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

• I have added at HomePage in the section Discussion a new sentence with a new link:

If you do contribute to the nLab, you are strongly encouraged to similarly drop a short note there about what you have done – or maybe just about what you plan to do or even what you would like others to do. See Welcome to the nForum (nlabmeta) for more information.

I had completly forgotton about that page Welcome to the nForum (nlabmeta). I re-doscivered it only after my recent related comment here.

• Have added to HowTo a description for how to label equations

In the course of this I restructured the section “How to make links to subsections of a page” by giving it a few descriptively-titled subsections.

• this evident concept maybe deserves an entry of its own, for ease of linking.

• As a kind of supplement to Urs’s running topology series, I wrote an article colimits of normal spaces. Mainly I had wanted to write down a reasonably clean proof of the fact that CW-complexes are $T_4$ spaces, in particular Hausdorff, as called for on the page CW-complexes are paracompact Hausdorff spaces, but working in slightly greater generality. There are a whole bunch of links to stick in, which I plan to get to.

This page has taken me longer than I had first anticipated. Only after some struggle and reading around did I discover the power of the Tietze characterization of normality, which can be used to give a simple proof of the following general fact:

If $X, Y, Z$ are normal and if $h: X \to Z$ is a closed embedding and $f: X \to Y$ a continuous map, the attachment space = pushout $W = Y \cup_X Z$ is also normal.

This doesn’t seem so easy to prove with one’s bare hands (i.e., just using the usual definition of normality and reasoning away)!

Urs, after recent discussion with Richard about paracompactness, where do matters stand on the page CW-complexes are paracompact Hausdorff spaces? It would be nice to tie up whatever loose ends are still left hanging there.

• recorded the definition

• With Igor Khavkine we finally have a polished version of what is now “Part I” of a theory of variational calculus in a differentially cohesive $\infty$-topos. It’s now called:

Synthetic geometry of differential equations

• Part I. Jets and comonad structure

Abstract:

We give an abstract (synthetic) formulation of the formal theory of partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as super-geometry, higher (stacky) differential geometry, or even a combination of both. A motivation for such a level of generality is the eventual goal of solving the open problem of covariant geometric pre-quantization of locally variational field theories, which may include fermions and (higher) gauge fields.

A remarkable observation of Marvan 86 is that the jet bundle construction in ordinary differential geometry has the structure of a comonad, whose (Eilenberg-Moore) category of coalgebras is equivalent to Vinogradov’s category of PDEs. We give a synthetic generalization of the jet bundle construction and exhibit it as the base change comonad along the unit of the “infinitesimal shape” functor, the differential geometric analog of Simpson’s “de Rham shape” operation in algebraic geometry. This comonad structure coincides with Marvan’s on ordinary manifolds. This suggests to consider PDE theory in the more general context of any topos equipped with an “infinitesimal shape” monad (a “differentially cohesive” topos).

We give a new natural definition of a category of formally integrable PDEs at this level of generality and prove that it is always equivalent to the Eilenberg-Moore category over the synthetic jet comonad. When restricted to ordinary manifolds, Marvan’s result shows that our definition of the category of PDEs coincides with Vinogradov’s, meaning that it is a sensible generalization in the synthetic context.

Finally we observe that whenever the unit of the “infinitesimal shape” ℑ\Im operation is epimorphic, which it is in examples of interest, the category of formally integrable PDEs with independent variables ranging in Σ is also equivalent simply to the slice category over ℑΣ. This yields in particular a convenient site presentation of the categories of PDEs in general contexts.

• I added a note about what “projectively flat” means in the context of parabolic Cartan geometry. I would assume that this should mean the same thing as “a connection which is flat up to central term” as described in the first sentence?

• this wasn’t pointing anywhere. Made a minimum disambiguation page.

• Changed the page name because a name was misspelled.

• for discussion such as at enriched model category, we should really have a page on the special properties of categories that are both tensored and cotensored.

I made a start here. Recorded that in this case tensoring and cotensoring are adjoint to each other, and that initial/terminal objects are also enriched initial/terminal.

• I gave functorial factorization its own little entry, for ease of pointing to the precise definition.

This is for the moment just copied over from the corresponding paragraph at weak factorization system (where I have re-organized the sectzion outline slightly, for clarity). Also I added cross-links with some relevant entries.

• for the five topics listed at HomePage (joyalscatlab) I added references to the corresponding nLab entries

for instance for model categories here.

• Quick idea section for call-by-push-value

• Added link and short description of contents of Essays on the Theory of Numbers

• Added to Dedekind cut a short remark on the $\neg\neg$-stability of membership in the lower resp. the upper set of a Dedekind cut.

• Created, with so far just an overview of all the possibilities.

• I have

• touched the formatting of this ancient entry,

• expanded and streamlined the Idea-section a little,

• added illustrating diagrams for the definition of the sieves appearing in the definition (here) of local epis from a given site

• added the version of the definition for covergages instead of Grothendieck topologies (here, is this in Johnstone?)

• added statement of the proposition that the Cech nerve projection out of a local epi is a local weak equivalence of simplicial presheaves (here)

• Page created, but author did not leave any comments.

• Page created, but author did not leave any comments.

• I’ve started sufficiently cohesive topos. Here are a couple of remarks and questions:

1. The corresponding terminology in def. 2.13 at cohesive topos strikes me as odd: $p_!(\Omega)=1$ is connectedness not contractability.

2. It isn’t quite clear to me yet at which level of generality to optimally state the definition of ’sufficient cohesion’. It seems that what one wants to get here are the minimal assumptions ensuring that the connectedness of $\Omega$ is equivalent to its contractibility and this presumably requires only preservation of finite products by $p_!$ and not the Nullstellensatz (nor even the existence of $p^!$ !?).

3. Since the entry so far lives on the (0,1)Lab maybe somebody here has an idea what to say for the ($\infty$,1)-case e.g. assuming connectedness of the (higher) object classifier !?

• Page created, but author did not leave any comments.

• added pointer to the preprint Scharf 13, which is apparently what that unexpected extra chapter 6 in the latest edition of the book is based on.