General idea: take a thing we know and add structure to it such that things that were formerly properties become structures
Do in such a way as to be able to recover the thing we categorified by forgetting the new structure.
In categorified world, we have more structure available to talk about the relationships between objects.
An example is how we treated preorders as categories.
Ordinary categories are Set-categories
Categorification of arithmetic using the category FinSet
Replace natural numbers with arbtirary sets of that cardinality.
Replace \(+\) with coproduct.
This is good categorification because, with replacements, arithmatic truths are preserved: \(\bar{5}\sqcup \bar{3} \cong \bar{8}\)
To do...
Just like preorders are special kinds of categories, symmetric monoidal preorders are special kinds of monoidal categories.
Just as we can consider \(\mathcal{V}\) categories for a symmetric monoidal preorder, we can consider \(\mathcal{V}\) categories when \(\mathcal{V}\) is a monoidal category.
One difference is that associativity is up to isomorphism: e.g. products in set \(S \times (T \times U)\) vs \((S \times T) \times U\)
When the isomorphisms of a symmetric monoidal category are replaced with equalities, we call it strict
Due to "Mac Lane’s coherence theorem" we can basically treat all as strict...something we implicitly do when writing wiring diagrams.
A rough definition of a symmetric monoidal structure on a category \(\mathcal{C}\)
Two additional constituents
An object \(I \in Ob(\mathcal{C})\) called the monoidal unit
A functor \(\mathcal{C}\times \mathcal{C}\xrightarrow{\otimes}\mathcal{C}\) called the monoidal product
Subject to the well-behaved, natural isomorphisms
\(I \otimes c \overset{\lambda_c}\cong c\)
\(c \otimes I \overset{\rho_c}\cong c\)
\((c \otimes d)\otimes e \overset{\alpha_{c,d,e}}\cong c \otimes (d\otimes e)\)
\(c \otimes d \overset{\sigma_{c,d}}\cong d \otimes c\)
A category equipped with these is a symmetric monoidal category
Monoidal structure on Set
Let \(I\) be any singleton, say \(\{1\}\) and the monoidal product is the cartesian product.
This means that \(\times\) is a functor:
For any pair of sets in \((S,T) \in Ob(\mathbf{Set}\times\mathbf{Set})\), one obtains a set \(S \times T \in Ob(\mathbf{Set})\).
For any pair of morphisms (functions) one obtains a function \((f\times g)\) which works pointwise: \((f\times g)(s,t):=(f(s),g(t))\) which preserves identities and composition.
The bookkeeping isomorphisms are obvious in Set
Check that monoidal categories generalize monoidal preorders: a monoidal preorder is a monoidal category \((P,I,\otimes)\) where \(P(p,q)\) has at most one element.
TODO
Consider the monoidal category \((\mathbf{Set},1,\times)\) together with the following diagram TODO - NEED TO COPY HERE
\(A=B=C=D=F=G=\mathbb{Z}\) and \(E=\mathbb{B}\)
\(f_C(a)=|a|\),
\(f_D(a)=a*5\),
\(g_E(d,b)=d\leq b\)
\(g_F(d,b)=d-b\)
\(h(c,e)=\text{if }e\text{ then }c\text{ else }1-c\)
Suppose the whole diagram defines a function \(A \times B \xrightarrow{q} G \times F\)
Answer:
What are \(g_E(5,3)\) and \(g_F(5,3)\)?
What are \(g_E(3,5)\) and \(g_F(3,5)\)?
What is \(h(5,true)\)?
What is \(h(-5,true)\)?
What is \(h(-5,false)\)?
What are \(q_G(-2,3)\) and \(q_F(-2,3)\)?
What are \(q_G(2,3)\) and \(q_F(2,3)\)?
\(False,\ 2\)
\(True,\ -2\)
\(5\)
\(-5\)
\(6\)
\((2,-13)\) ... \(a\mapsto -2,\ b \mapsto 3,\ c\mapsto 2,\ d\mapsto -10,\ e\mapsto true,\ f\mapsto -13, g \mapsto 2\)
\((-1,7)\) ... \(a\mapsto 2,\ b \mapsto 3,\ c \mapsto 2,\ d\mapsto 10,\ e\mapsto false, f\mapsto 7, g\mapsto -1\)
We said that ordinary categories were just Set-categories, but our definition of \(\mathcal{V}\) categories required the \(\mathcal{V}\) to be a preorder!
We have to generalize (categorify) \(\mathcal{V}\) categories.
Symmetric monoidal preorders can be considered as symmetric monoidal categories, despite not providing the data for identities and composition (these are not needed because there is no choice).
Example of property becoming structure: \(I \leq \mathcal{X}(x,x)\) is a property of \(\mathcal{V}\) categories as defined earlier but become part of the structure in the categorified version of the definition.
A \(\mathcal{V}\) category (a category enriched in \(\mathcal{V}\)) where \(\mathcal{V}\) is a symmetric monoidal category.
Call the category \(\mathcal{X}\). There are four constituents:
A collection of objects, \(Ob(\mathcal{X})\)
For every pair in \(Ob(\mathcal{X})\) one specifies the hom-object \(X(x,y) \in \mathcal{V}\).
For every object, specify a \(I \xrightarrow{id_x}X(x,x) \in \mathcal{V}\) called the identity element
For every pair of compatible morphisms, a \(\mathcal{X}(x,y)\otimes\mathcal{X}(y,z)\xrightarrow{;}\mathcal{X}(x,z)\) called the composition morphism.
These satisfy the usual associative and unital laws.
Recall the example with Set as a symmetric moniodal category. Apply the definition of a \(\mathcal{V}\) category and see if this agrees with the definition of an ordinary category. Is there a subtle difference?
We’ve replaced the identity morphisms with maps from the monoidal unit, but that is functionally equivalent to ‘just picking one’ given that the initial object is a singleton.
What are identity elements in Lawvere metric spaces (Cost-categories)? How do we interpret this in terms of distances?
\(0\) was the monoidal unit - the distance from an object to itself.