Definition of category without requirement of sets Mor(A,B) to be disjoint.

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Definition of category and precategory
Categorical terms and symbols in different contexts
More definitions

During my category theory research I after having adultery against category theory attempting to set my own way in place of category theory (see my self invented non-standard terminology of category theory and my algebraic model for category theory), I have indeed returned back.

I just have got one of standard definitions of a category but just removed the requirement that the sets $Mor (A_{1}, B_{1})$ and $Mor (A_{2}, B_{2})$ should be disjoint if $A_{1} \neq A_{2} \lor B_{1} \neq B_{2}$ .

Definition of category and precategory

So this is definition of category (well, first of precategory):

A precategory is a system of:

a set $Ob$ (elements of which are called objects);
a set $Mor$ (elements of which are called morphisms);
a function (also denoted $Mor$ ) from pairs of objects (the first element of the pair called source and the second destination) to sets of morphisms.
a partial binary operation $\circ$ (composition of morphisms) defined for such pairs $f$ , $g$ of morphisms that there exist objects $A$ , $B$ , $C$ such that $f \in Mor (A, B)$ , $g \in Mor (B, C)$ ; $g \circ f \in Mor (A, C)$ for any such $A$ , $B$ , $C$ .

A category is a precategory with additional operation $A \mapsto 1_{A}$ from the set of objects to the set of morphisms such that:

$1_{A} \in Mor (A, A)$ for any object $A$ ;
$f \circ 1_{A} = f$ , $1_{B} \circ f = f$ for any $f \in Mor (A, B)$ (for any objects $A$ and $B$ ).

Remark This definition is equivalent to the second (with binary relations of objects associated with every morphism) definition of category in myDefinition of category with multiple domains and ranges. [TODO: Prove equivalence.]

We can transform this into the standard definition of a category (with every morphism having certain source and destination) if we will consider triples $(A, B) f$ (typed morphisms) instead of morphisms and define composition of typed morphisms by the formula $(B, C) f \circ (A, B) f = (A, C) (g \circ f)$ .

Categorical terms and symbols in different contexts

Dependently on context $Mor$ will denote either a set of such triples or a function from pairs of objects to sets of morphisms.

Likewise the word morphism by itself will denote a triple, but morphism from A to B will denote an element of the set $Mor (A, B)$ (not a triple). (The same applies for usage of words isomorphism, epimorphism, monomorphism, etc.)

However, for example, morphism whose source is $A$ and destination is $B$ denotes a triple (unlike morphism from $A$ to $B$ ).

More definitions

We can defined an endomorphism as a morphism which is a member of $Mor (A, A)$ for some object $A$ (or equivalently as a morphism $f$ for which $f \circ f$ is defined.

[TODO: Define functors, prefunctors, isomorphisms, epimorphisms, monomorphisms, etc.]

These definitions are quite analogous to the standard category theory definitions, taking in account that in a morphism is missing information about its source and destination, which may be supplied separately, in the triple.

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