/sci/ - Science & Math

[math]\mathbf{M\text{-}Set}[/math] is the category of actions of a monoid [math]\mathbf{M}[/math] on a set. This category is a topos. It has a subobject classifier [math]\Omega=(L_M, \omega)[/math] where [math]L_M[/math] is the set of left ideals of [math]\mathbf{M}[/math], and [math]\omega[/math] is the action-map [math]\omega : M \times L_M \rightarrow L_M[/math] where [math]\omega(m, B) = \{n : n * m \in B\}[/math].
Maps between objects in [math]\mathbf{M\text{-}Set}[/math] are equivariant morphisms, that is, they preserve monoid actions such that they are invariant to application order (before or after traversing the map).
The truth-values of this topos are then the maps [math]1 \rightarrow \Omega[/math] which are just the left ideals.
The characteristic morphism describes the extent to which some object is equivalent to another object. In the context of [math]\mathbf{M\text{-}Set}[/math] say we have an inclusion [math](X, \lambda)\rightarrow (Y, \mu)[/math] where the first element is a set and the second its associated monoid action. Then the characteristic morphism describing [math]X[/math]'s membership in [math]Y[/math] is [math]\chi_f : Y \rightarrow L_M[/math] where [math]\chi_f(y)=\{m:\mu(m, y)\in X\}[/math].

This is in fact the only morphism that will commute the diagram shown to the right. Why?
Right answer gets infinite tendies in heaven