How Many Categories Exist with One Object and Two Morphisms?

[Singularity]

Homework Statement

Show that there are only two possible categories with one object and two morphisms.

Homework Equations

None

The Attempt at a Solution

My thinking here is as follows : Let's say that the object in our category is A, and that the two morphisms are f and 1, where 1 is the identity. Obviously they both have domain = codomain = A. So the only way that I see to make a difference between the categories is to define the composition differently, i.e. to make either ff = f or ff = 1. Is this enough to say that the two are different?

 

[mighty2000]

Your reasoning is correct. The two possible categories with one object and two morphisms are indeed different depending on how the composition is defined. If we define the composition such that ff = f, then this category would be a monoid, where f is the identity element. On the other hand, if we define the composition such that ff = 1, then this category would be a group, where f is the inverse of itself.

In both cases, we have a trivial category with only one object and two morphisms, but the different definitions of composition give rise to different algebraic structures. Therefore, we can conclude that there are only two possible categories with one object and two morphisms.