Category theory basics
A category \(\catC=(\Ob,\Hom,id,\cdot)\) is:
- A collection of objects \(\Ob\)
- For any two objects \(d\) and \(c\), a set \(\Hom_\catC(d,c)\) of arrows (also called morphisms) from the domain \(d\) to the codomain \(c\)
- For each object \(x\), an identity arrow \(id_x\)
- For each pair of morphisms \(f:a\rightarrow b\) and \(g: b \rightarrow c\), a designated composite, \(f\cdot g: a \rightarrow c\)
- The composition must be associative and respect identity morphisms
A preorder \((X,\leq)\) is a category which has \(\Ob:=X\) and exactly one morphism \(a\rightarrow b\) if \(a\leq b\), zero morphisms otherwise.
- A category with \(\Ob:=\NN\) and \(a \leq b := a\) is a divisor of \(b\)
- A category with \(\Ob:=\ZZ\) and \(a \leq b\) being the usual ordering
The category \(\Set\) has sets as objects and functions as morphisms.
The category \(\CAT{Pos}\) has preorders as objects and monotone maps as morphisms.
It’s beyond the scope of a short talk to give a full technical introduction to category theory, so I will try my best to highlight some aspects of it that are relevant.
Relationships between categories
Each category is like a different mathematical universe. A natural way to move between such universes is a functor.
A functor \(F:\catC\rightarrow\catD := (F_0:\Ob_\catC\rightarrow \Ob_\catD,\ F_1:\Hom_\catC \rightarrow \Hom_\catD)\) between categories \(\catC\) and \(\catD\) is:
- A choice, for each \(x \in \Ob_\catC\) of an object \(F(x) \in \Ob_\catD\)
- A choice, for each \(f \in \Hom_\catC(a,b)\), of a morphism \(F(f) \in \Hom_\catD(F(a),F(b))\)
- Preservation of identities \(F(1_a) = 1_{F(a)}\)
- Preservation of composites \(F(f\cdot_\catC g) = F(f)\cdot_\catD F(g)\)
A functor between preorders (viewed as categories) is a monotone map.
There is a functor \(\CAT{Pos}\rightarrow \Cat\) which makes explicit interpreting each preorder as a category.
\(\Cat\) is a category whose objects are categories and morphisms are functors.
Given a functor \(\catC\rightarrow\catD\) which is injective on objects and morphisms, \(\catC\) is said to be a subcategory of \(\catD\).
An adjunction between cancellative monoids and incompatibility frames
Let \(U: \CAT{IF}\rightarrow \CAT{CancMon}\) forget the distinguished subset. Let \(F: \CAT{Mon}\rightarrow\CAT{IF}\) take a monoid and produce a trivial implication frame (with \(\II = \{0\}\)).
Maps \(\CAT{IF}((X,0_X,+_X,\{0\}),(Y,0,+_Y,\II_Y))\) are naturally identified with monoid homomorphisms \(\CAT{CancMon}((X,0_X,+_X),(Y,0,+_Y))\), as the constraints
Moving between incompatibility frames of different monoids
Given an implication frame \(F=(S,+_S,0_S,\II)\) and a monoid homomorphism \(f: (S,+_S,0_S)\rightarrow(T,+_T,0_T)\) we can freely construct an implication frame \(f(F) = (T,+,0,f(\II))\)
We have a monoid homomorphism, so we just need to check that the local and global properties are satisfied: by fiat, we declare \(\II'=f(\II)\). To see that this entails, the global property, we check
\[\forall p,q \in X: (\forall r\in X : p+r\in\II \implies q+r\in\II) \implies (\forall s \in Y: f(p)+s \in f(\II) \implies f(q)+s \in f(\II))\]
We note that \(f(p)+s \in f(\II)\) means that \(\exists r \in X: f(r)=s\).
(Ok is this actually true?)