polished [[category of sheaves]] slightly

]]>created over-topos

]]>Adeel Khan created *sheaf of meromorphic functions*.

(He currently has problems logging into here, that’s why I am posting this for the moment.)

]]>edited [[classifying topos]] and added three bits to it. They are each marked with a comment "check the following".

This is in reaction to a discussion Mike and I are having with Richard Williamson by email.

]]>at internal hom the following discussion was sitting. I hereby move it from there to here

Here's some discussion on notation:

*Ronnie*: I have found it convenient in a number of categories to use the convention that if say the set of morphisms is $hom(x,y)$ then the internal hom when it exists is $HOM(x,y)$. In particular we have the exponential law for categories

Then one can get versions such as $CAT_a(y,z)$ if $y,z$ are objects over $a$.

Of course to use this the name of the category needs more than one letter. Also it obviates the use of those fonts which do not have upper and lower case, so I have tended to use mathsf, which does not work here!

How do people like this? Of course, panaceas do not exist.

*Toby*: I see, that fits with using $\CAT$ as the $2$-category of categories but $\Cat$ as the category of categories. (But I'm not sure if that's a good thing, since I never liked that convention much.) I only used ’Hom’ for the external hom here since Urs had already used ’hom’ for the internal hom.

Most of the time, I would actually use the same symbol for both, just as I use the same symbol for both a group and its underlying set. Every closed category is a concrete category (represented by $I$), and the underlying set of the internal hom is the external hom. So I would distinguish them only when looking at the theorems that relate them, much as I would bother parenthesising an expression like $a b c$ only when stating the associative law.

*Ronnie*: In the case of crossed complexes it would be possible to use $Crs_*(B,C)$ for the internal hom and then $Crs_0(B,C)$ is the actual set of morphisms, with $Crs_1(B,C)$ being the (left 1-) homotopies.

But if $G$ is a groupoid does $x \in G$ mean $x$ is an arrow or an object? The group example is special because a group has only one object.

If $G$ is a group I like to distinguish between the group $Aut(G)$ of automorphisms, and the crossed module $AUT(G)$, some people call it the *actor*, which is given by the inner automorphism map $G \to Aut(G)$, and this seems convenient. Similarly if $G$ is a groupoid we have a group $Aut(G)$ of automorphisms but also a group groupoid, and so crossed module, $AUT(G)$, which can be described as the maximal subgroup object of the monoid object $GPD(G,G)$ in the cartesian closed closed category of groupoids.

*Toby*: ’But if $G$ is a groupoid does $x \in G$ mean $x$ is an arrow or an object?’: I would take it to mean that $x$ is an object, but I also use $\mathbf{B}G$ for the pointed connected groupoid associated to a group $G$; I know that groupoid theorists descended from Brandt wouldn't like that. I would use $x \in \Arr(G)$, where $\Arr(G)$ is the arrow category (also a groupoid now) of $G$, if you want $x$ to be an arrow. (Actually I don't like to use $\in$ at all to introduce a variable, preferring the type theorist's colon. Then $x: G$ introduces $x$ as an object of the known groupoid $G$, $f: x \to y$ introduces $f$ as a morphism between the known objects $x$ and $y$, and $f: x \to y: G$ introduces all three variables. This generalises consistently to higher morphisms, and of course it invites a new notation for a hom-set: $x \to y$.)

continued in next comment…

]]>started *Galois cohomology*

somebody asked me for the proof of the claim at *canonical topology* that for a Grothendieck topos $\mathbf{H}$ we have $\mathbf{H} \simeq Sh_{can}(\mathbf{H})$.

I have added to the entry pointers to the proof in Johnstone’s book, and to related discussion for $\infty$-toposes. Myself I don’t have more time right now, but maybe somebody feels inspired to write out some details in the nLab entry itself?

]]>Just heard a nice talk by Simon Henry about measure theory set up in Boolean topos theory (his main result is to identify Tomita-Takesaki-Connes’ canonical outer automorphisms on $W^\ast$-algebras in the topos language really nicely…).

I have to rush to the dinner now. But to remind myself, I have added cross-links between *Boolean topos* and *measurable space* and for the moment pointed to

- Matthew Jackson,
*A sheaf-theoretic approach to measure theory*, 2006 (pdf)

for more. Simon Henry’s thesis will be out soon.

Have to rush now…

]]>Have added to *cyclic set* a pointer to notes from 1996 by Ieke Moerdijk where the theory classified by the topos of cyclic sets is identified (abstract circles).

This is an unpublished note, but on request I have now uploaded it to the nLab

- Ieke Moerdijk,
*Cyclic sets as a classifying topos*, 1996 (pdf)

I have also added a corresponding brief section to *classifying topos*.

By the way, there is an old query box with an exchange between Mike and Zoran at *cyclic set*. It seems to me that this has been resolved and the query box could be removed (to make the entry read more smoothly). Maybe Mike and/or Zoran could briefly look into this.

quickly added at [[accessible category]] parts of the MO discussion here. Since Mike participated there, I am hoping he could add more, if necessary.

]]>I have expanded Lawvere-Tierney topology, also reorganized it in the process

]]>stub for *constructible sheaf*

I have polished a little at *geometry of physics – smooth sets*, in reaction to feedback that Arnold Neumaier provided over on PF here.

I gave *Sets for Mathematics* a category:reference entry and linked to it from *ETCS* and from *set theory*, to start with.

David Corfield kindly alerts me, which I had missed before, that appendix C.1 there has a clear statement of Lawvere’s proposal from 94 of how to think of categorical logic as formalizing objective and subjective logic (to which enty I have now added the relevant quotes).

]]>stub for *arithmetic pretopos*, just to record the reference

added some actual text to *Verdier duality* (in the Idea-section). But it’s no really good yet. More later…

I was looking for a place to record a somewhat more global overview of the notion of *locally presentable category*, its related notions and its generalizations to higher category theory. But somehow all of the existing entries feel too narrow in focus to accomodate this. So I ended up creating now a new entry titled

Think of this as accomodating material such as one might present in a seminar talk that is meant to bring people with some basic background up to speed with the relevant notions, without going into the wealth of technical lemmas.

I only just started. Will continue in a moment after a short break…

]]>expanded the section *Idea – In brief* at *Bohr topos* just a little bit, in order to amplify the relation to Jordan algebras better (which previously was a bit hidden in entry).

slightly edited *AT category* to make the definition/lemma/proposition-numbering and cross-referencing to them come out.

Probably Todd should have a look over it to see if he agrees.

]]>I noticed that at *ΠW-pretopos* a pointer to the h-sets in HoTT was missing, have added a brief mentioning here.

I gave *sheaf with transfer* an Idea-section

(the entry used to me named “Nisnevich sheaves with transfer”. I have renamed it to singular to stay with our convention and removed the “Nisnevich” from the title, as the concept of transfer as such is really not specific to the Nisnevich topology).

The idea section now is the following. (Experts please complain, and I will try to fine tune further):

Given some category (site) $S$ of test spaces, suppose one fixes some category $Corr_p(S)$ of correspondences in $S$ equipped with certain cohomological data on their correspondence space. Then a *sheaf with transfer* on $S$ is a contravariant functor on $Corr_p(S)$ such that the restriction along the canonical embedding $S \to Corr_p(S)$ makes the resulting presheaf a sheaf.

Traditionally this is considered for $S$ the Nisnevich site and $Corr_p(S)$ constructed from correspondences equipped with algebraic cycles as discussed at *pure motive*, (e.g. Voevodsky, 2.1 and def. 3.1.1).

The idea is that, looking at it the other way around, the extension of a sheaf to a sheaf with transfer defines a kind of Umkehr map/fiber integration by which the sheaf is not only pulled back along maps, but also pushed forward, hence “transferred” (this concept of course makes sense rather generally in cohomology, see e.g. Piacenza 84, 1.1).

The derived categories those abelian sheaves with transfers for the Nisnevich site with are A1-homotopy invariant provides a model for motives known as *Voevodsky motives* or similar (Voevodsky, p. 20).

I have splitt off from *classifying topos* an entry *classifying topos for the theory of objects* and added the statement about the relation to finitary monads.

started [[locally connected topos]]

]]>at *sober space* the only class of examples mentioned are Hausdorff spaces. What’s a good class of non-Hausdorff sober spaces to add to the list?

I have expanded a bit at *Serre-Swan theorem*: gave it an actual Idea-section, mentioned more variants (over general ringed spaces, in higher geometry) and added more references.