Originally posted by loop quantum gravity
my question: does the morphism above imply the other way around, ie, y->x?
A morphism is a mathematical concept that describes a structure-preserving mapping between two mathematical objects. It can be seen as a generalization of functions, where the objects involved can be more complex than numbers or sets.
A morphism is defined as a mapping between two objects that preserves their algebraic structure. This means that the relationship between the objects is maintained under the mapping, and the operations on the objects are preserved.
The implication of a morphism is that it establishes a relationship between two objects based on their algebraic structure. This relationship can be used to understand the properties of the objects and to make predictions about their behavior.
No, the direction of a morphism does not necessarily imply the opposite direction. It is possible for a morphism to be one-way, where the mapping only goes from one object to the other. However, in some cases, there may be a bijective morphism where the mapping works both ways.
To determine if a morphism works both ways, we can use the concept of invertibility. If a morphism is invertible, then it has an inverse mapping that goes in the opposite direction, and the relationship between the objects is maintained in both directions. However, not all morphisms are invertible, and some may only work in one direction.