Is there any theory in which a point has a definition? What is the definition of "object" as seen in category theory?
Definition of a point and object
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elementary-set-theory
category-theory
definition
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0In your first question, what do you mean by a *point*? Geometric points? – 2012-03-09
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0"point" as in elementary set theory, I think geometric points are these kind of points – 2012-03-09
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0I suspect you won't like the definitions for 'object.' The power of category theory is in its abstraction, so something is an object and maps are morphisms if they satisfy the axioms to be objects in a category, and so on. – 2012-03-09
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0A category is a kind of enriched graph, and an object is just a vertex of this graph. The name is meant to suggest that there is a deeper meaning, but there really isn't. – 2012-03-09
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1This is sort of like asking about the definition of "element" in set theory, or "vector" in linear algebra. What we really care about are the collections of things (sets / vector spaces / categories) and the structure of these collections. Whatever things happen to be *in* the collections, we call them elements / vectors / objects. But they're not really important in their own right. – 2012-03-09