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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