Re: Category Theory for the Sciences

[Vaughan Pratt <pratt@cs.stanford.edu>]

On 1/30/2015 11:03 AM, Fred E.J. Linton wrote:
> Just see how late, and how lackadaisically, Yoneda's Lemma enters into [Spivak's book].

As those who heard my CT'2011 talk (which grew out of my CT'2004 talk on
communes) may recall, I'm in favour of exploiting the Yoneda Lemma in
the way automobile manufacturers exploit the internal combustion engine:
not as something whose mechanism is to be understood but merely as a
means of propulsion controlled by the accelerator pedal.

Such an approach could potentially make it accessible to more than just
physicists, in particular to a wide range of workers in the social sciences.

To that end, define a Sigma-category (C,Sigma) to be any category C
equipped with a distinguished set Sigma of objects of C.  In the obvious
(to this audience) way, this determines a multisorted unary theory T,
namely T = J' as the opposite of the full subcategory J of C with ob(J)
= Sigma.

In T, the objects represent the sorts and the morphisms the operations
of the theory.

In C, every object represents some model of T and every morphism
represents some homomorphism of those models, not necessarily faithfully
(a homomorphism may have more than one representative).

What I find particularly appealing about this presentation of
multisorted unary theories and (some of) their models and homomorphisms
is that it extends so straightforwardly to Sigma-Pi-categories
(C,Sigma,Pi).  Here Pi is a second subset of ob(C) dual to Sigma in the
sense that

(a) whereas Sigma consists of the *sorts* of T, Pi consists of its
*properties*; and

(b) whereas the *elements* of a model M are the morphisms from Sigma to
M, with a: s --> M being an element of sort s, the *states* of M are the
morphisms from M to Pi, with x: M --> p being a state for property p.


[Two asides:

1.  There is a nice alliteration pun here between the duality of sorts
and properties and that between sums and products.

2.  Up to equivalence there is an obvious notion of maximal
Sigma-Pi-category subject to leaving elements and states invariant.
With that notion, the following special cases arise:

(i) for Pi empty: the presheaf category Set^T;

(ii) for Sigma = {I}, Pi = {_|_}, rigid in the sense that |C(I,I)| =
|C(_|_,_|_)| = 1: the (ordinary) Chu category Chu(Set, C(I, _|_)); and

(iii) for Sigma = Pi: what Bill Lawvere has called the Isbell envelope
E(J) (J as above).]


One social science that could find Sigma-Pi-categories useful is
philosophy, which could find in them a single mathematical home for all
three of the problems of

(a) Cartesian dualism;

(b) extensionality of properties (as the sets of states of a model); and

(c) logical consistency of qualia (as morphisms from Sigma to Pi).

(Slightly) more on this application in Section 3.3 of Fundamenta
Informaticae 103 (2010) 203?218 at

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.395.2995&rep=rep1&type=pdf

Vaughan Pratt


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