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• $\ \exists_{S\,\in\,Obj(\mathbf C)}\\ \ |\,MOR(S\ B)\,|\ =\ |\,MOR(S\ A)\,|\ =\ |\,MOR(A\ C)\,|\ =\ |\,MOR(B\ C\,|\ =\ 1$
• whenever $\ D\ $ is like $\ C\ $ above then $\ |\,MOR(C\ D)\,|\ =\ 1$.

• $\ \exists_{S\,\in\,Obj(\mathbf C)}\\ \ \quad |\,MOR(S\ A)\,|\ =\ |\,MOR(S\ B)\,|\ =\ |\,MOR(A\ C)\,|\ =\ |\,MOR(B\ C\,|\ =\ 1$
• whenever $\ D\ $ is like $\ C\ $ above then $\ |\,MOR(C\ D)\,|\ =\ 1$.

INTERPRETATION: The objects of a connectivity category play the role of non-empty connected spaces.

DEFINITION 3  of a connectivity graph:

• $\ \mathbf C\ $ is vague;

• every union of objects of $\ \mathbf C\ $ is a merger;

• if $\ A\ B\ $ merge, and if $\ |\,MOR(A\ A')\,|\ 1\ $ then $\ A'\ $ and $\ B\ $ merge too;

• if $\ \mathbf D\ $ is a non-empty family of objects such that every two of them merge then there exists an object $\ C\ $ which is a merger of $\ \mathbf D,\ $ meaning that the following two conditions hold:

• $\ \forall_{A\in\mathbf D}\ \ |\,MOR(A\ C)\,|\ =\ 1$

• whenever $\ D\ $ is like $\ C\ $ then $\ \,MOR(C\ D)\,|\ =\ 1$

DEFINITION 3  of a connectivity category:

1. $\ \mathbf C\ $ is vague;

2. every union of objects of $\ \mathbf C\ $ is a merger;

3. if $\ A\ B\ $ merge, and if $\ |\,MOR(A\ A')\,|\ 1\ $ then $\ A'\ $ and $\ B\ $ merge too;

4. if $\ \mathbf D\ $ is a non-empty family of objects such that the full induced merger subgraph $\ \mathbf D\ $ is connected then there exists an object $\ C\ $ which is a merger of $\ \mathbf D,\ $ meaning that the following two conditions hold:

• $\ \forall_{A\in\mathbf D}\ \ |\,MOR(A\ C)\,|\ =\ 1$
• whenever $\ D\ $ is like $\ C\ $ then $\ \,MOR(C\ D)\,|\ =\ 1$

NOTATION   If $\ |\,MOR(A\ B)\,|\ =\ 1\ $   then   $\ i_{AB}\in MOR(A\ B)$.

TERMINOLOGY   A category $\ \mathbf C\ $ is called vague $\ \Leftarrow:\Rightarrow\,\ \forall_{X\ Y\,\in\,Obj(\mathbf C)}\ |\,MOR(X\ Y)\,|\ \le\ 1$

DEFINITION 2  of merger   Let category $\ \mathbf C\ $ be vague. Objects $\ A\ B\ $ merge* into an object $\ C\ \Leftarrow:\Rightarrow\ $ two conditions hold:

TERMINOLOGY   A category $\ \mathbf C\ $ is called vague $\ \Leftarrow:\Rightarrow\,\ \forall_{X\ Y\,\in\,Obj(\mathbf C)}\ |\,MOR(X\ Y)\,|\ \le\ 1$

DEFINITION 2  of merger   Let category $\ \mathbf C\ $ be vague. Objects $\ A\ B\ $ merge into an object $\ C\ \Leftarrow:\Rightarrow\ $ two conditions hold: