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

• 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.

• 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.

• 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.

• I created a stub page for Douglas Bridges. I linked to his home page but also to a page on FAQs in constructive mathematics. He seems to have other stuff there and there may be other useful links worth creating.

• 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.

• 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).

• 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?

• 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.

• I thought up until just a few minutes ago that I had proved that WISC was equivalent to local essential smallness of $Cat_{ana}$. 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.