A category is a collection of objects and morphisms (best to think of them as arrows) between them. For example you can think of a monoid (like the natural numbers(no zero) with multiplication) the class of objects being the set and the the collection of morphisms being the described by the binary operation by some element of the set(*,n for all n in the natural numbers). What happens when you want to move between categories? Well then you introduce functors, maps between categories. Full means for every pair of objects x,y in C the mapping f to F(f) defines a surjection Mor(x,y) to Mor(Fx, Fy); faithful means basically the same but injection not surjection, combined full and faithful is like injectivity, you need a third thing; essentially surjective. Essentially surjective means for every object d in D there is some c in C st Fc isom d. If you have fully faithful and essentially surjective you have an isomorphism on catagories.For example in the degenerate example of the category of the natural numbers with multiplication by a natural number, and the category of the integers, addition and multiplication by an integer, the identity from the natural numbers to the integers is a faithful functor trivially. For a less degenerate example think of k dimensional vector (obj) spaces and k linear maps (mor). This is not nessasarialy surjective on objects. What are some functors is a natural question. Now onto my favorite functor: the forgetful functor. The forgetful functor: U, from the catagory to the set maps objects to their underlying sets and morphisms to their underlying set-maps. it tosses out the structure of the catagory, hence forgetful. Its really fucking dope ngl. to define the next functor we need to define the C^{op}, the opposite category, it has the same object but with flipped arrows(swap the sources and targets of C's morphisms). There is a functor C to C^{op} that takes objects to themselves and arrows to their "flip". For the third example we will talk about based spaces. Let (X,x) be a based space, (X is a space, x in X). we call the fundemental group \pi_1(X,x) the set of based continous maps l:[0,1] to X st l(0)= l(1) = x, moding out homotopy equivilence. The group structure is loop conatenation: given l, l':[0,1] to X, define l'l:[0,1] to X to do one loop over [0.5] then one loop over [0.5, 1]. then given a based map f:(X,x) to (Y,y), there is aninduced map f_*:\pi_1 X to \pi_1 Y given by l to f \circ l. This defines a functor: \pi_1 (-): Top_* to Grp. for the forth and final example for this post for each n, singular homology defines a functor H_n(-): Top to Ab. Simularly, singular cohomology defines a functor H^n(-): Top^{op} to Ab. Next post we will talk about natural transormations. they are rather hype ngl.
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