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:

Composition of morphism is associative

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?

If you remove those axioms, how many interesting results can you prove? — Randall, Apr 12 '22 at 15:24
Well that I am not sure but I was trying to see if category can be thought to build up from more basic structure like how we say a set builds a magma nad then magma becomes a group @Randall — tryst with freedom, Apr 12 '22 at 15:35
You might be interested in semicategories and paracategories. — vikraman, Apr 12 '22 at 15:44

score 3 · Accepted Answer · answered Apr 12 '22 at 16:23

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.

score 0 · Answer 2 · answered Apr 12 '22 at 16:24

It seems to me that you're trying to fill in the analogy "magma:group::___:category" - or in other words, you're looking for a standard name for a "partial magma" (= set equipped with a partial binary function). As far as I know this is too weak a notion to have a commonly used name. However, in Bruck's A survey of binary systems, the term "halfgroupoid" is used, with "groupoid" being used synonymously with "magma" (contra the more current usage).

If I were really pressed to use a term to describe this, I would go with "partial magma" - even if uncommon, its meaning is quite clear. If I wanted to connect explicitly with Bruck's book and the various sources cited therein, I might compromise and go with "halfmagma," but I would never use "halfgroupoid" given the ambiguity around the term "groupoid."

Is there a weaker structure a category is made out of?

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