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*: TS –> T1
(where TS 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*: TS –> 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 –> TS
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 (TS and T1) and three functions
d*: TS –> T1
c*: TS –> T1
s*: T1 –> TS
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 = 1B
Now we have to show that the three functions
d*: TS –> T1
c*: TS –> T1
s*: T1 –> TS
corresponding to the T-field satisfy
d*s* = c*s* = 1T1
d*s* (u) = d* (us) = usd = u11 = u
since sd = 1 – d –> S – s –> 1 = 11 and d*s* = 1T1
c*s* (u) = c* (us) = usc = u11 = u
since sc = 1 – c –> S – s –> 1 = 11 and c*s* = 1T1
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.