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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 , we probably have generators for . Later, if we have some quotient , the same set of generators will work. The trouble comes when we have a subobject , which (given typical bad luck) probably misses every one of our generators. We need theorems to find generators for subobjects.
August 17, 2015
A Wrinkle in the Mathematical Universe
Posted by John Baez
Of all the permutation groups, only 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 be the groupoid of -element sets and bijections between these. Only for 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 has an outer isomorphism.
And here’s another way to play with this idea:
Given any category , let be the category where objects are equivalences and morphisms are natural isomorphisms between these. This is like a group, since composition gives a functor
which acts like the multiplication in a group. It’s like the symmetry group of . But it’s not a group: it’s a ‘2-group’, or categorical group. It’s called the automorphism 2-group of .
By calling it a 2-group, I mean that 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 , the groupoid of -element sets and bijections between them. Up to equivalence, we can describe as follows. The objects are just automorphisms of , while a morphism from an automorphism to an automorphism is an element that conjugates one automorphism to give the other:
So, if all automorphisms of are inner, all objects of are isomorphic to the unit object, and thus to each other.
Puzzle 1. For , all automorphisms of are inner. What are the connected pointed homotopy 2-types corresponding to in these cases?
Puzzle 2. The permutation group has an outer automorphism of order 2, and indeed What is the connected pointed homotopy 2-type corresponding to ?
Puzzle 3. Let be the groupoid where objects are finite sets and morphisms are bijections. is the coproduct of all the groupoids where :
Give a concrete description of the 2-group , up to equivalence. What is the corresponding pointed connected homotopy 2-type?
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’.