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Little page to focus on this important notion, as opposed to the general remarks at walking structure.
Added alternative terminology “local right adjoint” and “strongly cartesian monad” from Berger-Mellies-Weber. They claim the former “has become the more accepted terminology” than “parametric right adjoint”; does anyone know other references to support this? (I think it’s certainly more logical, in that it fits with the general principle of “local” meaning “on slice categories” — not to be confused with the different general principle of “local” meaning “in hom-objects”.)
Stub. For the moment just for providing a place to record this reference:
I have removed the following discussion box from stuff, structure, property – because the entry text above it no longer contained the word that the discussion is about :-)
[begin forwarded discussion]
+–{: .query} Mike: Maybe you all had this out somewhere that I haven’t read, but in the English I am accustomed to speak, “property” is not a mass noun. So you can “forget a property” or “forget properties” but you can’t “forget property.”
Toby: Well, ’property’ can be a mass noun in English, but not in this sense. Also, if we were to invent an entirely new word for the concept, it would surely be a mass noun. Together, these may explain why it's easy to slip into talking this way, but I agree that it's probably better to use the plural count noun here. =–
[end forwarded discussion]
created stub for dimension
I felt like the (∞,1) section should give an abstract description rather than a model specific one, so I’ve done so, and proved in the abstract the equivalence between the hom-space and slice-category characterizations.
It feels like cheating to invoke the Grothendieck construction for it; can the argument be made just as cleanly without it?
… I’m having trouble with the formatting, so I’m going to do some bisection to track down the issue….
I have expanded the Idea section in localization of model categories as it previously seemed to be a stub. (It said: A localisation for model categories. Doh!) I have given a quote from Hirschhorn’s book.
started working on superalgebra. But have to interrupt now.
added to Yang-Mills instanton a discussion of instantons as tunnelings between Chern-Simons vacua.
Added:
the entry Galois theory used to be a stub with only some links. I have now added plenty of details.
Started this page to record various facts about large cocompletions as I survey some of the literature. It’s possible that some of these concepts should have their own pages eventually, but at the moment this seems awkward as many don’t already have names in the literature (or have names that are not ideal for various reasons).
http://ncatlab.org/nlab/show/Isbell+duality
Suggests that Stone, Gelfand, … duality are special cases of the adjunction between CoPresheaves and Presheaves. A similar question is raised here. http://mathoverflow.net/questions/84641/theme-of-isbell-duality
However, this paper http://www.emis.ams.org/journals/TAC/volumes/20/15/20-15.pdf
seems to use another definition. Could someone please clarify?
at total category I have added after the definition and after the first remark these two further remarks:
+– {: .num_remark}
Since the Yoneda embedding is a full and faithful functor, a total category induces an idempotent monad on its category of presheaves, hence a modality. One says that is a totally distributive category if this modality is itself the right adjoint of an adjoint modality.
=–
+– {: .num_remark}
The -adjunction of a total category is closely related to the -adjunction discussed at Isbell duality and at function algebras on ∞-stacks. In that context the -modality deserves to be called the affine modality.
=–
Created a seperate page with material mostly copied over from generalized Reddy category plus some additional references.
This article describes left Bousfield localizations. Why is it separate from left Bousfield localization? I suggest deleting this article.
How would people feel about renaming distributor to profunctor? I seem to recall that when this came up on the Cafe, I was the main proponent of the former over the latter, and I've since changed my mind.
I added some material to Mal’cev variety, namely proofs showing the various characterizations are equivalent, and a brief Examples section.
all of
[[!redirects coreflector]]
[[!redirects coreflectors]]
[[!redirects coreflection]]
[[!redirects coreflections]]
[[!redirects coreflective subcategory]]
[[!redirects coreflective subcategories]]
used to still be in reflective subcategory. I have removed it there and instead included these redirects here
Created arity class. Added links from a few places, but there are probably others I didn’t think of.
added pointer to
for review of how Galois representations are arithmetic incarnations of local systems/flat connections. Added the same also to local system and maybe elsewhere.
added references to Lean
I have expanded symplectic manifold a little
added to generalized Reedy category a bunch of definitions and propositions from Cisinski’s article, concerning the notion of normal morphisms of presheaves over a generalized Reedy category.
tried to polish one-point compactification. I think in the process I actually corrected it, too. Please somebody have a close look.
added to modality a minimum of pointers to the meaning in philosophy (Kant).
added to exact functor the characterization of left exact functors as those preserving terminal object and pullbacks. This was previously stated only at finitely complete category.
Added a reference.
Can we say exactly what kind of pretopos the category of small presheaves on a category C is?
Is it a ΠW-pretopos, provided that PC is complete?
Added a new Properties section to connected object. Including a theorem which is a bit of a hack (where I leave it to others to decide if ’hack’ should be interpreted positively or negatively!).
I fixed a trivial typo in adjoint functor theorem but left wondering about this:
… the limit
over the comma category (whose objects are pairs and whose morphisms are arrows in making the obvious triangle commute in ) of the projection functor
I don’t really understand this (and while I could figure it out, it’s probably not good to make readers do so). At first it sounds like someone is saying “the limit over the comma category of the projection functor ”, which would be circular. But it must be that both formulas are intended as synonymous definitions of . At that point one is left wondering why one has a backwards arrow under it and the other does not. I guess old-fashioned people prefer writing limits with backwards arrows under them, so someone is trying to cater to all tastes? I think it’s better in this website to use and for limit and colimit.
I could probably guess how to fix this, but I won’t since I might screw something up.