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

• 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?

• 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.

• 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…

Nobuo Yoneda (kanji: 米田, 信夫, katakana: ヨネダ, ノブオ) was a Japanese mathematician.

He got his PhD degree in 1961 from the University of Tokyo, supervised by Shokichi Iyanaga.

• starting an entry, for the moment mainly in order to record the fact that “crossed homomorphisms” are equivalently homomorphic sections of the corresponding semidirect product group projection. This is obvious, but is there a reference that makes it explicit?

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

• There’s a first stab at it.

• I understood that the old terminology was ’projective system’, and ’projective limit’ refereed to the limit of a projective system. Can anyone confirm that? if I am right the present entry is slightly incorrect, but this needs checking first before changing it.

• brief category:people-entry for hyperlinking references at CW complex

• and properly expanded out the page title

• I edited The Joy of Cats to link to metacategory and to disambiguate quasicategory, as twice now someone on MO has used the term ’quasicategory’ to talk about (very) large categories. This way, if people find the book using the nLab page they are forewarned.

I also edited quasicategory to move the terminological warning up to the idea section where it is immediately visible, rather than in the second section, below the definition.

• added to equalizer statement and proof that a category has equalizers if it has pullbcks and products

• Added material on diagonal maps and the product functor, mentioning for instance the fact that the product functor is right adjoint to a diagonal functor.

• added a Properties-section to pullback

• I wrote out a proof that geometric realization of simplicial sets valued in compactly generated Hausdorff spaces is left exact, using essentially the observation that simplicial sets are the classifying topos for intervals, combined with various soft topological arguments. I left a hole to be plugged, that geometric realizations are CW complexes. I also added a touch to filtered limit, and removed a query of mine from triangulation.

I wanted a “pretty proof” for this result on geometric realization, centered on the basic topos observation (due to Joyal). I was hoping Johnstone did this himself in his paper on “a topological topos”, but I couldn’t quite put it together on the basis of what he wrote, so my proof is sort of “homemade”. I wouldn’t be surprised if it could be made prettier still. [Of course, “pretty” is in the eye of the beholder; mainly I want conceptual arguments which avoid fiddling around with the combinatorics of shuffle products (which is what I’m guessing Gabriel and Zisman did), decomposing products of simplices into simplices.]

• The entry Clifford algebra used to state the classification and Bott periodicty over the complex numbers, but not over the real numbers. I have added in now the relevant statements, straight from Lawson-Michelson:

Just the bare statements so far.