Download Basic Category Theory

1. Categories and Functors
e) Given a set A, consider the set A~ of strings a1 : : :an on the alphabet
A [ A 1 (A 1 consists of elements a 1 for each element a of A; the sets
A and A 1 are disjoint and in 1-1 correspondence with each other), such
that for no x 2 A, xx 1 or x 1x is a substring of a1 : : :an . Given two
such strings ~a = a1 : : :an; ~b = b1 : : :bm , let ~a ? ~b the string formed by rst
taking a1 : : :an b1 : : :bm and then removing from this string, successively,
~
substrings of form xx 1 or x 1x, until one has an element of A.
~ A~ is called the free group on the set
This denes a group structure on A.
A.
Exercise 3. Prove this, and prove that the assignment A 7! A~ is part of a
functor: Set ! Grp. This functor is called the free functor.
f) Every directed graph can be made into a category as follows: the objects
are the vertices of the graph and the arrows are paths in the graph. This
denes a functor from the category Dgrph of directed graphs to Cat. The
image of a directed graph D under this functor is called the category
generated by the graph D.
g) Quotient categories. Given a category C , a congruence relation on C
species, for each pair of objects X; Y , an equivalence relation X;Y on
the class of arrows C (X; Y ) which have domain X and codomain Y , such
that
for f; g : X ! Y and h : Y ! Z, if f X;Y g then hf X;Z hg;
for f : X ! Y and g; h : Y ! Z, if g Y;Z h then gf X;Z hf.
Given such a congruence relation on C , one can form the quotient category C = which has the same objects as C , and arrows X ! Y are
X;Y -equivalence classes of arrows X ! Y in C .
Exercise 4. Show this and show that there is a functor C ! C =, which takes
each arrow of C to its equivalence class.
h) An example of this is the following (\homotopy"). Given a topological
space X and points x; y 2 X, a path from x to y is a continuous mapping
f from some closed interval [0; a] to X with f(0) = x and f(a) = y. If
f : [0; a] ! X is a path from x to y and g : [0; b] ! X is a path from y to z
ta
there is a path gf : [0; a+b] ! X (dened by gf(t) = f(t)
g(t a) else )
from x to z. This makes X into a category, the path category of X,
and of course this also denes a functor Top ! Cat. Now given paths
f : [0; a] ! X, g : [0; b] ! X, both from x to y, one can dene f x;y g if
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