III. Categorified Arithmetic 0. Introduction Naive introduction to categorified arithmetic, with pictures of: addition of finite sets, empty set, associative law, commutative law... multiplication of finite sets, one-element set, associative law, commutative law distributive law The problems of subtraction and division. Why division is easier! The weak quotient, division, and groupoid cardinality Euler characteristic and negative sets Structure types: A few fun combinatorial puzzles, solved via structure types Homotopy cardinality Stuff types and n-stuff types Categorifying quantum mechanics... just a taste! 1. Various kinds of categories with colimits Various ways of categorifying addition or multiplication as a property that a category might have, including on the additive side: a. Categories with finite coproducts free category on 1 with finite coproducts on 1 is FinSet The adjunction between Cat and FinCoprodCat. This gives an example of a "KZ doctrine", i.e. a pseudomonad on Cat with some extra properties, as discussed in F. Marolejo, Doctrines whose structure forms a fully faithful adjoint string, TAC 3 (1997), 24-44 and the references. Do existing coproducts get replaced by new "formal" ones in this construction, or are there two constructions, one of which keeps the coproducts that already exist? b. Categories with finite colimits free category on 1 with finite colimits is FinSet The adjunction between Cat and CoLex c. Categories with all small colimits (cocomplete categories) free cocomplete category on 1 is Set The free cocomplete category on a category C: presheaves on C or $\hat{C}$ The Yoneda embedding is the unit of this adjunction; we can think of \hat as giving a pseudomonad on Cat! Limits work dually. 2. Various kinds of monoidal categories Various ways of categorifying addition or multiplication as a structure that a category might have, including: a. Monoidal categories The free monoidal category on a category C is $\bar{C}$ (or whatever they call it, borrowing the notation from the computer science notation for the set of words in a given alphabet): the category with finite lists of objects in C as objects, and lists of morphisms in C as morphisms. $\bar{1} \iso 1Braid_2 \sim \N$ The adjunction between Cat and MonCat. A functor from 1 into the the underlying category of a monoidal category C, i.e. an object of C, is "the same" is a monoidal functor from $\N$ into C. The wreath product of a category and a concrete category; $\bar{C}$ as the wreath product of $C$ and the discrete category $\N$, which is just $\bar{1}$ b. Braided monoidal categories The free braided monoidal category on a category C is br(C), the category with lists of objects in C as objects, and "3d braids labelled with morphisms in C" as morphisms. $br(1) \iso 1Braid_3$ The adjunction between Cat and BrMonCat: $br{C}$ as the wreath product of $C$ and $1Braid_3$, which is just $br{1}$ c. Symmetric monoidal categories The free symmetric monoidal category on a category C is fam(C), the category with lists of objects in C as objects, and "4d braids labelled with morphisms in C" as morphisms. fam(1) \iso 1Braid_4 \sim FinSet_0 The adjunction between Cat and SymmMonCat - take advantage of Hyland-Fiore work on species here! $br{C}$ as the wreath product of $C$ and $1Braid_4$, which is just $\FinSet_0$ 3. Categorified Rigs Examples of categorified rigs: FinSet Set MSet for a monoid M - the Burnside 2-rig of M MVect for a monoid M - the representation 2-rig of M RBiMod for a ring R RMod for a commutative rig R Vector bundles over a space - same as projective modules over the algebra of continuous (or smooth) functions Some generalization of all of these, e.g. hom(C,V) where is a nice sort of monoidal category, perhaps a 2-rig itself! For example: SimpSet Other examples: Top Diff AMod for a bialgebra A Putting both additive and multiplicative structures/properties on a category we get various concepts of categorified rig, including: a. Distributive Categories Here everything is property: we have a category with finite products and coproducts, one distributing over the other... or is it finite limits and colimits? There are lots of variations! The adjunction between Cat and DistCat The free distributive category on ... is FinSet? Just a little taste of Schanuel's theory of the free distributive category on one object x with x = 2x + 1. b. 2-Rigs Here addition is property: we have a monoidal cocomplete category, with the tensor product distributing over colimits. One can also define braided and symmetric 2-rigs. Starting with a monoidal category, we can obtain a 2-rig by taking presheaves on this category. The tensor product in this 2-rig goes by the name of "Day convolution". What we're really doing here is applying the "free cocompletion" 2-functor Presheaves: Cat -> CocompleteCat to get a 2-functor Presheaves: MonCat -> 2Rig We also get Presheaves: BrMonCat -> Br2Rig and Presheaves: SymmMonCat -> Symm2Rig Composing the last one with the "free symmetric monoidal category" 2-functor \fam: Cat -> SymmMonCat we get the "free symmetric 2-rig" 2-functor from Cat to Symm2Rig. Another way to get the free symmetric 2-rig on a category C is to to apply the "direct sum of symmetric tensor powers" 2-functor to the category of presheaves on C. Here we see a "distributive law" at work - for categorified monads. What's the free symmetric 2-rig on nothing? Set? The free symmetric 2-rig on 1 is the category of structure types. The golden 2-rig. c. Rig categories Here everything is structure; we have lots of coherence laws to deal with, a la Kelly--Laplaza. What's the free rig category on nothing? FinSet_0? 3. Structure types in combinatorics Theory and examples of ordinary and exponential generating functions 4. Homotopy Cardinality Groupoid cardinality and weak quotients Homotopy cardinality and homotopy colimits 5. Stuff Types and n-Stuff Types 6. Case Studies a. Feynman Diagrams The free symmetric 2-rig on one generator (the category of structure types) as a categorified version of Fock space. The symmetric 2-rig of n-stuff types as an improved version of this. Wick powers and Feynman diagrams. M-stuff types where M is a monoid, e.g. U(1). b. Representations of the Classical Groups Characterization in terms of universal properties of the representation 2-rigs of the classical groups: GL(n,k) SL(n,k) O(n) U(n) Sp(n) SO(n) SU(n) Also for unitary representations? c. Schur-Weyl Theory The map from the free symmetric 2-rig on one generator (the category of structure types) to the free symmetric k-linear abelian 2-rig on one generator (the category Rep[GL(infinity)], also known as the category of representations of the symmetric groups d. q-Deformation The Hecke algebras as a q-deformation of the group algebras of the symmetric group; the q-deformed version of the free symmetric k-linear abelian 2-rig on one generator. The work of Joyal and Street on this topic. d. Euler characteristic versus homotopy cardinality Schanuel's theory of the Euler characteristic: the free distributive category on an object x with x = 2x + 1. Mysterious relations between homotopy cardinality and Euler characteristic.

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