### GRT Wonderland

John Baez recommends further installments of Geometric Representation Theory. In Lecture 17 see James Dolan explain how degroupoidification is related to logos theory! If a map taking sets to the trivial category (which we think of as a -1 category) is decategorification, then an enriched version takes us from categories to categories enriched in the trivial category, which are just sets (with Boolean truth values). Imagine a whole recursion process of decategorifications!

In Lecture 17 you can also see homology and cohomology rear their Medusa's heads! These functors give a way to turn spans (the natural morphisms which have been floating around) into matrices. Then Baez's Lecture 18 looks at the example of the groupoid of finite sets (and bijections), before explaining how groupoid cardinality can be a fraction! Recall that this came up when we looked at Abel sums and counting trees, not to mention that Euler characteristics for orbifolds are secretly this kind of number! That's one of the ways we counted the number of particle generations in M theory.

Our preference for operadification and cooperadification should be viewed with these new ingredients in mind. Remember that an operad is a one object multicategory. An example of an arrow in a multicategory is a cospan diagram, which is made of two arrows with the same target. Multicategories generalise to allow arrows with an arbitrary number of inputs and outputs. Fixing our attention on the 1-operad of associahedra, recall that the coherence law dimension is associated to the number of inputs. Thus spans and cospans are naturally associated with two dimensional structures underlying duality. An instance of duality can be seen in the cardinalities for $Z_{2}$ (appearing in Lecture 18), namely 2 as a set and $\frac{1}{2}$ as a groupoid! Decategorification takes groupoids to vector spaces (or sets), and cardinality is thus reduced to an integer.

In Lecture 17 you can also see homology and cohomology rear their Medusa's heads! These functors give a way to turn spans (the natural morphisms which have been floating around) into matrices. Then Baez's Lecture 18 looks at the example of the groupoid of finite sets (and bijections), before explaining how groupoid cardinality can be a fraction! Recall that this came up when we looked at Abel sums and counting trees, not to mention that Euler characteristics for orbifolds are secretly this kind of number! That's one of the ways we counted the number of particle generations in M theory.

Our preference for operadification and cooperadification should be viewed with these new ingredients in mind. Remember that an operad is a one object multicategory. An example of an arrow in a multicategory is a cospan diagram, which is made of two arrows with the same target. Multicategories generalise to allow arrows with an arbitrary number of inputs and outputs. Fixing our attention on the 1-operad of associahedra, recall that the coherence law dimension is associated to the number of inputs. Thus spans and cospans are naturally associated with two dimensional structures underlying duality. An instance of duality can be seen in the cardinalities for $Z_{2}$ (appearing in Lecture 18), namely 2 as a set and $\frac{1}{2}$ as a groupoid! Decategorification takes groupoids to vector spaces (or sets), and cardinality is thus reduced to an integer.

## 2 Comments:

Hi Kea. You write 'Multicategories generalise to allow arrows with an arbitrary number of inputs and outputs.' An arrow in a multicategory does indeed have an arbitrary number of inputs, but it only has one output.

For instance, a one-object multicategory is exactly an operad, and operads are made up of operations that can take any number of inputs, but always have just one output.

The Catsters have a nice introduction to multicategories on YouTube.

People

havestudied structures in which arrows can have multiple outputs as well as multiple inputs. There are several variations, depending on what kinds of diagram you want to be able to compose. Polycategories are one approach. Another approach deserves to be called 'modular multicategories', although as far as I know there's only been stuff published on the one-object case, modular operads.Yeah, sorry about being a bit casual with the terminology. Welcome to AF by the way.

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