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Stream: learning: questions

Topic: Connected Category

Frank Tsai (Jun 09 2023 at 21:24):

Category theory in context: A category is connected if any pair of objects can be connected by a finite zig-zag of morphisms.
If I understand this correctly, for some objects $x$ and $y$, we don't necessarily have a morphism $x \to y$ because they are connected by a finite zig-zag of morphisms?

John Baez (Jun 09 2023 at 21:32):

Yes, if you have a category with just two objects $x$ and $y$ and one morphism $f: y \to x$ together with the two identity morphisms, then $x$ can be connected by a finite zig-zag of morphisms to $y$, even though there is no morphism from $x$ to $y$.

John Baez (Jun 09 2023 at 21:34):

(In this example you can use a finite zig-zag that's very short, consisting of a single zag and no zig.)