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HOW MINDS THEORIZE: I. Opposites

January 27, 2014

Theories are made—made by us—in our struggle to come to terms with the given realityIn the universe of mathematical discourse, Lawvere’s Functorial Semantics objectifies abstraction as a functor.  Here we construct examples of the abstraction functor, which can inform the scientific discourse on mind—as it reaches for generals.

The abstraction functor

STop → T

(where the domain category STop is the category of models [in the category of sets S] of the codomain category T of theories) and its ‘opposite’ interpretation functor

T → STop

are the focus of our study (Sets for Mathematics, pp. 154 – 5, 248 – 50).  Beginning with simple cases such as T = 1, a category with one morphism, we slowly unravel the mathematical content of the opposite pair—abstraction, interpretation—of functors.

The category 1 with one morphism has one object and the definition of CATEGORY forces the one morphism to be the identity morphism on the one object.  The opposite category 1op has same objects as 1 but with arrows reversed (Sets for Mathematics, page 25).  Since reversing the identity morphism results in the same identity morphism, we have 1op = 1.

The category

STop

with T = 1 is the category

S1op

of functors

1opS

from the category 1op (with exactly one identity morphism) to the category of sets S.

A functor is an assignment of objects and maps in the codomain category to the objects and maps, respectively, of the domain category in a way respectful of the domain, codomain, composition, and identity of the domain category.  So an assignment of a set and its identity function in S to the only object and the only identity morphism, respectively, of 1op constitutes a functor

X: 1opS

Denoting the only object of 1op with

and its identity morphism with

• — 1 —> •

we find

X (•) = X (a set in S)

X (1) = 1X (identity function on X)

So we say that the functors

X: 1opS

correspond to sets in S.

A natural transformation

f: X → Y

from a functor

X: 1opS

to a functor

Y: 1opS

is an assignment of a function

X (•) — f —> Y (•)

to the only object (•) in 1op subject to the condition that the square

X (•) — f —> Y (•)

^                      ^

|                       |

X (1)               Y (1)

|                       |

X (•) — f —> Y (•)

assigned to the only map (1) in 1op commutes.

Upon substituting

X (•) = X (a set in S)

X (1) = 1X (identity function on X)

Y (•) = Y

Y (1) = 1Y

we find that the natural transformations

f: X → Y

correspond to functions in S.  So we say that the category

S1op

whose objects and morphisms are functors

X: 1opS

and natural transformations

f: X → Y

respectively, corresponds to the category of sets S.  With this identification, we can study the opposition between abstraction

STop → T

and interpretation

T → STop

With T = 1 and STop = S, we have an adjoint situation

A: S1

I: 1S

Since there is only one object (•) in the category 1, all sets of S are mapped to that only object and all functions (in S) are mapped to the only morphism (1) in the category 1 and that is the only functor (A) from the category of sets S to the category 1.  In contrast to this situation, there are many functors from 1 to the category of sets; each functor corresponding to a set (in S).  Out of these, there’s one

I: 1S

whose value at the only object (•) in the category 1 is the terminal set 1 (= {•}) in the category of sets S.

I (•) = 1 (in S)

I (1) = 11

and this interpretation functor

I: 1S

is the right adjoint to the abstraction functor

A: S1

Some more work is required to show that I is right adjoint to A, and I need a break ;)  More importantly, you are probably not feelin the connection between all this (assuming I didn’t screw-up the math big-time) and your mind busy theorizing.  Well, you will once we get to somewhat more involved categories such as

T = 2 (= [0 → 1])

T = G (= [D = •, A = • → •, s, t: D → A])

embodying correspondingly richer theories.  In the meantime, you may want to check out the Child’s problem (Conceptual Mathematics, page 96) and a simple and disturbingly delightful (deep) illustration of how an abstract structure arises from a particular example (Conceptual Mathematics, page 150).

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