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I'm learning about category theory from this video: He says:

The purpose of objects is to mark the start and end of morphisms.

This seems somewhat strange.

It's like morphisms and objects are in some type of fabricated interdependence. We can't have morphisms without objects and we can't have objects with no morphisms. And this chicken and egg like situation creates a category.

Is this correct? Or i'm just reading too much into it?

Can a category have just objects and no morphisms? But then the objects should not be there in the first place. There is no end to mark if there is no morphism. So there should be no objects. Which maybe implies no category.

So are there categories with no morphisms? Or at least one category with no morphisms? I'm trying to make sure i rely understand what a category is. Sorry if this sounds like a dumb question. Thanks.

AIon
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2 Answers2

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Morphisms are an integral part of the definition of a category - note that every object has its identity morphism, so the only category without any morphisms is the empty category, which of course doesn't have any objects either.

Of course, we can consider a category $C$ where there are no non-identity morphisms, as Micah suggests above, but such categories are essentially just sets.

Actually, the real situation is the other way around: it's objects that are relatively unimportant! Indeed, we can dispense with objects entirely and just talk about morphisms; see this old answer of mine. This is really a key point of the category-theoretic philosophy: we care less about what an object "is" than how it interacts with other objects, and this latter behavior is captured (to a certain extent anyways) by the appropriate maps between the objects in question.

Noah Schweber
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If a category has an object, then it must have at least one morphism, since each object has an identity morphism. There is a category with no morphisms, though: namely, the empty category which has no objects or morphisms.

You shouldn't take statements about "purposes" like this too literally. They are meant to give you some intuition about the concepts, but if you want to see what is possible or not and what you can actually prove, you should always look instead to the actual definitions. The definition of a category is precise and tell you exactly what a category is; anything else is just a guide to help you think about the definition. (And if you're learning from a source like that video that doesn't clearly state the definition, look up another one that does! You can find definitions of most basic mathematical concepts on Wikipedia, for instance.)

Eric Wofsey
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