This is distinguished from the geometric theory of motion in a Lorentzian manifold. In that context, the laws of motion dictate that objects can only move along time-like future-pointing curves. This creates a geometric preorder on a Lorentzian manifold. As a consequence, there are both algebraic and geometric theories of motion.
Transformations:
Functions are among the most basic units of mathematics alongside sets. We will start by describing the changes in a set by functions from the set back to itself. This is generalized to include partial transformations, which allow us to describe changes on only parts of a set back to themselves.- Transformations
- Permutations
- Partial transformations
- Charts
Monotone Galois connections:
The different types of actions by transformations, permutations, charts, etc all necessarily produce preorders that describe how they move the elements of the sets they act upon.Definitions. let $X$ be a set then for any set of partial transformations $S$ we define $Rel(S)$ to be the preorder closure of the set of all ordered pairs of $S$. For a preorder $R$ we define $Full(R)$ to be the full set of all partial transformations whose ordered pairs are in $R$. \[ Rel : Sub(PT_X) \to Po(X) \] \[ Full : Po(X) \to Sub(PT_X) \] Then these two form adjoints of one another between the lattice of preorders $Po(X)$ and the lattice of partial transformation semigroups $Sub(PT_X)$. $Rel$ is a lower adjoint and $Full$ is an upper adjoint.
Theorem. $Rel$ and $Full$ are adjoints of one another. \[ S \subseteq Full(R) \Leftrightarrow Rel(S) \subseteq R \] Proof. (1) suppose that $Rel(S) \subseteq R$ then every partial transformation of $S$ is included in $R$, so that $S \subseteq Full(R)$. (2) suppose that $S \subseteq Full(R)$ then every partial transformation of $S$ is included in $R$, which implies that $Rel(S) \subseteq S$. $\square$
The monotonicity of $Rel$ means that the larger a set of motions, the larger the corresponding preorder it moves in. The same general principle is true for any kind of motion. This is very useful in keeping track of the directions of special types of actions.
The monotone Galois connection between preorders and partial transformations can be restricted to other sets of transformations. Each of the different types of transformations has a corresponding complete system of transformations associated to a preorder: \[ Full : Po(X) \to Sub(PT_X) \] \[ FT : Po(X) \to Sub(T_X) \] \[ FPS : Po(X) \to Sub(PS_X) \] \[ FS : Po(X) \to Sub(S_X) \] These four have the obvious inclusions: $FT(X) \subseteq Full(X)$, $FPS(X) \subseteq Full(X)$, $FS(X) \subseteq FPS(X)$, and $FS(X) \subseteq FPS(X)$. In the case of actions by charts and permutations, we know they often form inverse semigroups or groups.
Theorem. let $E$ be a symmetric preorder on $X$, then $FPS(E)$ is an inverse subsemigroup of $PS_X$ and $FS(X)$ is a subgroup of $S_X$.
Proof. suppose that $p \subseteq E$ then $p^{-1} \subseteq E^{-1}$ because the transpose relation is monotone. $E^{-1} = E$ because $E$ is symmetric, so that $p^{-1} \subseteq E$, which means that $FPS(X)$ is inverse closed and $FS(X)$ is as well. $\square$
There is therefore a natural adjointness relationship between symmetric preorders and orbit symmetric permutation groups, which are the maximal permutation groups with a given orbit. This demonstrates the various ways in which we can get different types of actions from preorders, but there is yet one more.
A preorder $R$ on a set $X$ is a set of ordered pairs $(a,b)$ but an ordered pair is also an atomic partial transformation $\{(a,b)\}$ with a single element. This produces a partial semigroup action associated with any preorder $R$ on its ground set: \[ f: R \to PS_X \] By considering a preorder as an action on a set by atomic partial transformations, we can see that the underlying action preorder of an action is simply a way of reducing a transformation system to its simplest components: which are the ordered pairs that define movements from one point to another.
Monoid actions and categories:
The concept of a transformation semigroup can naturally be generalized to a monoid action. Instead of a fixed set of transformatinos, a monoid action can have a set of elements that transform another set.* A monoid action is the action of a total semigroup on a set by a set of total transformations. \[ f : M \to T_X \] * A category is the action of a partial semigroup on a set by a set of atomic partial transformations. \[ f : Arrows(C) \to PT_{Ob(C)} \] This demonstrates that the morphisms of a category act on objects. A morphism $f : A \to B$ can move an object from point $A$ to point $B$, which is an atomic partial transformation.
| Elements | Actions | |
|---|---|---|
| Monoid action | Elements | Transformations |
| Category | Objects | Morphisms |
Definition. a faithful category is a preorder.
A preorder is a faithful category, because each morphism $f : A \to B$ produces a different atomic partial transformation $\{(A,B)\}$. Faithful categories can therefore be identified uniquely by their action preorders. The opposite notion, a completely faithless category is a trivial monoid action on a single object. In that case, every action of a morphism moves an object back to itself.
The idea of defining actions by atomic partial transformations is so fundamental that categories play an important role in the algebraic theory of motion and change. There closest relatives, as we have seen here are the monoid actions which also play an important role in models of change.
We have described both monoid actions and categories by their actions on an underlying set of objects. But another aspect of the algebraic laws of motion in monoid actions and categories, is that transformations and morphisms can act on themselves.
Definition. let $f : M \to T_X$ be a monoid action of $M$ on $X$. Then $M$ also acts on itself by the left, or right, or two sided actions. Dually for a category $f : Arrows(C) \to PS_{OB(C)}$. Green's preorders are the action preorders of these self-induced actions and Green's relations are defined from them.
The Green's relations merely described the algebraic laws of motion of a monoid, in therefore makes sense that in general they are the most important property of a given monoid. As they are very general concept dealing with the dynamics of motion, they can naturally be generalized to categories.
This produces the left, right, and two sided action preorders of a category. Furthermore, we can get subpreorders by subcategories like the mono preorder and the epi preorder. These produce the mono input action and epi input action preorders whose condensations are the posets of subobjects and quotients of a category.
An immediate difference between monoid actions and categroies is that the former form topoi, and the later do not. But the topoi of monoid actions comes by fixing a given ground monoid $M$ and then considering only $M$ sets, where categories can be constructed from many different kinds of partial semigroups.
A framework for higher category theory:
A general model of algebraic motions arises by keeping track of types of objects and how they can act on each other. In the simplest models of algebraic motion, we only have two types of objects: elements and transformations which can only act on each other in a single way but there is no reason that this cannot be generalized.A 2-category is an algebraic system of motion, in which 2-morphisms can act on 1-morphisms to move them from point to another, 1-morphisms can act on objects. This is further generalized to tricategories, which are categories that are enriched over 2-categories and so on. In each case, we have a number of types that act on each other.
See also:
[1] Categories for order theorists
[2] Semigroup methods in category theory
References:
[1] Categories
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