Props - definition and first examples

Each object is the n-fold monoidal product of the generating object \(1\).

To specify a prop \(P\) it is enough to specify 5 things (and check they satisfy the rules for symmetric monoidal categories):

A set \(\mathcal{C}(m,n))\) of morphisms \(m \rightarrow n\) for \(m,n \in \mathbb{N}\).
For all naturals, an identity map \(n \xrightarrow{id_n} n\)
For all \(m,n \in \mathbb{N}\), a symmetry map \(m+n \xrightarrow{\sigma_{m,n}} n+m\)
A composition rule: given \(m \xrightarrow f n\) and \(n \xrightarrow g p\), a map \(m \xrightarrow{f;g} p\)
A monoidal product on morphisms: given \(m \xrightarrow f m'\) and \(n \xrightarrow g n'\), a map \(m+n \xrightarrow{f+g} m' +n'\)

Prop(1)

A prop

A symmetric strict monoidal category \((\mathcal{C}, 0,+)\) for which \(Ob(\mathcal{C})=\mathbb{N}\) and the monoidal product on objects is given by addition.

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Prop functor(1)

A \(\ref{Prop|\emph{prop|referenced}}\) functor \(\mathcal{C} \xrightarrow F \mathcal{D}\) A functor for which

\(\forall n \in \mathbb{N}: F(n)=n\)
For all \(m_1 \xrightarrow f m_2\) and \(n_1 \xrightarrow g n_2 \in \mathcal{C}\): \(F(f))+F(g)=F(f+g) \in \mathcal{D}\)

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FinSet as prop(1)

The prop FinSet
Morphisms \(m \xrightarrow f n\) are ?functions from \(\bar m\) to \(\bar n\).
The identities, symmetries, and composition rule are obvious.
The monoidal product on functions is given by ?the disjoint union.

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Bij as prop(1)

Recall that a bijection is both surjective and injective.
There is a prop Bij where morphisms are bijections
\(m\ne n \implies\) ?\(\mathbf{Bij}(m,n)=\varnothing)\)
Bij is a ?subcategory of \(\ref{FinSet as prop|\textbf{FinSet|referenced}}\), so we can use the same monoidal product.

Correl as prop(1)

The compact closed category \(\ref{Correl as CCC|\textbf{Corel|referenced}}\) is a ?prop.

Rel as prop(1)

There is a prop Rel
Morphisms are relations \(R \subseteq \bar m \times \bar n\)
Composition with \(S \subseteq \bar n \times \bar p\) is
- \(\{(i, k)\ |\ \exists (i, j) \in R \land \exists (j,k) \in S\}\)
Monoidal product is given by the coproduct, which amounts to placing the two relations side-by-side.

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Exercise 5-10: referenced

Bij FinSet prop functor(1)

The inclusion \(\mathbf{Bij} \overset{i}{\hookrightarrow} \mathbf{FinSet}\) is a prop functor.
There is a prop functor \(\mathbf{Bij} \xrightarrow F \mathbf{Rel_{Fin}}\) defined by \(F(\bar{m} \xrightarrow{f} \bar{n}):=\{(i,j)\ |\ f(i)=j\} \in \bar m \times \bar n\)

Exercise 5-10(2)

For the prop \(\ref{Rel as prop|\textbf{Rel|referenced}}\), provide the five aspects of props described in the notes.

Solution(1)

TODO

Exercise 5-5(1)

Draw a morphism \(3 \xrightarrow f 2\) and a morphism \(2 \xrightarrow g 4\) in \(\ref{FinSet as prop|\textbf{FinSet|referenced}}\)
Draw \(f+g\)
What is the natural composition rule for morphisms in \(\ref{FinSet as prop|\textbf{FinSet|referenced}}\)
What are the identities in \(\ref{FinSet as prop|\textbf{FinSet|referenced}}\)
Draw a symmetry map \(\sigma_{m,n}\) for some \(m,n\) in \(\ref{FinSet as prop|\textbf{FinSet|referenced}}\).

Solution(0)

TODO

Exercise 5-9(2)

A posetal prop is a prop that is also a poset.
In other words, a symmetric monoidal preorder \((\mathbb{N}, \leq)\) for some posetal relation \(\leq\), where the monoidal product is addition.
Give three examples of a posetal prop.

Solution(1)

The normal meaning of \(\leq\) as less than or equal to
The divisibility relation
The opposite of the first example (greater than or equal to).

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