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

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

Posted at 8:50 PM UTC | Permalink | Followups (5)

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

$a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)$

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

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

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

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 $\mathbf{FDVect}$ of finite-dimensional vector spaces, what would lead you to identify the spectrum as an interesting invariant of endomorphisms in $\mathbf{FDVect}$?

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 $(\infty, 1)$-category. Their 1973 book also asked and answered fundamental questions like this:

Given a topological group $X$ and a homotopy equivalence between $X$ and another space $Y$, what structure does $Y$ acquire?

Clearly $Y$ 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 $X$, we probably have generators for $X$. Later, if we have some quotient $X / \sim$, the same set of generators will work. The trouble comes when we have a subobject $Y \subseteq X$, 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 $S_6$ 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 $Bij_n$ be the groupoid of $n$-element sets and bijections between these. Only for $n = 6$ 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 $S_6$ has an outer isomorphism.

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

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

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

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

By calling it a 2-group, I mean that $Aut(X)$ 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 $Bij_n$, the groupoid of $n$-element sets and bijections between them. Up to equivalence, we can describe $Aut(Bij_n)$ as follows. The objects are just automorphisms of $S_n$, while a morphism from an automorphism $f: S_n \to S_n$ to an automorphism $f' : S_n \to S_n$ is an element $g \in S_n$ that conjugates one automorphism to give the other:

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

So, if all automorphisms of $S_n$ are inner, all objects of $Aut(Bij_n)$ are isomorphic to the unit object, and thus to each other.

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

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

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

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

Give a concrete description of the 2-group $Aut(Bij)$, 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)

July 28, 2015

Internal Languages of Higher Categories

Posted by Mike Shulman

(guest post by Chris Kapulkin)

I recently posted the following preprint to the arXiv:

Btw if you’re not the kind of person who likes to read mathematical papers, I also gave a talk about the above mentioned work in Oxford, so you may prefer to watch it instead. (-:

I see this work as contributing to the idea/program of HoTT as the internal language of higher categories. In the last few weeks, there has been a lot of talk about it, prompted by somewhat provocative posts on Michael Harris’ blog.

My goal in this post is to survey the state of the art in the area, as I know it. In particular, I am not going to argue that internal languages are a solution to many of the problems of higher category theory or that they are not. Instead, I just want to explain the basic idea of internal languages and what we know about them as far as HoTT and higher category theory are concerned.

Disclaimer. The syntactic rules of dependent type theory look a lot like a multi-sorted essentially algebraic theory. If you think of sorts called types and terms then you can think of rules like $\Sigma$-types and $\Pi$-types as algebraic operations defined on these sorts. Although the syntactic presentation of type theory does not quite give an algebraic theory (because of complexities such as variable binding), it is possible to formulate dependent type theory as an essentially algebraic theory. However, actually showing that these two presentations are equivalent has proven complicated and it’s a subject of ongoing work. Thus, for the purpose of this post, I will take dependent type theories to be defined in terms of contextual categories (a.k.a. C-systems), which are the models for this algebraic theory (thus leaving aside the Initiality Conjecture). Ultimately, we would certainly like to know that these statements hold for syntactically-presented type theories; but that is a very different question from the $\infty$-categorical aspects I will discuss here.

A final comment before we begin: this post derives greatly from my (many) conversations with Peter Lumsdaine. In particular, the two of us together went through the existing literature to understand precisely what’s known and what’s not. So big thanks to Peter for all his help!

Posted at 3:30 AM UTC | Permalink | Followups (22)

July 19, 2015

Category Theory 2015

Posted by John Baez

Just a quick note: you can see lots of talk slides here:

Category Theory 2015, Aveiro, Portugal, June 14-19, 2015.

The Giry monad, tangent categories, Hopf monoids in duoidal categories, model categories, topoi… and much more!

Posted at 9:19 AM UTC | Permalink | Followups (1)

July 8, 2015

Mary Shelley on Invention

Posted by Tom Leinster

From the 1831 introduction to Frankenstein:

Invention, it must be humbly admitted, does not consist in creating out of void, but out of chaos; the materials must, in the first place, be afforded: it can give form to dark, shapeless substances, but cannot bring into being the substance itself. […] Invention consists in the capacity of seizing on the capabilities of a subject, and in the power of moulding and fashioning ideas suggested to it.

She’s talking about literary invention, but it immediately struck me that her words are true for mathematical invention too.

Except that I can’t think of a part of mathematics I’d call “dark” or “shapeless”.

Posted at 12:54 PM UTC | Permalink | Followups (4)

June 29, 2015

What is a Reedy Category?

Posted by Mike Shulman

I’ve just posted the following preprint, which has apparently quite little to do with homotopy type theory.

The notion of Reedy category is common and useful in homotopy theory; but from a category-theoretic point of view it is odd-looking. This paper suggests a category-theoretic understanding of Reedy categories, which I find more satisfying than any other I’ve seen.

Posted at 6:33 PM UTC | Permalink | Followups (20)

Feynman’s Fabulous Formula

Posted by Simon Willerton

Guest post by Bruce Bartlett.

There is a beautiful formula at the heart of the Ising model; a formula emblematic of all of quantum field theory. Feynman, the king of diagrammatic expansions, recognized its importance, and distilled it down to the following combinatorial-geometric statement. He didn’t prove it though — but Sherman did.

Feynman’s formula. Let $G$ be a planar finite graph, with each edge $e$ regarded as a formal variable denoted $x_e$. Then the following two polynomials are equal:

$\displaystyle \sum_{H \subseteq_{even} G} x(H) = \prod_{[\vec{\gamma}] \in P(G)} \left(1 - (-1)^{w[\vec{\gamma}]} x[\vec{\gamma}]\right)$

I will explain this formula and its history below. Then I’ll explain a beautiful generalization of it to arbitrary finite graphs, expressed in a form given by Cimasoni.

Posted at 7:47 AM UTC | Permalink | Followups (19)

June 21, 2015

Posted by Mike Shulman

In my last post I promised to follow up by explaining something about the relationship between homotopy type theory (HoTT) and computer formalization. (I’m getting tired of writing “publicity”, so this will probably be my last post for a while in this vein — for which I expect that some readers will be as grateful as I).

As a potential foundation for mathematics, HoTT/UF is a formal system existing at the same level as set theory (ZFC) and first-order logic: it’s a collection of rules for manipulating syntax, into which we can encode most or all of mathematics. No such formal system requires computer formalization, and conversely any such system can be used for computer formalization. For example, the HoTT Book was intentionally written to make the point that HoTT can be done without a computer, while the Mizar project has formalized huge amounts of mathematics in a ZFC-like system.

Why, then, does HoTT/UF seem so closely connected to computer formalization? Why do the overwhelming majority of publications in HoTT/UF come with computer formalizations, when such is still the exception rather than the rule in mathematics as a whole? And why are so many of the people working on HoTT/UF computer scientists or advocates of computer formalization?

Posted at 6:47 AM UTC | Permalink | Followups (10)

June 12, 2015

Carnap and the Invariance of Logical Truth

Posted by David Corfield

I see Steve Awodey has a paper just out Carnap and the invariance of logical truth. We briefy discussed this idea in the context of Mautner’s 1946 article back here.

Steve ends the article by portraying homotopy type theory as following in the same tradition, but now where invariance is under homotopy equivalence. I wonder if we’ll see some variant of the model/theory duality he and Forssell found in the case of HoTT.

Posted at 12:09 PM UTC | Permalink | Followups (2)