Abstract. Equivalence is an important idea in mathematics. What does it mean for two structures to be considered equivalent? What is required for this equivalence to be established? The answers to these questions depend on the structure. For example, for two groups to be equivalent requires notions additional than for two sets to be equivalent.
We consider some equivelances and a more general concept: homotopy. One could suggest that homotopy is the most general equivelance, though through further
study we’ll see that this statement makes no sense. Levels of equivelance would likely be homotopy equivalent themselves.