#classfunctions

[m---a---x]

Can we slightly weaken the properties to make it interesting? In general, what are the objects here:

R is a class of pointed endofunctions from X to Y (or morphisms in case of a non-Set topos). This seems like a good start to build something interesting, let's see what conditions it must fulfill.

The first condition says that the point of any of our pointed functions must be reachable from its image by f.

The second condition says that each function has (if at all) one unique point associated to it. We will call that point the center of the function going forward. Considering that we do not have any given additional structure on X that is an odd choice of a property.

The third condition in conjunction with the pervious one says that Y is a singleton. This makes it uninteresting, let's weaken it instead to only xRg, so the center of a function is a fixpoint.

The fourth condition is interesting. It says that any point is the center of a function that when extending f (now possibly only a partial function, so we must by the definition of R have Y=X) has its center reachable by a finite power of f, the latter part of which holds (for m=0) iff any point is the center of a function that extending f has a center.

The last condition says that f contracts less than q many points to any center of a function, which by the former property means that it contracts less than q many points to any point. In particular this means if our class only contains continuous functions on a metric space it must be injective.

So in conclusion a pseudo-scromulate is a tuple (X,R,f,q) representing a

class of endofunctions on X with fixed points,

a "center" classfunction from this class to X, such that the image of a function is a fixed point of the function,

an endofunction f which is not too contracting (<q to each point) and the centers of all functions such that extending f has a center is the entire space.

How interesting this is depends on what our notion of center is and what f represents. For example our class might be rotations of a geometric space where the center is the center of rotation and f is a translation.

A scromulate is an ordered tuple (X, Y, R, f, n, q) where X is a set, Y is a subset of X, R is a relation on X × Y^X, f : Y -> X is a function, and n and q are integers such that

for all x in X and g : X -> Y, if xRg then there exists some x' in X such that f(g(x'))=x

for any g : X -> Y and any a, b in Y, if aRg and bRg then a=b

for all x in X and g : X -> Y, we have g(x)Rg

for every x in X there exists some m < n, some y in Y and some g : X -> Y such that xRg and f^m(y)Rgf

for any g : X -> Y we have |{y in Y : f(y)Rg}| < q

Scromulates are not interesting. I made them up just now and pulled all the conditions out of my ass.

[marcglanville]

Lil bro decided to pull up and sit in the teachers lap 😶😐😶🤦����‍♂️. #ClassFunctions @z.oe.z get ya brother! --- -- #theMasonCruz #CruzControl2014 #cruzWorldOrder #lilHomie #ClassinSession

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