[FRIAM] How is a vector space like an evolutionary function?

[jon zingale]

I probably should have added an abstract :) What I am getting at here is that
*free* object constructions build higher-order structures relative to those
they are built from and relative to the target of their associated
right-adjoint (often the category of sets via a forgetful functor). That
these higher-order structures then support notions that may not exist in a
direct way relative to the structures they are built from, one can view
these newly supported notions as a kind of *epiphenomena* relative to the
underlying structure.

While it is not meaningful to speak about the length of a set, we can
support a higher-order monoidal structure where length is a meaningful
notion. For a monoid relevant example[𝜆], consider the following recursive
definition of the length of a list:

len :: [a] -> Int
len [] = 0
len [t] = 1
len (t:ts) = len [t] + len ts

What I seek to show is that this definition follows directly from an
adjunction that constructs *free* monoids from sets.

A functor F :: Set -> Mon builds from the elements of a set X, the set of
all possible words on X and equips this set with a multiplication (++) and
an identity element ([]). This functor has a right 'forgetful' functor which
forgets the monoidal structure and returns the set of all possible words on
X, X*. We can then look at morphisms from this object (X*, ++, []) into a
natural number monoid (ℕ, +, 0), taking concatenation to addition and the
empty word to the additive identity (the first and third rule in the
definition above). Now, the second rule (len [t] = 1) is a little arbitrary
and finds its meaning in the interaction between the underlying category of
sets and the higher-order monoidal category. There in Set, we find a
composition that ultimately gives len its character. The composition of
Functors (G∘F) gives a natural map, η, which includes X into X* as an
inclusion of generators. Looking at η: X -> X* and the map const1 :: X -> ℕ
(Where ℕ is given by the same forgetful functor G and all elements of X are
mapped to the number 1), we find that the unique function G(len) :: X* -> ℕ
that makes the triangle commute and respects the functorial conditions gives
via the monoidal structure the notion of length we seek. The structural
functors (G and F) can be seen as *founding* a category of monoids upon a
category of sets, and dually *structuring* the category of sets by the
category of monoids.

[𝜆] This example is from Benjamin Pierce's 'Category Theory for Computer
Scientists'.



--
Sent from: http://friam.471366.n2.nabble.com/