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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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
2 Answers
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.
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.