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Is there any theory in which a point has a definition? What is the definition of "object" as seen in category theory?

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In your first question, what do you mean by a point? Geometric points? –  user2468 Mar 9 '12 at 5:42
"point" as in elementary set theory, I think geometric points are these kind of points –  Jr. Mar 9 '12 at 5:46
I 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. –  mixedmath Mar 9 '12 at 5:55
A 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. –  Zhen Lin Mar 9 '12 at 7:33
This 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. –  Rahul Mar 9 '12 at 10:33

2 Answers 2

An object is just an element of the set of things named objects in the definition of a category ("a category is a collection of objects together with arrows between those objects...").

What you want to call a point depends a lot on what kind of category you're working in. The most restrictive definition I know is in terms of a terminal object $1$; we say that a global point of an object $X$ is a morphism $1 \to X$.

Example. In $\text{Set}$, the terminal object is the one-element set $1$ and a global point is a point in the ordinary sense.

Example. In $G\text{-Set}$, the category of group actions of a group $G$, the terminal object is the one-element set $1$ with the trivial action, and a global point is a fixed point of $G$.

Example. In $\text{Aff}$, the category of affine schemes, the terminal object is $\text{Spec } \mathbb{Z}$ and a global point is a morphism $\text{Spec } \mathbb{Z} \to \text{Spec } R$, or in the opposite category a map $R \to \mathbb{Z}$. For example, if $R = \mathbb{Z}[x_1, ... x_n]/(f_i)$ where $f_i$ is a system of polynomial equations over $\mathbb{Z}$, then a global point of $\text{Spec } R$ is precisely a solution to the corresponding system of Diophantine equations over $\mathbb{Z}$.

Example. In $\text{Grp}$, the category of groups, the terminal object is the trivial group, so every object has a unique global point.

Example. Let $X$ be a topological space and $\text{Sh}(X)$ the category of sheaves of sets on $X$. The terminal object is the constant sheaf $1$ and a global point is a global section.

As you can see, often global points are not enough to completely describe an object; more precisely, often the functor $\text{Hom}(1, -)$ is not faithful. In that case one might want to admit a broader notion of point.

The least restrictive definition I know is the following: an $R$-valued point of an object $X$ is a morphism $R \to X$. You can run through the examples above to see how this plays out in practice. Unlike global points, $R$-valued points always completely determine an object (in the appropriate sense) by the Yoneda lemma.

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Quite simply, "object" is defined separately for each category. The definition of a category always states what its objects are: Rng is defined as the category whose objects are rings and whose morphisms are homomorphisms; Top is defined as the category whose objects are topological spaces and whose morphisms are continuous maps; and so on.

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