Monoidal categories

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.

SMC(1)

A rough definition of a symmetric monoidal structure on a category \(\mathcal{C}\)

Two additional constituents
1. An object ?\(I \in Ob(\mathcal{C})\) called the monoidal unit
2. A functor ?\(\mathcal{C}\times \mathcal{C}\xrightarrow{\otimes}\mathcal{C}\) called the monoidal product
Subject to the well-behaved, natural isomorphisms
1. ?\(I \otimes c \overset{\lambda_c}\cong c\)
2. ?\(c \otimes I \overset{\rho_c}\cong c\)
3. ?\((c \otimes d)\otimes e \overset{\alpha_{c,d,e}}\cong c \otimes (d\otimes e)\)
4. ?\(c \otimes d \overset{\sigma_{c,d}}\cong d \otimes c\)
A category equipped with these is a symmetric monoidal category

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Set as SMC(1)

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

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Exercise 4-52: referenced

Exercise 4-48(1)

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.

Solution(0)

TODO

Exercise 4-50(2)

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:
1. What are \(g_E(5,3)\) and \(g_F(5,3)\)?
2. What are \(g_E(3,5)\) and \(g_F(3,5)\)?
3. What is \(h(5,true)\)?
4. What is \(h(-5,true)\)?
5. What is \(h(-5,false)\)?
6. What are \(q_G(-2,3)\) and \(q_F(-2,3)\)?
7. What are \(q_G(2,3)\) and \(q_F(2,3)\)?

Solution(1)

\(False,\ 2\)
\(True,\ -2\)
\(5\)
\(-5\)
\(6\)
\((2,-13)\) ... \(a\mapsto -2,\ b \mapsto 3,\ c\mapsto 2,\ d\mapsto -10,\ e\mapsto true,\ f\mapsto -13, g \mapsto 2\)
\((-1,7)\) ... \(a\mapsto 2,\ b \mapsto 3,\ c \mapsto 2,\ d\mapsto 10,\ e\mapsto false, f\mapsto 7, g\mapsto -1\)