-- Yuxi Liu ]]>

I can’t edit because of “invalid LaTeX block” in space and quantity. I saved the edit I was trying to make as follows. Also it would be really good if there’s a preview button in the editors.

“‘ which underlies much of mathematics is at its heart controlled by the following elementary category theoretic reasoning:

Let $S$ be some category whose objects we want to think of as certain simple spaces on which we want to model more general kinds of spaces. For instance $S = \Delta$, the simplicial category, or $S =$ CartSp, the category of $\mathbb{R}^n$ and smooth maps between them.

An ordinary manifold, for instance, is a space required to be *locally isomorphic* to an object in $S = CartSp$. But more generally, a space $X$ modeled on $S$ need only be *probeable* by objects of $S$, giving a rule with which, to each test object $U \in S$, we assign the set of probing maps from $U$ to $X$, such that this assignment is well-behaved with respect to morphisms in $S$. Such an assignment is nothing but a presheaf on $S$, i.e. a contravariant functor

Therefore general spaces modeled on $S$ are nothing but presheaves on $S$:

$Spaces_S := PSh(S) \,.$Of course this is an extremely general notion of spaces modeled on $S$.

For example, any smooth manifold $M$ is a presheaf on CartSp by $F_M:= Hom_{Man}(-, M): CartSp^{op} \leftarrow Set$, where we consider CartSp$as a full subcategory of Man, the category of [[smooth manifold|smooth manifolds]] and smooth maps between them.$F_M$maps each$\mathbb{R}^n$to the set of all the smooth ways that$\mathbb{R}^n$can probe$M$.

In particular, every object in $S$ is a space modeled on $S$, by the Yoneda embedding $S \hookrightarrow Spaces_S$, whereby every object $X$ in $S$ is embedded as $Hom_S(-, X)$. That is, any object in $S$ is nothing but a consistent way to be probed by all the objects in $S$.

Now take a space $X$ modeled on $S$, and consider the set of *quantities* on $X$ with values in $U \in S$. It should be

This defines a covariant functor $C(X) := Hom_{Spaces_S}(X,-): S \to Sets$. More generally, we can consider the S-valued quantities on $X$ to be a copresheaf on $S$, namely a covariant functor

$C(X): S \to Sets \,.$One can think of $C(X)$ as a generalized quantity which may be *co-probed* by objects of $S$.

In this vein, one can say, generally, that copresheaves on $S$ are generalized quantities modeled on $S$, and we write

$Quantities_S := CoPSh(S) \,.$Given any such generalized quantity $A \in Quantities_S$, we can ask which generalized space it behaves like the algebra of functions on. This generalized space should be called $Spec(A)$ and can be defined as a presheaf by the assignment

$Spec(A) : U \mapsto Hom_{Quantities_S}(A, C(U)) \,.$In total this yields an adjoint pair of contravariant functors between generalized spaces and generalized quantities:

$Spaces_S^{op} \stackrel{\stackrel{C(-)}{\to}}{\stackrel{Spec(-)}{\leftarrow}} Quantities_S \,.$(That this is an adjunction can be understood as a special case of abstract Stone duality induced by a dualizing object.)

Lawvere refers to this adjoint pair as **Isbell conjugation**.

“‘

]]>Sorry, I correct all my suggestions from #73 again to:

```
p { margin-bottom:.5em; }
li p { margin-top:.5em; }
```

and deleting

```
li p { margin-bottom:.5em; }
```

The problem with the current margin comes from the fact that the text in lists is put in `p`

s (in contrast to how tables of content are generated). But `p`

s have top/bottom margins `=0`

.

Speaking of margins, I would also suggest

```
h6 {
margin-top:2em;
margin-bottom:1em;
}
```

The only two examples using `h6`

I am aware of are Science of Logic and Critique of Pure Reason.

I have to correct my suggestion:

```
li ul, li ol { margin-top:.25em; }
li li: { margin-bottom:.25em; }
```

The is another point about first order lists. There is too much spacing when a list follows a paragraph, e.g.pseudofunctor. My suggestion

```
p { margin-bottom:.5em; }
ul, ol { margin-top:.5em; }
```

]]>
I find spacing at the beginning of lists in lists not pleasing, e.g. lax functor. I would suggest `li li:first-child { margin-top:.25em; }`

.

Thanks for notifying! I made the change you suggested; it looks better at Giry monad now, but I guess not perfect yet. I ran out of time to experiment further; will do so when I get the chance.

]]>Footnotes increase line height, e.g. see Giry monad. Reason is that line height in normal text is not 1em but 1.3em. Thus the `<sup>`

-tag shifts this extra space to the top creating too much space. Remedy: `sup { line-height: 1em; }`

.

It’s been happening sporadically and I could definitely believe it was Google A/B testing something on me that wasn’t working as intended.

]]>Hi, I’ve not noticed this myself. There have not been any changes to the nLab software which should lead to this. I also could not reproduce it in the browsers I use. Let me know if it looks like there is something we can do on the nLab side about it.

]]>Recently, when visiting the nLab from Google, extra text has been added to the URL, and the corresponding text highlighted on the nLab, which seems to slow everything down, and also looks ugly. An example of what I mean:

https://ncatlab.org/nlab/show/over+category#:~:text=Definition,-The%20slice%20category&text=The%20slice%20category%20is%20a,g%3AX%E2%86%92X%E2%80%B2

I’m not sure who’s responsible (Google/nLab). Is anyone else experiencing this?

]]>Thank you! I’ll take a look at these things as soon as I get a chance.

]]>Hi Richard,

I have now a “working” example of the bug with modifying the redirects:

I have just started to add a new subsection to *diffeological space*. In the source code it’s visible as

```
### Relation to topological spaces
```

Then I added one redirect to the page

```
[[!redirects D-topology]]
```

After that, the added material no longer appears in the output of the page.

$\,$

Experience shows that if next I made any other edit to the page the output will appear. But I’ll leave it as is for the moment in order to demonstrate the bug.

]]>Hi Richard,

on a related note, I just encountered a bug with the `imagefromfile`

-environment:

A minimal example is now in the Sandbox:

The source code of the Sandbox page right now is this:

```
\begin{center}
\begin{imagefromfile}
"file_name": "HardWallModelPredictions.jpg",
"width": 440,
"caption": "From [Erlich 09, section 1.2](#Erlich09)"
\end{imagefromfile}
\end{center}
```

while the output of the Sandbox page is this error message:

```
XML Parsing Error: mismatched tag. Expected: </div>.
Location: https://ncatlab.org/nlab/show/Sandbox
Line Number 314, Column 35:
<div style="text-align: center"></p>
----------------------------------^
```

The error goes away when removing *either* (or both, of course) of

the

`center`

-environment,the image

`caption`

.

Hi Richard,

does the `imagefromfile`

-environment admit specifying a URL to make the image be hyperlinked?

(I have tried some evident choices for tags, such as `URL`

and `href`

, without success)

Re #59 Thanks!!

]]>Re #48, this is a bug in Maruku, the markdown parser which Instiki uses. I will see if I can find a way around it when I get a chance.

]]>Re #54 and #55, I have fixed this in the database now. I do this whenever I see this problem occurring, merging the newly created thread with the old one. Another case where it occurs is where the original thread has a different category than ’Latest changes’.

Just post here if you see other examples of this which I overlook.

]]>Fixed now. In this case the problem was that someone had tried to include vertical space by an empty LaTeX block, which I removed and replaced by `\linebreak`

.

This usually occurs when LaTeX is not closed properly, i.e. missing a dollar sign. The old Instiki permitted this (with consequent erroneous rendering of the page), the ’new’/current one does not. I will take a look when I get a chance.

]]>The page cartesian bicategory appears to be uneditable: upon save I get “Invalid LaTeX Block:” with no further information.

]]>Just a quick note that I have seen the above posts, thanks for raising. I have been tied up with other things and not been able to take a look yet.

]]>Yes, but only by the author of the first message in the thread:

Clicking “edit” on that first message (which it’s original author can do) gives an edit pane for the message itself, but also for the title of the thread.

]]>Is it possible to rename nForum posts?

For instance, this post https://nforum.ncatlab.org/discussion/3179/measurable-locales/#Item_7 is title “Measurable locales”, which means that when somebody edits the article measurable locale, the software will create a new thread, ignoring the old one.

]]>