Exercise 30 (Conceptual Mathematics, p. 151)

If S, d, c is a given bipointed object

d: 1 –> S

c: 1 –> S

in a category X, then for each object X of X, the graph of ‘X fields’ on S is actually a reflexive graph, and for each map f: X –> Y in X, the induced maps on sets constitute a map of reflexive graphs.

Given a space S representing a room and a map

t: S –> T

(where T is the temperature line) specifying the temperature at each point in the room, the ‘temperature field’ has reflexive graph structure when two points

d: 1 –> S

c: 1 –> S

in the space S are distinguished (1 is a one-point space).  Since the domain set of the function

t: S –> T

is same as the codomain set of

d: 1 –> S

we can compose them to get a point on the temperature line T

1 – d –> S – t –> T = 1 – td –> T

which is the temperature at the point

d: 1 –> S

in the room S.   So, corresponding to each temperature field in the room S i.e. corresponding to each function

t: S –> T

we have a temperature

td: 1 –> T

at the point

d: 1 –> S

in S.  In other words, the function

d: 1 –> S

induces a function

d*: T^S –> T¹

(where T^S is the set of all possible temperature fields on S and T¹ is the set of all temperature values on T) with

d* (t: S –> T) = td: 1 –> T

giving the temperature td at the point d in S for a given temperature field t on S.  Along the same lines, the second distinguished point

c: 1 –> S

induces another function

c*: T^S –> T¹

with

c* (t) = tc

giving the temperature tc at point c in S.  Furthermore, the unique function

s: S –> 1

(mapping all points in S to the only point in 1) induces another function

s*: T¹ –> T^S

mapping each point on the temperature line to a temperature field which has that temperature at every point in S.

The above temperature-field or T-field consisting of two sets (T^S and T¹) and three functions

d*: T^S –> T¹

c*: T^S –> T¹

s*: T¹ –> T^S

has the structure of reflexive graphs as shown below.

A reflexive graph consists of two sets and three functions

p: A –> B

q: A –> B

r: B –> A

such that the function r is the common section of p and q.  In other words,

pr = qr = 1_B

Now we have to show that the three functions

d*: T^S –> T¹

c*: T^S –> T¹

s*: T¹ –> T^S

corresponding to the T-field satisfy

d*s* = c*s* = 1_T¹

d*s* (u) = d* (us) = usd = u1₁ = u

since sd = 1 – d –> S – s –> 1 = 1₁ and d*s* = 1_T¹

c*s* (u) = c* (us) = usc = u1₁ = u

since sc = 1 – c –> S – s –> 1 = 1₁ and c*s* = 1_T¹

Thus for each physical quantity, such as temperature T, we obtain a reflexive graph.  Transformations from one physical quantity into another induce transformations of the corresponding reflexive graphs.  For example, given a transformation

f: T –> V

we can pre-compose it with the T-field

t: S –> T

to obtain a V-field

ft: S –> V

on the space S.  Pre-composing the V-field with a point

d: 1 –> S

gives the value of the V-field

(ft)d: 1 –> V

at that point in space S.  Alternatively, we could first find the value of the T-field

t: S –> T

at the point

d: 1 –> S

i.e.

td: 1 –> T

and post-compose it with the given transformation of fields

f: T –> V

to obtain the value

f(td): 1 –> V

of the V-field at the point

d: 1 –> S

in the space S.  The associativity of composition of functions i.e.

(ft)d = f(td)

preserves the reflexive graph structure.  In addition to the above equation corresponding to

d: 1 –> S

we have two more equations

(ft)c = f(tc)

f(us) = (fu)s

corresponding to

c: 1 –> S

s: S –> 1

all of which together constitute a structure-preserving transformation of reflexive graphs.  Thus fields (such as temperature T on a space S with two distinguished points and a retraction of the space to a one-point space) along with their transformations (such as from T to V) can be construed as a category of reflexive graphs.