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

• 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.

• I’ve expanded the section on morphisms in Banach space, because the new page on isomorphism classes of Banach spaces refers to a different notion of isomorphism than what the Banach space page previously called the “usual” notion of isomorphism. (The issue is that what’s usual seems to be different for analysts and category theorists.)

• 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.

• I added the following remark to classifying topos of a localic groupoid.
It would be nice if somebody more competent in this area expanded it.

The above equivalence of categories can in fact be lifted to an equivalence between the bicategory of localic groupoids, complete flat bispaces, and their morphisms and the bicategory of Grothendieck toposes, geometric morphisms, and natural transformations. The equivalence is implemented by the classifying topos functor, as explained in

Ieke Moerdijk, The classifying topos of a continuous groupoid II,
Cahiers de topologie et géométrie différentielle catégoriques 31, no. 2 (1990), 137–168.