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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• changed broken website link to new departmental entry. Note the name change, some redirection may be useful.

• I worked on brushing up (infinity,1)-category a little

• mostly I added in a section on homotopical categories, using some paragraphs from Andre Joyal's message to the CatTheory mailing list.

• in this context I also rearranged the order of the subsections

• I removed in the introduction the link to the page "Why (oo,1)-categories" and instead expanded the Idea section a bit.

• added a paragraph to the beginning of the subsection on model categories

• added the new Dugger/Spivak references on the relation between quasi-cats and SSet-cats (added that also to quasi-category and to relation between quasi-categories and simplicial categories)

• there had been no references at Hilbert space, I have added the following, focusing on the origin and application in quantum mechanics:

• John von Neumann, Mathematische Grundlagen der Quantenmechanik. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.

• George Mackey, The Mathematical Foundations of Quamtum Mechanics A Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963

• E. Prugovecki, Quantum mechanics in Hilbert Space. Academic Press, 1971.

• There’s a paper out characterising the category of continuous linear functions between Hilbert spaces

• Chris Heunen, Andre Kornell, Axioms for the category of Hilbert spaces (arXiv:2109.07418)

But Hilb concerns short linear maps between Hilbert spaces. Should we have a page for the former category?

• brief category:people-entry for hyperlinking references at PU(ℋ)

• I created homotopy extension property and homotopy lifting property. If somebody wonders why I made identical copy of one of them on my personal nlab part is because there I want to keep conservative page for students and here in the main nlab I expect more vigorous extensions by others. On the other hand, I would like to have under homotopy lifting property mention of various variants like "soft map" homotopy lifting property, the homotopical variant of Dold etc. all in one place.

• starting something, with a hat-tip to Charles Rezk

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

• (Hi, I’m new)

I added some examples relating too simple to be simple to the idea of unbiased definitions. The point is that we often define things to be simple whenever they are not a non-trivial (co)product of two objects, and we can extend this definition to cover the “to simple to be simple case” by removing the word “two”. The trivial object is often the empty (co)product. If we had been using an unbiased definition we would have automatically covered this case from the beginning.

I also noticed that the page about the empty space referred to the naive definition of connectedness as being

“a space is connected if it cannot be partitioned into disjoint nonempty open subsets”

but this misses out the word “two” and so is accidentally giving the sophisticated definition! I’ve now corrected it to make it wrong (as it were).

• added a second equivalent definition at quasi-category , one that may be easier to motivate

• am finally giving this its own entry (this used to be treated within the entry on Elmendorf’s theorem)

but just a stub for the moment

• The cut rule for linear logic used to be stated as

If $\Gamma \vdash A$ and $A \vdash \Delta$, then $\Gamma \vdash \Delta$.

I don’t think this is general enough, so I corrected it to

If $\Gamma \vdash A, \Phi$ and $\Psi,A \vdash \Delta$, then $\Psi,\Gamma \vdash \Delta,\Phi$.

• Page created and some notes added

• essentially just a stub entry, for the moment just to make links work

• typo list: - closed \$ for page rendering about 60% through file - invalid mathcal(G)

ccg

Anonymous

• wrote a definition and short discussion of covariant derivative in the spirit of oo-Chern-Weil theory

• The cube diagram on this page is MIA, in case anyone feels like investing a few minutes of tikzcd fun to fix it.

• 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.

The approach is echoed in Riehl & Verity 13 with Cat enhanced to the homotopy 2-category of (∞,1)-categories.

Also touched some wording further below (“is very difficult to read” $\mapsto$ “may be difficult to read”)

Finally, I made “formal category theory” a redirect to this page (this would deserve to point to a dedicated page, but as long as that doesn’t exit, it’s good to have it redirect here)

• A stub to record findings from wondering whether such a thing exists.

• a stub entry, for the moment just to record some references

• Added a reference to the following which provides a proof of the Arnold conjecture

• Mohammed Abouzaid, Andrew J. Blumberg, Arnold Conjecture and Morava K-theory, (arXiv:2103.01507)
• Entry on lattice ordered groups. Work in progress.

• made explicit (here) that simplicial $\infty$-colimits are $\infty$-sifted

• added the previously missing pointer to HTT 5.5.8.4 for the statement that $\Delta^{op}$ is $\infty$-sifted.

• Stub to come.

• Karoubian category

Added the definitions of Karoubian category and Karoubi envelope that appear in (an exercise in) SGA 4.

A stupid question: why do they call that difference kernel the image of p? In what sense is it the image?

• I was a bit confused about a search for “pullback lemma” not returning any result, hence this redirect

• Created to give active reference link.

• Added some remark on the order of a semiring. Actually, does anybody know if any semiring embedds into a semifield?

• Page to complete stubs on residuated things (more could be added).

• New entry to complete references.

• Entry to fill in some of the ideas from a poset viewpoint. Note the use of the term ’residual’ for the left adjoint. It seems that this use is really traditional coming from the sense that a ’residue’ is the bit left ove. The link with ‘internal homs’ is then a categorication of that, which puts a different light on internal homs!

• Updated Mark Lawson’s information.

• I had started an entry “exponentiation” but then thought better of it and instead expanded the existing exponential object: added an examples-section specifically for $Set$ and made some remarks on exponentiation of numbers.

• Started an article on monoidal monad. An earlier redirect had sent it over to Hopf monad which is something that Zoran was working on, but I think it deserves an article to itself, with discussion of the relation to commutative monads, etc. (which I have started).

• 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…

• created equivariant de Rham cohomology with a brief note on the Cartan model.

(I seem to remember that we had discussion of this in the general context of Lie algebroids elsewhere already, several years back. But now I cannot find it….)