An (elementary) topos is a category that:
 Is finitely complete
 Is finitely cocomplete
 Has exponentiation
 Has a subobject classifier
This is equivalent to:
 Being Cartesian closed
 Has a subobject classifier
Topos and Logic
A logic is defined over a type. So if we have a type, say 'set', then its substucture will correspond to a logic as described below. A different type, say 'POSET', will have a different logic.
Elementary (set) Topos
The main example of a topos is set, in other words an (elementary) topos is a setlike category.
In order to understand sets we need to describe the structure of subsets. We need to define an element of the powerset. There are at least two ways to do this:
One way to define S as a subset of X An injective function is one where no two distinct inputs give the same output. 

It would not work to have a map in the other direction because elements in S are not uniquely identified with elements in X.  
However we can define a subset with a map going out of X. This maps onto a two point set 2 or {0,1}. Those elements that are in the subset map to 1 and those that are not map to 0. This is called the characteristic function.  
We can complete the diagram by adding an object representing the 'true' element. All elements in S map to it. As we can see, the diagram commutes.  
So, in the general case for sets, we have this diagram which commutes. The diagram is also a pullback.  
The diagram relates the set category to the logic of boolean algebra but without the law of the excluded middle, that is intuitional or constructive logic. 
Alternative Viewpoint
We have two sets, 'A' and 'B', with an arbitrary mapping 'F' between them. We can define a subset of set 'B' by using the predicate 'P'. For every subset of 'B' defined by 'P' there exists a subset of 'A' defined by 'R'.  
However this does not necessarily work in the other direction. Here is an example of a subset of 'A' which does not map to a subset of 'B'. 
So the map 'F' between objects induces a contravarient map between arrows P>R.
See also Yoneda embedding.
Generalisation to Nonsets
We can genarise from elementary topos based on sets to other topoi.
We generalise this to categories that are not sets.


Graph Theory
Graph theory is an example of a category with a logic not based on sets. Possible values for Ω in a graph:
For nodes:  For edges: 


Graphical representatin of Ω: 
Subobject Classifier
See page here for a noncategorical approach to subobjects.
If the arrow 'f' is monic (injective) then it maps to a subobject (subset) of 'B'. That is: there is a subobject of B that is isomorphic to A.
The inclusion relation on subobjects is:
 reflexive
 transitive
In order to define the subobject we must define the monic 'upto equality' which is not in the spirit of category theory. We will go on to define a 'subobject classifier' which defines a subobject, in a much more category theoretic way, by the composition of maps (and also relates the whole subject to logic).
Naming Arrows
In category theory we don't usually identify elements, such as elements of a set, because we only tend to determine things 'upto isomorphism'. The objects in a category are whole 'structures', they don't represent individual elements.
However, there is the possibility to indicate elements indirectly using arrows. For instance, if we want to enter a specific element we can use:
1→A
because, in sets, 1 is the single element set so it will indicate an element uniquely.
We can also use naming arrows. An arrow:
f: A→B
Is a subset of a function spacef:{0}→B^{A}
fis called the 'name' of the function.
Classifier
In the opposite direction there is a correspondence between subsets of B and functions:
B →2
This is in the space 2^{B} which corresponds to the powerset(B), that is, all the possible subsets of B.
The characteristic arrow: x_{f} classifies objects of B to determine if they are images of A. This is a pullback square. The object 2 has two values: 0 and 1. We also use the symbol Ω and call the elements 'true' and 'false'. 
This means the topos theory is related to logic. (see also 'characteristic function' in number theory).
Example in Set
Example in set  

Set A (blue) is subset of B (red).  
If we take the inverse of f we get the concept of a 'bundle' as discussed on this page.  
classifier for set.

Predicate Category
The relationship SX above is a predicate. We can form a category of predicates on sets as follows:
Objects  predicates are pairs (S,X) where X(i) implies an element iS is a free variable in S. 

Morphisms  (S,X) > (T,Y) where: u:S> T and X(i) implies: Y(u(i)) 
See also  Predicate logic.
Family of Sets
A family of sets consists of an index set I and, for each element k of I, a set S_{k}.
There seems to be two ways to conceptulise this:
 As a 2category. A set of sets.
 As disjoint sets.
Modest Sets
Set theoretic model of polymorphism.
Indexed Categories
(fibred category)
example
many sorted algebra ∑
Generalisation from Sets to Categories
Here we have illustrated some basic concepts using sets but they can be generalised to categories as shown here:
Sets  Categories 

Injective Function No two distinct inputs give the same output. We can define 'A' a subset of 'B' by using an injective function from C

Monic Arrow Whenever f•g = f•h then g=h 
Here we are showing this as an inclusion function because we are interested in subset classifiers. Injective function f: C >>B determines A as a subset of B (AB) Set inclusion is a partial odering on the Power set 

Surjective function 
Epic Arrow 
Inverse Functions 

Power Set 
Power Object subobjects of d form a poset 
Topological Space
In many cases the concept of a metric space is unnessary, however we still need the concept of 'nearness' and hence 'continuity'. 'Topological space' based on the 'topological open set' is the most general way we can do this. This allows us to define nearness purely using the concept of a subset.
Hausdorff Space
Hausdorff space is a bit more specific than general topological space. Space is Hausdorff if, in addition to being a topological open set, for any two points:
x_{1}, x_{2}X there are disjoint open sets U_{1}, U_{2} that contain x_{1}and x_{2}.
Basis
A basis is a subcollection B= U
where
 U is the open set
 B is the basis
 = a subcollection (subset)  I need to add the proper symbol for this.
such that evey element of U is a union of open sets in B.