2023

- Discussion Type
- discussion topicfiber bundle
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 8
- Last comment by Urs
- Last Active May 9th 2023

- Discussion Type
- discussion topicproper model category
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by Urs
- Last Active May 9th 2023

I expanded proper model category a bit.

In particular I added statement and (simple) proof that in a left proper model category pushouts along cofibrations out of cofibrants are homotopy pushouts. This is at Proper model category -- properties

On page 9 here Clark Barwick supposedly proves the stronger statement that pushouts along all cofibrations in a left proper model category are homotopy pushouts, but for the time being I am failing to follow his proof.

(??)

- Discussion Type
- discussion topicReedy model structure
- Category Latest Changes
- Started by Urs
- Comments 19
- Last comment by Urs
- Last Active May 9th 2023

- Discussion Type
- discussion topicnormal form
- Category Latest Changes
- Started by David_Corfield
- Comments 5
- Last comment by ncfavier
- Last Active May 9th 2023

Started this page normal form, but I see there might be a difference between the no-further-rewrites idea and the designated set of normal terms idea (as in disjunctive normal form).

- Discussion Type
- discussion topicKaroubian category
- Category Latest Changes
- Started by adeelkh
- Comments 22
- Last comment by Urs
- Last Active May 9th 2023

Added the definitions of Karoubian category and Karoubi envelope that appear in (an exercise in) SGA 4.

A stupid question: why do they call that difference kernel the

*image*of p? In what sense is it the image?

- Discussion Type
- discussion topicKaroubi envelope
- Category Latest Changes
- Started by David_Corfield
- Comments 7
- Last comment by Urs
- Last Active May 9th 2023

Added reference to a generalization of the Karoubi envelope for n-categories in

- Davide Gaiotto, Theo Johnson-Freyd,
*Condensations in higher categories*, (arXiv:1905.09566)

- Davide Gaiotto, Theo Johnson-Freyd,

- Discussion Type
- discussion topicSusumu Hayashi
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 9th 2023

- Discussion Type
- discussion topicA-infinity-module
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active May 9th 2023

- Discussion Type
- discussion topicTarski-Grothendieck set theory
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active May 9th 2023

- Discussion Type
- discussion topicsimplicial model category
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Urs
- Last Active May 9th 2023

added to simplicial model category a handful of theorems that state when and how a model category is Quillen equivalent to a simplicial model category.

My motivation for filling this in was actually that I was reading van den Berg/Garner types are weak omega-groupoids and my impression was that the main theorem there is

*morally*the usual simplicial resolution technique in model categories, only that instead of simplicial objects they use globular objects.The other main statement in there I hope we can isolate in some other entry (and it may go back to other authors?): that the context categories of certain type theoreies with identity types naturally carry the structure of somthing close to a category with fibrant objects.

- Discussion Type
- discussion topicclosed monoidal category
- Category Latest Changes
- Started by Mike Shulman
- Comments 25
- Last comment by Urs
- Last Active May 9th 2023

added to closed monoidal category a proof that the pointwise tensor product on a functor category with complete codomain is closed.

- Discussion Type
- discussion topicmodel structure on chain complexes
- Category Latest Changes
- Started by Tim_Porter
- Comments 35
- Last comment by Urs
- Last Active May 8th 2023

At model structure on chain complexes, an ’anonymous editor’ suggests that a line saying ’blah blah’ should be completed to something more illuminating!

- Discussion Type
- discussion topicdiscrete fibration
- Category Latest Changes
- Started by zskoda
- Comments 13
- Last comment by BryceClarke
- Last Active May 8th 2023

In discrete fibration I added a new section on the Street’s definition of a discrete fibration from $A$ to $B$, that is the version

**for spans of internal categories**. I do not really understand this added definition, so if somebody has comments or further clarifications…

- Discussion Type
- discussion topiclist of journals publishing homotopy theory and category theory
- Category Latest Changes
- Started by Mike Shulman
- Comments 66
- Last comment by BryceClarke
- Last Active May 8th 2023

- Discussion Type
- discussion topicMahmoud Zeinalian
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Urs
- Last Active May 8th 2023

- Discussion Type
- discussion topicLepage form
- Category Latest Changes
- Started by perezl.alonso
- Comments 1
- Last comment by perezl.alonso
- Last Active May 8th 2023

- Discussion Type
- discussion topicantilinear map
- Category Latest Changes
- Started by gregprice
- Comments 5
- Last comment by Urs
- Last Active May 8th 2023

Add “Idea” section, and define in more generality as needed by star-algebra.

- Discussion Type
- discussion topicEmmy Noether
- Category Latest Changes
- Started by DavidRoberts
- Comments 8
- Last comment by Dmitri Pavlov
- Last Active May 7th 2023

Added some more content, most particularly the abstract of her talk from 1925 introducing homology

*groups*, as a form of categorification:Ableitung der Elementarteilertheorie aus den Gruppentheorie. Die Elementarteilertheorie gibt bekanntlich für Moduln aus ganzzahligen Linearformen eine Normalbasis von der Form $(e_1y_1, e_2y_2, ..., e_ry_r)$, wo jedes $e$ durch das folgende teilbar ist; die $e$ sind dadurch bis aufs Vorzeichen eindeutig festgelegt. Da jede Abelsche Gruppe mit endlich vielen Erzeugenden dem Restklassensystem nach einem solchen Modul isomorph ist, ist dadurch der Zerlegungssatz dieser Gruppen als direkte Summe größter zyklischer mitbewiesen. Es wird nun umgekehrt der Zerlegungssatz rein gruppentheoretisch direkt gewonnen, in Verallgemeinerung des für endliche Gruppen üblichen Beweises, und daraus durch Übergang vom Restklassensystem zum Modul selbst die Elementarteilertheorie abgeleitet. Der Gruppensatz erweist sich so als der einfachere Satz; in den Anwendungen des Gruppensatzes — z.B. Bettische und Torsionszahlen in der Topologie — is somit ein Zurückgehen auf die Elementarteilertheorie nich erforderlich.