Is there a weaker structure a category is made out of

Question

(Category). A category $C$ consists of the following components:

• A class $\text{Ob}(C)$; the elements of $\text{Ob}(C)$ are objects of $C$.

• A set $\text{Mor}_C(X, Y)$ for each choice of objects $X, Y ∈ \text{Ob}(C)$; elements of $\text{Mor}_C(X, Y)$ are called morphisms from $X$ to $Y$. (We implicitly assume that morphism sets between different pairs of objects are disjoint.)

• For all objects $X, Y, Z ∈ \text{Ob}(C)$ a composition$ ◦: \text{Mor}_C(Y, Z) × \text{Mor}_C(X, Y) −→ \text{Mor}_C(X, Z) $

$$ (g, f) −→ g ◦ f$$
of morphisms.

The above data must satisfy two condition:

1. Composition of morphism is associative

2. Existence of identity morphism in the automorphism set of an object.

Suppose I had a class and a set of morphism assigned to that class (similar to first part of category definition), then would there be any name to this structure?

In otherwords, for a group, it is fundamentally made of a set and a binary operation with additional axioms. So, in the category definition if I removed the additional axioms, would I get any meaningful object?

Answer 1

Loosely speaking:

• A category without identities or composites corresponds to a directed multigraph (in category theory, often called a quiver).
• A category without identities corresponds to a semicategory.
• A category with only partial nonassociative composition corresponds to a neocategory.
• A category with only partial composition corresponds to a paracategory.

Note that not all of this terminology is very common/standard.