From: firstname.lastname@example.org (Mark Hopkins)
Subject: Time as a Category (was: What is Time?)
Date: 28 Aug 1997 22:50:32 GMT
To elaborate a little on my comment:
> In the case where two events are 'ordered' one way with respect to one
> observer and another with respect to a second, then NEITHER is before the
> other. The partial ordering is what distinguishes relativity's space-time
> from Newton's.
> If you allow for closed time loops, then things turn into a mess...
What happens then is that you then have points A, B in space-time and time-
like curves f and g of the form:
With respect to f, B comes after A, but with respect to g, A comes after B.
So now the 'path-dependence' has to remain explicit.
A convenient way to write this is to say "f: A -> B", if A is before B on f.
We can define the null 'path' I_A: A -> A, and define composition fg: A -> C,
where f: B -> C and g: A -> B in the obvious way, with the following
properties trivially following:
I_B f = f = f I_A, when f: A -> B
(fg)h = f(gh), when f: C -> D, g: B -> C and h: A -> B
Thus, you have a category.
>SPK: How do we distinguish the null "path" I_A:A ->A from the null ray light cone structure?
The analogue of a metric is then a "time-measure" on the category: one with
Tf >= 0
T I_A = 0
T (fg) = Tf + Tg
>SPK: It is easy to see that this "time-measure" does not resolve the question of "becoming" that we experience. Do you have any thoughts on this? Typically we think about time in terms of clocks. Could we consider a time measure to be defined by first assuming that all morphisms g are actions, e.g. "the moving hands of the clock" and gauge g against a reference set A that serve as the face of the clock? Perhaps the reference set could be the dual of g:
(Work on this with V. Pratt’s stuff.)
`Also we need a way to relate the thermodynamic arrow of time with that of our subjective experience.
A 'path' p can be defined as a set of morphisms subject to the following
a) If f is in p with f: A -> B, then I_A and I_B are in p
b) p is closed under composition
c) If I_A, I_B are in p, then there exists an f in p such
that f: A -> B or f: B -> A
d) If f1 and f2 are in p with f1: A -> B and f2: A -> B
then f1 = f2.
This allows one to define a local time measure on a path as follows:
T_p(A, B) = Tf if f is in p with f: A -> B
= -Tf if f is in p with f: B -> A
In the case where one has f, g in p with f: A -> B and g: B -> A, it
follows by a) and d) that fg = I_B, so that 0 = T I_B = T(fg) = Tf + Tg
or that Tf = Tg = 0. Therefore T_p(A, B) = 0 is uniquely defined.
This enables us to include light-like curves.
By fixing an origin p_0 in p, one can define T_p(A) = T_p(p_0, A)
and it will follow that T_p(A, B) = T_p(B) - T_p(A), independently of
the origin p_0.
The morphism can be interpreted in various ways: as a time-like/light-like
curve, or a "transition" or "change" in the abstract between "events" A
and B. If you populate your space with enough of these, and of the right
kind, it should be possible to develop a finite axiom system of Riemannian
Geometry in terms of these category-theoretic primitives.
Nothing is actually mentioned about the algebraic structure of the values
Tf takes on. It could be any linearly ordered algebra with an addition
operation and 0. Particular, one which satisfies the properties:
(a+b)+c = a+(b+c); a+0=a=0+a; a+b=b+a
a+c=b+c -> a=b; a+b=0 -> a=0=b
there exists c such that a+c=b or b+c=a
with the ordering relation (a <= b) defined by (a + c = b for some c).
This yields the notion of congruence, via the equivalence
f == g <--> Tf = Tg
and a system of axioms for congruence could then be developed which capture
the algebraic properties listed above; thus taking us one level further back
to the primitives: category + congruence.
This then yields probably the most natural generalization of the partial
order before-after to something more elaborate when there are closed time