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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)