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)