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

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

]]>polished [[category of sheaves]] slightly

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

]]>created *Hodge structure*. Currently with nothing but a pointer to this nice book:

- Chris Peters, Jozef Steenbrink,
*Mixed Hodge Structures*, Ergebisse der Mathematik (2007) (pdf)

Eventually I’d think we should move over Hodge-structure articles from *Hodge theory* to here. But not tonight.

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.

]]>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).

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

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…

]]>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.

added pointer to

- Tom Lovering,
*Etale cohomology and Galois Representations*, 2012 (pdf)

for review of how Galois representations are arithmetic incarnations of local systems/flat connections. Added the same also to *local system* and maybe elsewhere.

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?

]]>I needed an entry for Charles Rezk’s term *model topos* in order to complete *locally presentable categories - table*, and so I created it. But we should have had that anyway. I have also cross-linked it with relevant entries

back in “The point of pointless topology” Peter Johnstone suggested that localic homotopy theory ought to be developed:

So far, relatively little work has been done on specific applications of locale theory in contexts like these; so it is perhaps appropriate to conclude this article by mentioning some areas which (in the writer’s opinion, at least) seem ripe for study in this way. One is homotopy theory: the work of Joyal, Fourman and Hyland [15] shows that in a constructive context it may be necessary to regard the real Une as a (nonspatial) locale, at least if we wish to retain the Heine-Borel theorem that its closed bounded subsets are compact. So there is scope for developing the basic ideas of homotopy theory for locales, starting from the localic notion of the unit interval; when interpreted in the two contexts mentioned above, it should yield results in the “Ex-homotopy theory” and “equivariant homotopy theory” that have been studied in recent years by James [27, 28]

Has anything been done in this direction?

]]>It seems that only now did I come across

- Garth Warner,
*Homotopical topos theory*(pdf)

So I went to record it in the References-section at *simplicial sheaf*, only to notice that this entry had never existed as a decent entry. I edited just a little for the moment.

started [[locally connected topos]]

]]>quick entry for *pullback of differential forms*, to be further expanded

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.

]]>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.)

]]>started *Galois cohomology*

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…

]]>stub for *constructible sheaf*

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).

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