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• Created page, work in progress

• An attempt to create this page was made by Paulo Perrone, but the creation was not successful. Am creating the page without any content beyond ’TODO’ now as a test.

• I fixed a broken link to Guy Moore’s lectures

• am starting some minimum here. Have been trying to read up on this topic. This will likely become huge towards beginning of next year

• stub entry, for the moment just so as to satisfy links

• 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

$Cat(x \times y,z) \cong Cat(x,CAT(y,z)).$

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…

• added a graphics showing length scales of fundamental physics in the observable universe

• starting something

• brief category:people-entry for hyperlinking references at kaon and at flavour anomaly

• A stub for M-theory. What’s supposed to be so mysterious about it? Is it that people don’t even know what form it would take?

• added English translation of this bit

PN§260 Der Raum ist in sich selbst der Widerspruch des gleichgültigen Auseinanderseins und der unterschiedlosen Kontinuität, die reine Negativität seiner selbst und das Übergehen zunächst in die Zeit. Ebenso ist die Zeit, da deren in Eins zusammengehaltene entgegengesetzte Momente sich unmittelbar aufheben, das unmittelbare Zusammenfallen in die Indifferenz, in das ununterschiedene Außereinander oder den Raum.

Space is in itself the contradiction of the indifferent being-apart and of the difference-less continuity, the pure negativity of itself and the transformation, first of all, to time. In the same manner time – since its opposite moments, held together in unity, immeditely sublate themselves – is the undifferentiated being-apart or: space.

And polished a little around and following this bit.

• some bare minimum, for the moment just a glorified list of references

• added to quantum anomaly

• an uncommented link to Liouville cocycle

• a paragraph with the basic idea of fermioninc anomalies

• the missing reference to Witten’s old article on spin structures and fermioninc anomalies.

The entry is still way, way, stubby. But now a little bit less than a minute ago ;-

• Todd points out elsewhere that there is a problem with the following sentence in the section Smallness in the context of universes:

$C$ is essentially $U$-small if there is a bijection from its set of morphisms to an element of $U$ (the same for the set of objects follows); this condition is non-evil.

(introduced in revision 11).

It looks to me that first of all this is not the right condition – the right condition must mention equivalence of categories to a U-small category.

• Correct the characterization of nerves of groupoids.

• starting something – not done yet

• I have expanded the Idea section at state on a star-algebra and added a bunch of references.

The entry used to be called “state on an operator algebra”, but I renamed it (keeping the redirect) because part of the whole point of the definition is that it makes sense without necessarily having represented the “abstract” star-algebra as a C*-algebra of linear operators.

• To fulfil a link.

• I came to think that the term geometric type theory for the type theory internal toi sheaf toposes should exists. Thanks to Bas Spitter for pointing out that Steve Vickers had already had the same idea (now linked to at the above entry).

Also created geometric homotopy type theory in this vein, with some evident comments.

• This is a base topic of my contribution. It introduces a new function that gives series whose coefficients are powers of fine structure constant. Furthermore each member represents natural physical interaction. It can be treated as natural physics that introduces natural particles.

May be I made a lot of mistakes. I will correct them.

• added a further quote from

interview with Mike Duff by Graham Fermelo, The universe speaks in numbers – Interview 14 (web):

(7:04) The problem we face is that we have a patchwork understanding of M-theory, like a quilt. We understand this corner and that corner, but what’s lacking is the overarching big picture. So directly or indirectly, my research hopes to explain what M-theory really is. We don’t know what it is.

In a certain sense, and this is not a popular statement, I think it’s premature to be asking: “What are the empirical consequences”, because it’s not yet in a mature enough state, where we can sensibly make falsifiable prediction.