| I must confess, that I too was overwhelmed with topos when I first saw it too. Anyone who understands topos at first glance really doesn't understand topos at all!
Well, in the beginning there are sets. Essentially a set is a set of elements represented by these barckets "{" and "}", the elements are listed in between them (e.g. {1, 3, ..., 6}).
Now, if we had two sets, say {1, 2, 3} and {5, 3, 4}. The first set is set A, the second is set B. If we had a function "f" that maps from A to B, i.e. changing the elements in set A to be identical in value (although not in order) as set B; this is represented by f : A --> B.
In my example, f(A)=A+2; {1+2, 2+2, 3+2} are all equal in value as set B, although not in the same order as set B.
Alternatively, one can construct a set with only a few things known. The notation for building a set is {X : (conditions)}, the set X is such that the conditions are true.
Now, given a set A and a binary operator *, a group is {A, *}. A binary operator is something like addition, subtraction, multiplication, etc.
It's important to know that a Cartesian product is the product of two sets {a, b} and {1, 2, 3} such that it becomes a set of ordered pairs {(a,1),(a,2),(a,3),(b,1),(b,2),(b,3)}. A Cartesian product of two sets A and B is denoted by AxB.
A binary operation is one where * : AxA --> A described by (x, y) |--> x*y.
A Category consists of all sets G endowed with a binary operation satisfying a certain set of axioms.
Now, each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set theoretic mathematics. But one could instead choose to work with many alternate topoi.
It is also possible to encode a logical theory, such as the theory of all groups, in a topos. The individual models of the theory, i.e. the groups in our example, then correspond to functors from the encoding topos to the category of sets that respect the topos structure.
Functors can be thought of as morphisms in the category of small categories.
Suppose C and D are categories. A functor F from C to D is a mapping (or a function) that does two things:
1. it associates to each object X in C an object F(X) in D, and
2. it associates to each morphism f : X → Y in C a morphism F(f) : F(X) → F(Y) in D
So basically a functor translates the elements of one category into those of another.
But what the hell is a "morphism"?
A category C is given by two pieces of data: a class of objects and a class of morphisms.
There are two operations defined on every morphism, the domain (or source) and the codomain (or target).
Morphisms are often depicted as arrows from their domain to their codomain, e.g. if a morphism f has domain X and codomain Y, it is denoted f : X → Y.
That should explain it all: each topos completely defines its own mathematical framework. |