Made a start here.

]]>created *immersion of smooth manifolds* .

Stated also the definition that $f : X \to Y$ is an immersion precisely if the canonical morphism

$T X \to X \times_Y T Y =: f^* Y$is an injection.

This style of writing the conditon I have now also added to *submersion* (where this canonical morphism is a surjection) and to *local diffeomorphism* (where it is an iso).

This way of stating the condition makes most manifest that with respect to the infinitesimal cohesion $i :$ Smooth∞Grpd $\hookrightarrow$ SynthDiff∞Grpd we have

immersion $\Leftrightarrow$ formally unramified morphism

submersion $\Leftrightarrow$ formally smooth morphism

local diffeomorphism $\Leftrightarrow$ formally etale morphism

created *fixed point of an adjunction*, just minimally so that it is possible to link to it.

It seems strange that we wouldn’t have already an nLab entry on this, but after checking it seems to me that we didn’t.(?)

Needs to be expanded…

]]>For size reasons I need to split this section off and re-!include it into *geometry of physics – categories and toposes*. (Hence nothing to be seen here, this is not a stand-alone entry. I am announcing it only since the system is forcing me to do so.)

Latest Revisions is currntly giving me a “500 Internal Server Error” with both Firefox and Chrome.

]]>I am splitting off Gelfand duality from Gelfand spectrum. Want to state the actual equivalence theorem here. But just a moment…

]]>brief category:people entry for hyperlinking references at *topological algebra* and *Gelfand duality*

Since Dan and Felix and others asked me to list good problems (i.e. potential theorems of interest whose proof should be within reach) in cohesive/elastic/solid modal homotopy theory, I started making a list here. Just a start, for the moment.

In addition to the problem of formalizing the fundamental theorem of calculus (i.e. Stokes theorem, an issue we had discussed at some length here a while back) there is so far only one new item: To show that the bosonic body of a supermanifold is an ordinary manifold (here).

]]>Just collecting together things on the list to implement/look into, to make it a little easier and anybody else interested in contributing to keep an overview.

Import of HoTT special year pages into the nLab. Request of Mike over email. DONE.

Speed up of page loading on the nLab. Last touched on by Urs here. See also the experiments with a static frontend, discussed here.

Update nForum announcements made within 30 minutes rather than post new announcements, to mirror the behaviour of the nLab. Last touched on here.

Make tool for handling references in the nLab. Last discussed here.

Save and display nLab page title changes in the revision history. See here.

Fix bugs in the nLab’s display of the diff in mathematics the revision history. (See more or less any diff where mathematics is involved).

Automatically convert links posted verbatim in the nForum to actual links rather than plain text.

I think that’s everything that has been raised so far. Things may soon be added to the list, e.g. the thoughts Urs had in #68 and #70 here.

Let me know if there’s anything else I should add.

]]>I’ll be preparing here notes for my lectures *Categories and Toposes (schreiber)*, later this month.

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

]]>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.

]]>We have been given early warning by CMU that the building the nLab is in will be out of power all day on the 11th of August. We have enough time to put up a warning message on the nLab, but I wonder if we should use this as a spur to setup a second server on a virtual machine somewhere? We can use a pay-per-use machine, and then decide whether or not we keep it up after the downtime.

There will be quite a bit of work to do to set this up, so I will need a decision soon. If we go for it, we will also need to finance it. Maybe those willing can divide the sum between them, it should not be too much. I am willing to contribute.

If no-one objects too strongly, I would suggest on this occasion that we use Amazon, simply because I am already familiar with it. We can probably get by with a single EC2 instance of size something like t2.large (I can experiment a bit beforehand to get the right size machine). If we have it up for 48 hours, such a machine would cost 4.4544 dollars. I can pay for this myself, but it’d be nice if a few others also commit to paying, just so that we can split the costs if needed (the actual cost will be a little bit more, because I will need to test, and one pays every time one switches it on and off). We would also need a way to switch routing. To keep costs down, we could simply use an nginx server on the same EC2 instance that we hire, and manually switch. The more robust approach (but maybe too much for now) would be to use a load-balancer (which will come with some small additional costs).

There would be some side-benefits of doing this. One being that I will need to make the nLab easily deployable, e.g. build locally.

]]>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.

I have made a start on one of the Technical TODO list (nlabmeta) items, namely to create a dashboard with some statistics about the nLab, for example number of page views. I am not yet at the stage where we can get graphs of page views, but just to illustrate it a bit, here is a ’snapshot’ of how it looks at the moment.

https://ncatlab.org/grafana/dashboard/snapshot/a4K1AOyDB0YDq1tQv2RlwprCxsKWW3QB

There is one statistic there, which shows the number of page edits/creations over the last 24 hours. This snapshot will be deleted after 7 days from now.

The dashboard tool is called grafana. It is extremely powerful and flexible. In the snapshot, you cannot do much except look at the statistic, but the actual dashboard is live (but password protected), and one can create all kinds of graphs, etc, change the time over which one searches,etc.

To get statistics on page views, since we do not have any metrics inside Instiki itself, probably the quickest will be to use the nginx logs. But this will require a bit more infrastructure, so will take a bit more time.

The statistic that is there uses the nLab database as its ’data source’.

For now, let me know what kind of graphs/statistics you would be looking for, and I’ll see what I can do to create them.

Probably in the end we’ll host daily snapshots of the dashboard which anybody can see, and certain people will be given access to the live dashboard.

]]>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*

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.

]]>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.

]]>I have now removed the s5 slideshow functionality and the SVG editor from the nLab. Here is the github commit.

This was discussed briefly recently here.

It is quite therapeutic to finally remove these from the codebase!

If you notice any issues, please let me know.

]]>