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brief category:people-entry for hyperlinking references at spectrum (geometry) and at multi-adjoint
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.
expanded model structure on functors by adding a long list of properties
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.
I am splitting off homotopy category of a model category from model category. Have spelled out statement and proof of the localization construction there.
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.
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?
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.
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.
Created sound doctrine as a stub to record relevant references.
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
added to conservative functor the proposition saying that pullback along strong epis is a conservative functor (if strong epis pull back).
How about the -version?
I have created stratified space in order to collect some references
I have added a reference to Cheng-Gurski-Riehl to two-variable adjunction, and some comments about the cyclic action.
added this pointer:
expanded copower:
added an Idea-section, an Example-section, and a paragraph on copowers in higher category theory.
created algebraic model category
added pointer to
and publication data for this item:
since we were talking about rigged Hilbert spaces, I figured it was time to create an entry on John Roberts
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.
(??)
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).
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?
Added reference to a generalization of the Karoubi envelope for n-categories in
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.
added to closed monoidal category a proof that the pointwise tensor product on a functor category with complete codomain is closed.
At model structure on chain complexes, an ’anonymous editor’ suggests that a line saying ’blah blah’ should be completed to something more illuminating!
In discrete fibration I added a new section on the Street’s definition of a discrete fibration from to , 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…
Add “Idea” section, and define in more generality as needed by star-algebra.
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 , wo jedes durch das folgende teilbar ist; die 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.