Are reflexive and associative morphisms good enough?

Question

This is probably a trivial question so I apologize in advance If it has already been answered (I didn't find anything).

This issue reared its head in a context of preorders and thin categories. I am really not a trained mathematician and I am very self conscious with my math skills so I could really use all the help I can get. I am not trying to find a shortcut for study or get quick answers, I would be more interested in the process of figuring this out rather than getting a clear cut answer although I won't say no to that either.

Consider a type of morphisms$\color{red}*$ that is reflexive and associative. Does such a type of morphisms guarantee that a collection of objects connected with those morphisms will constitute a category?

My intuition is that a reflexive morphism can serve as the identity morphism for every object but associativity of morphisms does not guarantee composition; in light of the previous statements, my understanding is that a collection of objects equipped with that type of morphisms is not a category.

Any help is appreciated.

Edit: I provide below definitions for 'reflexive' and 'associative' morphisms.

For every object $a$ the identity morphism $id_a$ is reflexive ie $id_a°f= f°id_a$.

Morphisms $f,g,h$ associate over "°" ie $(h°g)°f=h°(g°f)=h°g°f$.

$\color{red}*$ should I call it a 'class'? I am afraid that I'm abusing terminology here.

Answer 1

It's a bit unclear what you mean, as it looks a bit like you've just restated the definition of category. But it seems like you might be interested in this: let $C$ be a class with a partially defined function $\circ:C\times C\to C$ such that

1. When defined, $f\circ(g\circ h)=(f\circ g)\circ h$.
2. For every $f$ there exist ``identity morphisms" $i,j$ such that $f\circ i$ and $j\circ f$ are defined and such that, whenever they're defined, $g\circ i=g$ and $j\circ g=g$.

Then there exists a category $C'$, unique up to isomorphism, whose objects are symbols $x_i$ for each ``identity morphism" $i,j,...$ in $C$ and whose morphisms $\mathrm{Hom}(x_i,x_j)$ are those elements $f\in C$ such that $f\circ i$ and $j\circ f$ are defined. The identity morphism of $x_i$ is $i$, the composition operation is just $\circ$ (we have to prove a lemma that $f\circ g$ is defined if and only if there's an identity morphism $j$ such that $f\circ j$ and $j\circ g$ are defined), and associativity is guaranteed by axiom 1.

This is a well-known way of axiomatizing a category without any direct reference to objects at all.