A Universe of Sorts

§ Covariant Hom is left exact

Let's say we have the exact sequence:

0 \rightarrow A \xrightarrow{i} B \xrightarrow{\pi} C \rightarrow 0

Where the first arrow is

i

for inclusion and the third is

\pi

for projection. We now want to consider what happens when we have

Hom(X, -)

for some target space

X

. The induced arrows are induced from composition; I will write

(f; g)(x) \equiv g(f(x))

to mean "first do

f

, then do

g

". Hence, my arrows linking

Hom(X, A)

Hom(X, B)

will be

-;i

: To first go from

X

A

, and then apply

i

to go from

A

B

. This gives us the sequence (which we are to check if it is exact, and which of the left and right arrow exist):

0 \xrightarrow{?} Hom(X, A) \xrightarrow{-;i} Hom(X, B) \xrightarrow{;-\pi} Hom(X, C) \xrightarrow{?} 0

As usual, we go to the classic exact sequence:

0 \xrightarrow 2Z \rightarrow{\pi} Z \rightarrow{\pi} Z/2Z \rightarrow 0

We now have three interesting choices for our

X

in relation to the above sequence: (a)

Z

, (b)

2Z

, (c)

Z/2Z

. Since

Z

and

2Z

are isomorphic as modules, let's study the case with

Z

and

Z/2Z