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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• added to diffeomorphism group statements and references for the case of 3-manifolds (Smale conjecture etc.)

• I added the definition of a filtered (infinity,1)-category from HTT. Since this is performed in a simplicial model which is supposedly not to be emphasized from the nPov and I felt that the below proposition should center this article I added a sentence indicating this in the ”Idea”.

• I was surprised to discover that we had no page finite (infinity,1)-limit yet, especially given that they are slightly subtle in relation to the 1-categorical version. So I made one.

• Theorem: The existence of arbitrarily large supercompact cardinals implies the statement:

Every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class.

In other words, if $L$ is a reflective localization functor on a balanced accessible category such that the unit morphism $X \to L X$ is an epimorphism for all $X$ and the class of $L$-local objects is defined by an absolute formula, then the existence of a suficciently large supercompact cardinal implies that $L$ is a localization with respect to some set of morphisms.

This is in BagariaCasacubertaMathias

Urs Schreiber: I am being told in prvivate communication that the assumption of epis can actually be dropped. A refined result is due out soon.

• this is a message to Zoran:

I have tried to refine the section-outline at localizing subcategory a bit. Can you live with the result? Let me know.

• 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 $X:\C$, $C(X,f)$ be full is not equivalent in $\Grpd$ to $f$ being full. Moreover, in $\Grpd$, this condition implies faithfulness. We will define (Definition 197) a notion of full arrow in a $\Grpd$-category which, in $\Grpd$ and $\Symm2\Grp$ (symmetric 2-groups), gives back the ordinary full functors.» Note that this works only for some good groupoid enriched categories, not for $\Cat$, 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 homotopy level

However, the instiki-table does not come out correctly yet. It did before I added the third column. These tables are the most delicate things. I never know why sometimes they display correctly and sometimes not.

• 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...)

• 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.

• 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!

• 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 “$\Omega\mathbf{C}(X,Y)\simeq \mathbf{C}(X,\Omega Y)$ in any (oo,1)-category with homotopy pullbacks” in loop space object.

• 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.

• 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.

• 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.

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.

• I have been working on the entry twisted bundle.

Apart from more literature, etc. I have started typing something like a first-principles discussion: first a general abstract definition from twisted cohomology in any cohesive $\infty$-topos, then unwinding this in special cases to obtain the traditional cocycle formulas found in the literature.

Needs more polishing here and there, but I have to pause now.

• created Lie bialgebra, but so far just a comment on their quantization.

• 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!).

• 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.

• Added to pasting diagram a reference to the bicategorical pasting theorem given by Verity in his thesis.

• In the Definition-section at reflective factorization system I found the “$\Psi$” and “$\Phi$” used in the text oppositely to how they appear in the displayed diagram. I think I have fixed this.

• 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 $E \to F$ between normed vector spaces is affine if $F$ 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 $l_2$ norm is a continuum, whereas for other $p$ in the range $1 \lt p \lt \infty$, 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.