Editing User:Michiexile/MATH198/Lecture 1 (section)

====Backwards!====

If we have a path given by the arrows <math>(f,g)</math> in <math>G_2</math>, we expect <math>f:A\to B</math> and <math>g:B\to C</math> to compose to something that goes <math>A\to C</math>. The origin of all these ideas lies in geometry and algebra, and so the abstract arrows in a category are ''supposed'' to behave like functions under function composition, even though we don't say it explicitly.

Now, we are used to writing function application as <math>f(x)</math> - and possibly, from Haskell, as <hask>f x</hask>. This way, the composition of two functions would read <math>g(f(x))</math>.

On the other hand, the way we write our paths, we'd read <math>f</math> then <math>g</math>. This juxtaposition makes one of the two ways we write things seem backwards. We can resolve it either by making our paths in the category go backwards, or by reversing how we write function application. 

In the latter case, we'd write <math>x.f</math>, say, for the application of <math>f</math> to <math>x</math>, and then write <math>x.f.g</math> for the composition. It all ends up looking a lot like Reverse Polish Notation, and has its strengths, but feels unnatural to most. It does, however, have the benefit that we can write out function composition as <math>(f,g) \mapsto f.g</math> and have everything still make sense in all notations.

In the former case, which is the most common in the field, we accept that paths as we read along the arrows and compositions look backwards, and so, if <math>f:A\to B</math> and <math>g:B\to C</math>, we write <math>g\circ f:A\to C</math>, remembering that ''elements'' are introduced from the right, and the functions have to consume the elements in the right order.

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The existence of the identity map can be captured in a function language as well: it is the existence of a function <math>u:G_0\to G_1</math>. 

Now for the remaining rules for composition. Whenever defined, we expect associativity - so that <math>h\circ(g\circ f)=(h\circ g)\circ f</math>. Furthermore, we expect:
# Composition respects sources and targets, so that:
#* <math>s(g\circ f) = s(f)</math>
#* <math>t(g\circ f) = t(g)</math>
# <math>s(u(x)) = t(u(x)) = x</math>

In a category, arrows are also called ''morphisms'', and nodes are also called ''objects''. This ties in with the algebraic roots of the field.

We denote by <math>Hom_C(A,B)</math>, or if <math>C</math> is obvious from context, just <math>Hom(A,B)</math>, the set of all arrows from <math>A</math> to <math>B</math>. This is the ''hom-set'' or ''set of morphisms'', and may also be denoted <math>C(A,B)</math>.

If a category is large or small or finite as a graph, it is called a large/small/finite category.

A category with objects a collection of sets and morphisms a selection from all possible set-valued functions such that the identity morphism for each object is a morphism, and composition in the category is just composition of functions is called ''concrete''. Concrete categories form a very rich source of examples, though far from all categories are concrete.

Again, the Wikipedia page on Category (mathematics) [[http://en.wikipedia.org/wiki/Category_%28mathematics%29]] is a good starting point for many things we will be looking at throughout this course.