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November 11, 2015

Burritos for Category Theorists

Posted by John Baez

You’ve probably heard of Lawvere’s Hegelian taco. Now here is a paper that introduces the burrito to category theorists:

The source of its versatility and popularity is revealed:

To wit, a burrito is just a strong monad in the symmetric monoidal category of food.

Posted at 5:24 PM UTC | Permalink | Followups (25)

November 10, 2015

Weil, Venting

Posted by Tom Leinster

From the introduction to André Weil’s Basic Number Theory:

It will be pointed out to me that many important facts and valuable results about local fields can be proved in a fully algebraic context, without any use being made of local compacity, and can thus be shown to preserve their validity under far more general conditions. May I be allowed to suggest that I am not unaware of this circumstance, nor of the possibility of similarly extending the scope of even such global results as the theorem of Riemann–Roch? We are dealing here with mathematics, not theology. Some mathematicians may think they can gain full insight into God’s own way of viewing their favorite topic; to me, this has always seemed a fruitless and a frivolous approach. My intentions in this book are more modest. I have tried to show that, from the point of view which I have adopted, one could give a coherent treatment, logically and aesthetically satisfying, of the topics I was dealing with. I shall be amply rewarded if I am found to have been even moderately successful in this attempt.

I was young when I discovered by harsh experience that even mathematicians with crashingly comprehensive establishment credentials can be as defensive and prickly as anyone. I was older when (and I only speak of my personal tastes) I got bored of tales of Grothendieck-era mathematical Paris.

Nonetheless, I find the second half of Weil’s paragraph challenging. Is there a tendency, in category theory, to imagine that there’s such a thing as “God’s own way of viewing” a topic? I don’t think that approach is fruitless. Is it frivolous?

Posted at 11:41 PM UTC | Permalink | Followups (14)

November 3, 2015

Cakes, Custard, Categories and Colbert

Posted by John Baez

As you probably know, Eugenia Cheng has written a book called Cakes, Custard and Category Theory: Easy Recipes for Understanding Complex Maths, which has gotten a lot of publicity. In the US it appeared under the title How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics, presumably because Americans are less familiar with category theory and custard (not to mention the peculiar British concept of “pudding”).

Tomorrow, Wednesday November 4th, Eugenia will appear on The Late Show with Stephen Colbert. There will also be another lesser-known guest who looks like this:

Apparently his name is Daniel Craig and he works on logic—he proved something called the Craig interpolation theorem. I hear he and Eugenia will have a duel from thirty paces to settle the question of the correct foundations of mathematics.

Anyway, it should be fun! If you think I’m making this all up, go here. She’s really going to be on that show.

Posted at 10:03 PM UTC | Permalink | Followups (16)

October 21, 2015

When Not To Use ‘The’

Posted by David Corfield

I’ve revised A Note on ‘The’ and ‘The Structure of’ in Homotopy Type Theory, which we discussed a few months ago – The Structure of A.

As Mike said back then, trying to define ‘structure of ’ in HoTT is a form of ‘noodling around’, and I rather think that working up a definition of ‘the’ is more important. The claim in the note is that we should only form a term ‘The AA’ for a type AA, if we have established the contractibility of AA. I claim that this makes sense of types which are singleton sets, as well as the application of ‘the’ in cases where category theorists see universal properties, such as ‘the product of…’.

Going down the hh-levels, contractible propositions are true ones. I think it’s not too much of a stretch to see the ‘the’ of ‘the fact that PP’ as an indication of the same principle.

But what of higher hh-levels? Is it the case that we don’t, or shouldn’t, use ‘the’ with types which are non-contractible groupoids? One case that came to mind is with algebraic closures of fields. Although people do say ‘the algebraic closure of a field FF’ since any two such are isomorphic, as André Henriques writes here, a warning is often felt necessary about the use of ‘the’ in that these isomorphisms are not canonical. Do people here also get a little nervous with ‘the universal cover of a space’? Perhaps intuitively one provides a little extra structure (map in or map out, say) which makes the isotropy trivial.

I was also wondering if we see traces of this phenomena in natural language, but I think the examples I’m coming up with (the way to hang a symmetrical painting, the left of a pair of identical socks) are better thought of as concerning the formation of terms in equivariant contexts (as at nLab: infinity-action), and the subject of a lengthy discussion a while ago on coloured balls.

Posted at 11:16 AM UTC | Permalink | Followups (32)

October 20, 2015

Four Tribes of Mathematicians

Posted by John Baez

Since category theorists love to talk about their peculiar role in the mathematics community, I thought you’d enjoy this blog article by David Mumford, which discusses four “tribes” of mathematicians with different motivations. I’ll quote just a bit, just to whet your appetite for the whole article:

The title refers to an “astonishing experimental investigation” of what your brain is doing when you experience mathematical beauty. This was carried out here:

But on to the four tribes….

Posted at 1:06 AM UTC | Permalink | Followups (3)

October 5, 2015

Configurations of Lines and Models of Lie Algebras (Part 2)

Posted by John Baez

To get deeper into this paper:

we should think about the 24-cell, the D 4\mathrm{D}_4 Dynkin diagram, and the Lie algebra 𝔰𝔬(8)\mathfrak{so}(8) that it describes. After all, its this stuff that underlies the octonions, which in turn underlies the exceptional Lie algebras, which are the main subject of Manivel’s paper.

Posted at 10:03 PM UTC | Permalink | Followups (19)

October 4, 2015

Configurations of Lines and Models of Lie Algebras (Part 1)

Posted by John Baez

I’m really enjoying this article, so I’d like to talk about it here at the n-Category Café:

It’s a bit intense, so it may take a series of posts, but let me just get started…

Posted at 11:06 PM UTC | Permalink | Followups (4)

September 30, 2015

An exact square from a Reedy category

Posted by Emily Riehl

I first learned about exact squares from a blog post written by Mike Shulman on the nn-Category Café.

Today I want to describe a family of exact squares, which are also homotopy exact, that I had not encountered previously. These make a brief appearance in a new preprint, A necessary and sufficient condition for induced model structures, by Kathryn Hess, Magdalena Kedziorek, Brooke Shipley, and myself.

Proposition. If RR is any (generalized) Reedy category, with R +RR^+ \subset R the direct subcategory of degree-increasing morphisms and R RR^- \subset R the inverse subcategory of degree-decreasing morphisms, then the pullback square: iso(R) R id R + R \array{ iso(R) & \to & R^- \\ \downarrow & \swArrow id & \downarrow \\ R^+ & \to & R} is (homotopy) exact.

In summary, a Reedy category (R,R +,R )(R,R^+,R^-) gives rise to a canonical exact square, which I’ll call the Reedy exact square.

Posted at 8:50 PM UTC | Permalink | Followups (9)

September 19, 2015

The Free Modular Lattice on 3 Generators

Posted by John Baez

The set of subspaces of a vector space, or submodules of some module of a ring, is a lattice. It’s typically not a distributive lattice. But it’s always modular, meaning that the distributive law

a(bc)=(ab)(ac) a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)

holds when aba \le b or aca \le c. Another way to say it is that a lattice is modular iff whenever you’ve got aaa \le a', then the existence of an element xx with

ax=axandax=ax a \wedge x = a' \wedge x \; \mathrm{and} \; a \vee x = a' \vee x

is enough to imply a=aa = a'. Yet another way to say it is that there’s an order-preserving map from the interval [ab,b][a \wedge b,b] to the interval [a,ab][a,a \vee b] that sends any element xx to xax \vee a, with an order-preserving inverse that sends yy to yby \wedge b:

Dedekind studied modular lattices near the end of the nineteenth century, and in 1900 he published a paper showing that the free modular lattice on 3 generators has 28 elements.

One reason this is interesting is that the free modular lattice on 4 or more generators is infinite. But the other interesting thing is that the free modular lattice on 3 generators has intimate relations with 8-dimensional space. I have some questions about this stuff.

Posted at 9:57 PM UTC | Permalink | Followups (69)

September 14, 2015

Where Does The Spectrum Come From?

Posted by Tom Leinster

Perhaps you, like me, are going to spend some of this semester teaching students about eigenvalues. At some point in our lives, we absorbed the lesson that eigenvalues are important, and we came to appreciate that the invariant par excellence of a linear operator on a finite-dimensional vector space is its spectrum: the set-with-multiplicities of eigenvalues. We duly transmit this to our students.

There are lots of good ways to motivate the concept of eigenvalue, from lots of points of view (geometric, algebraic, etc). But one might also seek a categorical explanation. In this post, I’ll address the following two related questions:

  1. If you’d never heard of eigenvalues and knew no linear algebra, and someone handed you the category FDVect\mathbf{FDVect} of finite-dimensional vector spaces, what would lead you to identify the spectrum as an interesting invariant of endomorphisms in FDVect\mathbf{FDVect}?

  2. What is the analogue of the spectrum in other categories?

I’ll give a fairly complete answer to question 1, and, with the help of that answer, speculate on question 2.

(New, simplified version posted at 22:55 UTC, 2015-09-14.)

Posted at 1:06 AM UTC | Permalink | Followups (37)

September 3, 2015

Rainer Vogt

Posted by Tom Leinster

I was sad to learn that Rainer Vogt died last month. He is probably best-known to Café readers for his work on homotopy-algebraic structures, especially his seminal 1973 book with Michael Boardman, Homotopy Invariant Algebraic Structures on Topological Spaces.

It was Boardman and Vogt who first studied simplicial sets satisfying what they called the “restricted Kan condition”, later renamed quasi-categories by André Joyal, and today (thanks especially to Jacob Lurie) the most deeply-explored incarnation of the notion of (,1)(\infty, 1)-category. Their 1973 book also asked and answered fundamental questions like this:

Given a topological group XX and a homotopy equivalence between XX and another space YY, what structure does YY acquire?

Clearly YY is some kind of “group up to homotopy” — but the details take some working out, and Boardman and Vogt did just that.

Martin Markl wrote a nice tribute to Vogt, which I reproduce with permission here:

Posted at 11:54 PM UTC | Permalink | Followups (3)

September 2, 2015

How Do You Handle Your Email?

Posted by Tom Leinster

The full-frontal assault of semester begins in Edinburgh in twelve days’ time, and I have been thinking hard about coping strategies. Perhaps the most incessantly attention-grabbing part of that assault is email.

Reader: how do you manage your email? You log on and you have, say, forty new mails. If your mail is like my mail, then that forty is an unholy mixture of teaching-related and admin-related mails, with a relatively small amount of spam and perhaps one or two interesting mails on actual research (at grave peril of being pushed aside by the rest).

So, you’re sitting there with your inbox in front of you. What, precisely, do you do next?

Posted at 12:47 PM UTC | Permalink | Followups (24)

August 31, 2015

Wrangling generators for subobjects

Posted by Emily Riehl

Guest post by John Wiltshire-Gordon

My new paper arXiv:1508.04107 contains a definition that may be of interest to category theorists. Emily Riehl has graciously offered me this chance to explain.

In algebra, if we have a firm grip on some object X X , we probably have generators for X X . Later, if we have some quotient X/ X / \sim , the same set of generators will work. The trouble comes when we have a subobject YX Y \subseteq X , which (given typical bad luck) probably misses every one of our generators. We need theorems to find generators for subobjects.

Posted at 5:33 PM UTC | Permalink | Followups (17)

August 17, 2015

A Wrinkle in the Mathematical Universe

Posted by John Baez

Of all the permutation groups, only S 6S_6 has an outer automorphism. This puts a kind of ‘wrinkle’ in the fabric of mathematics, which would be nice to explore using category theory.

For starters, let Bij nBij_n be the groupoid of nn-element sets and bijections between these. Only for n=6n = 6 is there an equivalence from this groupoid to itself that isn’t naturally isomorphic to the identity!

This is just another way to say that only S 6S_6 has an outer isomorphism.

And here’s another way to play with this idea:

Given any category XX, let Aut(X)Aut(X) be the category where objects are equivalences f:XXf : X \to X and morphisms are natural isomorphisms between these. This is like a group, since composition gives a functor

:Aut(X)×Aut(X)Aut(X) \circ : Aut(X) \times Aut(X) \to Aut(X)

which acts like the multiplication in a group. It’s like the symmetry group of XX. But it’s not a group: it’s a ‘2-group’, or categorical group. It’s called the automorphism 2-group of XX.

By calling it a 2-group, I mean that Aut(X)Aut(X) is a monoidal category where all objects have weak inverses with respect to the tensor product, and all morphisms are invertible. Any pointed space has a fundamental 2-group, and this sets up a correspondence between 2-groups and connected pointed homotopy 2-types. So, topologists can have some fun with 2-groups!

Now consider Bij nBij_n, the groupoid of nn-element sets and bijections between them. Up to equivalence, we can describe Aut(Bij n)Aut(Bij_n) as follows. The objects are just automorphisms of S nS_n, while a morphism from an automorphism f:S nS nf: S_n \to S_n to an automorphism f:S nS nf' : S_n \to S_n is an element gS ng \in S_n that conjugates one automorphism to give the other:

f(h)=gf(h)g 1hS n f'(h) = g f(h) g^{-1} \qquad \forall h \in S_n

So, if all automorphisms of S nS_n are inner, all objects of Aut(Bij n)Aut(Bij_n) are isomorphic to the unit object, and thus to each other.

Puzzle 1. For n6n \ne 6, all automorphisms of S nS_n are inner. What are the connected pointed homotopy 2-types corresponding to Aut(Bij n)Aut(Bij_n) in these cases?

Puzzle 2. The permutation group S 6S_6 has an outer automorphism of order 2, and indeed Out(S 6)= 2.Out(S_6) = \mathbb{Z}_2. What is the connected pointed homotopy 2-type corresponding to Aut(Bij 6)Aut(Bij_6)?

Puzzle 3. Let BijBij be the groupoid where objects are finite sets and morphisms are bijections. BijBij is the coproduct of all the groupoids Bij nBij_n where n0n \ge 0:

Bij= n=0 Bij n Bij = \sum_{n = 0}^\infty Bij_n

Give a concrete description of the 2-group Aut(Bij)Aut(Bij), up to equivalence. What is the corresponding pointed connected homotopy 2-type?

Posted at 9:45 AM UTC | Permalink | Followups (51)

August 9, 2015

Two Cryptomorphic Puzzles

Posted by John Baez

Here are two puzzles. One is from Alan Weinstein. I was able to solve it because I knew the answer to the other. These puzzles are ‘cryptomorphic’, in the vague sense of being ‘secretly the same’.

Posted at 7:21 AM UTC | Permalink | Followups (10)