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- Discussion Type
- discussion topicHomePage
- Category Latest Changes
- Started by Urs
- Comments 36
- Last comment by Todd_Trimble
- Last Active 9 hours ago

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.

- Discussion Type
- discussion topicHowTo
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Mike Shulman
- Last Active 12 hours ago

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.

- Discussion Type
- discussion topicQuillen adjoint triple
- Category Latest Changes
- Started by Urs
- Comments 35
- Last comment by Urs
- Last Active 13 hours ago

- Discussion Type
- discussion topiccolimits of normal spaces
- Category Latest Changes
- Started by Todd_Trimble
- Comments 19
- Last comment by Todd_Trimble
- Last Active 13 hours ago

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.

- Discussion Type
- discussion topicsmooth isotopy
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 23 hours ago

- Discussion Type
- discussion topicPDE theory in differential cohesion
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by David_Corfield
- Last Active 1 day ago

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*

We keep our latest version of the file

**here**.Comments are most welcome.

**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.

- Discussion Type
- discussion topicprojectively flat connection
- Category Latest Changes
- Started by Bartek
- Comments 1
- Last comment by Bartek
- Last Active 1 day ago

- Discussion Type
- discussion topicderived hom-functor
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicTaylor Dupuy
- Category Latest Changes
- Started by Todd_Trimble
- Comments 2
- Last comment by Todd_Trimble
- Last Active 1 day ago

- Discussion Type
- discussion topicpowered and copowered category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

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.

- Discussion Type
- discussion topicfunctorial factorization
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 2 days ago

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.

- Discussion Type
- discussion topicJoyal's Cat Lab
- Category Latest Changes
- Started by Urs
- Comments 31
- Last comment by Richard Williamson
- Last Active 2 days ago

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

for instance for model categories here.

- Discussion Type
- discussion topiccall-by-push-value
- Category Latest Changes
- Started by maxsnew
- Comments 3
- Last comment by David_Corfield
- Last Active 3 days ago

- Discussion Type
- discussion topicdouble-negation shift
- Category Latest Changes
- Started by Mike Shulman
- Comments 7
- Last comment by spitters
- Last Active 3 days ago

Created double-negation shift, with a proof that it is equivalent to double-negated excluded middle.

- Discussion Type
- discussion topicRichard Dedekind
- Category Latest Changes
- Started by DavidRoberts
- Comments 1
- Last comment by DavidRoberts
- Last Active 4 days ago

- Discussion Type
- discussion topicDedekind cut
- Category Latest Changes
- Started by IngoBlechschmidt
- Comments 5
- Last comment by DavidRoberts
- Last Active 4 days ago

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.

- Discussion Type
- discussion topiclax monoidal category
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by Richard Williamson
- Last Active 4 days ago

- Discussion Type
- discussion topiclocal epimorphism
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 4 days ago

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)

- Discussion Type
- discussion topictopological twist
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 4 days ago

added pointer to Elliott-Safronov 18

- Discussion Type
- discussion topicinhomogeneous cosmology
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 4 days ago

started something. For the moment really just a glorified pointer to Buchert et al. 15 and putting Scharf 13 into perspective

- Discussion Type
- discussion topicMasahico Saito
- Category Latest Changes
- Started by Tim_Porter
- Comments 10
- Last comment by Tim_Porter
- Last Active 5 days ago

- Discussion Type
- discussion topicChristof Wetterich
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

brief category:people entry for hyperlinking references at

*asymptotic safety*and*inhomogeneous cosmology*

- Discussion Type
- discussion topicidentity among the relations
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active 6 days ago

- Discussion Type
- discussion topicCayley graph
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by Tim_Porter
- Last Active 6 days ago

- Discussion Type
- discussion topicsufficiently cohesive topos
- Category Latest Changes
- Started by Thomas Holder
- Comments 16
- Last comment by Thomas Holder
- Last Active 7 days ago

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

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

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 $p^!$ !?).

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 !?

- Discussion Type
- discussion topicJohn Moffat
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

brief category:people entry for hyperlinking references at

*inhomogeneous cosmology*

- Discussion Type
- discussion topicCarlos Contou-Carrere
- Category Latest Changes
- Started by alex
- Comments 1
- Last comment by alex
- Last Active 7 days ago

- Discussion Type
- discussion topiccosmological constant
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 7 days ago

I have added a comment and collected some references on the renormalization freedom in the cosmological constant: here

I have cross-linked this with related entries:

*renormalization*,*perturbative quantum gravity*and*stress-energy tensor*

- Discussion Type
- discussion topicThomas Buchert
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

brief category:people entry for the purpose of hyperlinking references at

*inhomogeneous cosmology*

- Discussion Type
- discussion topicQuantum Gauge Theories -- A True Ghost Story
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago