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discovered the following remnant discussion at full functor, which hereby I move from there to here
Mathieu says: I agree that, for functors, there is no reason to say “fully faithful” rather than “full and faithful”. But for arrows in a 2-category (like in the new version of the entry on subcategories), there are reasons. I quote myself (from my thesis): «Remark: we say fully faithful and not full and faithful, because the condition that, for all , be full is not equivalent in to being full. Moreover, in , this condition implies faithfulness. We will define (Definition 197) a notion of full arrow in a -category which, in and (symmetric 2-groups), gives back the ordinary full functors.» Note that this works only for some good groupoid enriched categories, not for , for example.
Mike says: Do you have a reason to care about full functors which are not also faithful? I’ve never seen a very compelling one. (Maybe I should just read your thesis…) I agree that “full morphism” (in the representable sense) is not really a useful/correct concept in a general 2-category, and that therefore “full and faithful” is not entirely appropriate, so I usually use “ff” in that context. I’ve changed the entry above a bit to reflect your comment; is it satisfactory now? Maybe all this should actually go at full and faithful functor (and/or fully faithful functor)?
created line object
I have finally started – long overdue – an entry higher U(1)-gauge theory. What I really needed right now (as an entry) is the subentry higher electric background charge coupling, that I also started.
I am not done yet with what I wanted to do, but need to quit for tonight.
I am starting stubs
created an entry mapping cocone, following a suggestion by Zoran, that this is the right technical term for what is discussed in more detail at generalized universal bundle.
(the examples section needs more attention, though...)
Stubs for fibrant object and acyclic fibration (also redirecting their duals, for now).
I have created final lift, and added to adjoint triple a proof that in a fully faithful adjoint triple between cocomplete categories, the middle functor admits final lifts of small structured sinks (and dually). This means that it is kind of like a topological concrete category, except that the forgetful functor need not be faithful.
I find this interesting because it means that in the situation of axiomatic cohesion, where the forgetful functor from “spaces” is not necessarily faithful, we can still construct such “spaces” in “initial” and “final” ways, as long as we restrict to small sources and sinks.
I figured it was high time we had a general page on truncated objects. I think some number of links, and perhaps redirects, which currently point to n-truncated object of an (infinity,1)-category would more usefully point here, but I haven’t updated any yet.
I noticed that discrete object used to redirect to discrete morphism, where I expected it to take me at least to discrete space, if not to its own entry.
We should eventually disambiguate here and add some comments. For the moment I made it redirect to discrete space and added there a remark “to be merged with discrete morphism”.
There is a query on category of representations. Basically asking for some references to be added???
If you're not following the categories mailing list, then you're missing out on a great discussion of evil. Peter Selinger has come from the list to the Lab to discuss it here too!
New entry Banach bundle covering for now also more special notion of Hilbert bundle and a different notion of Banach algebraic bundle. Sanity check is welcome!
Thought I’d write up some old notes at symmetric product of circles (linked from unitary group, explanation to come on symmetric product of circles). Not finished yet, but have to leave it for now.
(I was incensed to discover that to look at the source article for the material for this to check that I’m remembering it right - I last looked at it about 10 years ago - I have to pay 30 UKP. The article is 3 pages long. That’s 10UKP per page! So I’m going from vague memories and “working it out afresh”.)
Added Thom-Federer and Gottlieb thorems to Eilenberg-MacLane space; added the remark “ in any (oo,1)-category with homotopy pullbacks” in loop space object.
At inhabited and (-1)-connected I have added cross-links pointing out the synonyms.
At local topos I have added a Properties-section stating that here every inhabited object is globally inhabited, which is a shadow of the homotopy dimension 0 of local (infinity,1)-toposes.
I noticed that cell complex was missing, so I created it
New entry fundamental vector field which covers also the somewhat dual notion fundamental differentiable form. Redirecting also fundamental form. The entry is partly intended to support the content in the entry Ehresmann connection. Please check the content.
Partially spurred on by an MO question, I have started an entry on simple homotopy theory. I am also intrigued as to whether there is a constructive simple homotopy theory that may apply in homotopy type theory, but know so little (as yet) about that subject that this may be far fetched.
Steve (Lack) has put a comment box on AT category. I have not been following that entry so am not able to reply to his point.
placeholder for 1d WZW model, to be expanded
I filled in content at n-truncated object of an (infinity,1)-category.
to go with my discussion with David Roberts. I had planned to go further and also write the entry on Postnikov twoers, but got distracted all day.
Apart from that I just added this link to Higher Topos Theory and did some editing there, added a table of contents, expanded the floating toc.
Heard the rumours and wondered what it is? Now you know.
I started editing the page on reflexive Banach spaces - in particular I corrected the definition and stuck in a mention of "James space". A link or reference is needed but I am currently a bit too frazzled/stressed to do further editing today.
I have expanded at DHR category the Idea-section and added more hyperlinks.
I have given the definition of localized endomorphism its own entry (it is otherwise sitting inside DHR superselection theory).
stub for Jacobian
Joel Hamkins and Andy Putman made some comments about the nLab on MathOverflow, beneath an answer by Andrew.
It’s interesting to know what people’s perceptions are, even if they’re wrong. (And I would think that Andy P’s perception is wrong.) I don’t know what Andrew S has in mind when he says that Joel’s point is extremely easy to answer.
Following a discussion on the algebraic topology list, I’ve written a proof of the contractibility of the space of embeddings of a smooth manifold in a reasonably arbitrary locally convex topological vector space. The details are on embedding of smooth manifolds and it also led to me creating shift space (I checked on MO to see if there was an existing name for this, and Bill Johnson said he hadn’t heard of it).
added the recent Barwick/Schommer-Pries preprint to (infinity,n)Cat, together with a few more brief remarks.
created Lie bialgebra, but so far just a comment on their quantization.
while polishing up type theory - contents I felt the need for entries called syntax and semantics. I have created these just so that the links to them are not grey, but I put in only something minimalistic . I could add some general blah-blah, but I’d rather hope some actual expert feels inspired to start with some decent paragraphs.
created proofs as programs
Added to pasting diagram a reference to the bicategorical pasting theorem given by Verity in his thesis.
just for completness, I have split off a brief entry (hyperconnected,localic) factorization system.
Created a stub for normal field extension. (and missed out the x in the title of this page.)
stub for indexed adjoint functor theorem
In the Definition-section at reflective factorization system I found the “” and “” used in the text oppositely to how they appear in the displayed diagram. I think I have fixed this.
I finally wrote uniformly continuous map. Pretty much just definitions.
After contributing to the article on parallelogram identity, I added to isometry and created Mazur-Ulam theorem. The easy proof added at isometry, that shows an isometry between normed vector spaces is affine if is strictly convex, might lead one to suspect that the proof under parallelogram identity was overkill, but I think that’s an illusion. Ultimately, I believe the parallelogram identity is secretly an expression of the perfect ’roundness’ of spheres, connected with the fact observed by Tom Leinster recently at the Café that the group of isometries for the norm is a continuum, whereas for other in the range , you get just a finite reflection group (this is for the finite-dimensional case, but there’s an analogue in the infinite-dimensional case as well).
The Mazur-Ulam theorem removes the strict convexity hypothesis, but adds the hypothesis that the isometry is surjective. The conclusion is generally false if this hypothesis is omitted.
I have started a separate entry on strong shape theory, but it is only a stub with references filtched from shape theory.
I have split off universal quantifier and existential quantifier from quantifier in order to expose the idea in a more pronounced way in dedicated entries.
Mainly I wanted to further amplify the idea of how these notions are modeled by adjunctions, and how, when formulated suitably, the whole concept immediately and seamlessly generalizes to (infinity,1)-logic.
But I am not a logic expert. Please check if I got all the terminology right, etc. Also, there is clearly much more room for expanding the discussion.
Thought I’d nick an another answer from MathOverflow and paste it to the nLab. Unfortunately, doing an internet search for “functional analysis type” or even cotype doesn’t look like I’m going to be able to figure out what those terms mean all that quickly …
Oops. Forgot the link: isomorphism classes of Banach spaces.
Added parallelogram identity since it was linked from type (functional analysis). Actually, I mostly stole the content from another wiki, (this one) but I don’t think that the original author will mind.
stub for cohomology of operads, so far just in order to record Charles Rezk’s thesis.
Bill Johnson kindly sent me an explanation of type and cotype for Banach spaces which I’ve mangled and put up at type (functional analysis).
I have created some genuine content at implicit function theorem. I’d like to hear the comments on the global variant, which is there, taken from Miščenko’s book on vector bundles in Russian (the other similar book of his in English, cited at vector bundle, is in fact quite different).
finally a stub for (infinity,1)-semitopos
I have created an entry notions of type to be included under “Related notions” in the relevant entries.
(I have managed to refrain from titling it “types of types”.)
Which notions of types are still missing in the table?
I have added some remarks to chain complex, model structure on chain complexes, homotopy limit, and derivator regarding the fact that every chain complex over a field is equivalent to its homology (regarded as a chain complex with zero differentials).
In reaction to the public demand exhibited by Guillaume Brunerie's comments I have created an entry
To replace some anonymous scribblings, I cribbed some definitions from Wikipedia to get a stub at deformation retraction.
quick note on 2-framing
I thought up until just a few minutes ago that I had proved that WISC was equivalent to local essential smallness of . Mike urged me to put my proof on the lab, but in doing so I discovered it was flawed. So now WISC just has a proof that the principle implies local essential smallness.
I noticed there is no entry electrodynamics so I “created” it as a redirect to the existing stub electromagnetism. Though I personally consider electromagnetism as a phenomenon in real world, while electrodynamics just as a theoretical formalism to describe it, i.e. a theory of electromagnetism. There is some overlap between existing entries, like there is another, rather stubby classical electrodynamics, and entry quantum electrodynamics. The real content is in electromagnetic field.
I am starting something at higher dimensional WZW theory
stub for false vacuum
created cohomology of local net of observables. I have included a brief Idea-section but mainly this is, for the moment, to record references.
New entry combinatorial Hopf algebra. Reference additions or updates in Hopf algebra, BV formalism, Hall algebra, graph homology, Marcelo Aguiar, renormalization.
Have a look at horizon.