May 25
Gosh these stable (∞, 1)-categories are so hot looking
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Gosh these stable (∞, 1)-categories are so hot looking
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How cool is it that a category object in the category of groups is the same thing as a group object in the category of categories (or even in category of groupoids)? There's just no obvious reason why swapping the words "groups" and "categories" should lead to equivalent definitions. I guess this is one of those cases in which category theory validating all the things you would want to be true.
I've been reading this short expository article, which explains the relationships between these and crossed modules, all of which are apparently incarnations of (strict?) 2-groups. I don't understand crossed modules at all yet, but I'm looking forward to working out for myself what the smallest finite strict 2-groups are, concretely. Maybe later this year I'll be able to read Baez & Lauda's notes on 2-groups.
It's crazy that category theorists use the term "2-group", though, since that has long meant something totally different in group theory: a 2-group in the original sense is a group whose elements all have order a power of 2, which is a completely unrelated concept.
One of the five great pains in my life is that globular infinity-categories are not the correct notion of an infinity-category it seems, yet globular infinity-categories feel to me like the most intuitive of the options as to what I would want an infinity-category to be.......
I had hopes for a moment when I read "informal and readable introduction to higher algebra". Back to nlab, I guess...
When you not just count, but also remember how you count, you arrive at the sphere [spectrum].
Lars Hesselholt
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Abstract: Topological quantum computation (TQC) is a new fault-tolerant approach to quantum information, where computations are carried out by braiding particles called anyons. Anyons are quasiparticles that exist in 2 + 1 dimensions and are neither bosons or fermions. Modular tensor categories capture the structure of anyon systems and thus serve as models for topological quantum computation. However, program of using category theory to find higher-level structures and protocols has yet to be applied to TQC due to the difficulty of working with modular tensor categories. This difficulty could be greatly mitigated by the development of a computer algebra system to represent such categories. Thus, a computer algebra system for representing modular tensor categories within the symmetric monoidal 2-category 2Vect was developed.
A general representation for 2Vect is described. This involves extending basic linear algebra operations to handle matrices of zero-dimension and 2-matrices (matrices of matrices). Then, representations for the 0-cells, 1-cells, and 2-cells of 2Vect are found. Due to the chosen representation, various structural isomorphisms are required to ensure equations expected of 1-categories hold in the 2-categories setting. These structural isomorphisms are explicitly constructed.
In order to identify special categories, such as modular tensor categories, a diagrammatic language of monoidal categories is extended for 2-categories. This language is used to construct the axioms for these special categories within 2Vect. Finally, a way of implementing these axioms within 2Vect in the computer algebra system is described. This system will now be used for the investigation of higher-level structures within modular tensor categories to better understand the structure of topological quantum computing.