Homework 1

An object $i$ in a category $C$ is called initial if for every object $c\in C$ there exists a unique arrow $i\to c$; in other words, the hom-sets ${\mathrm{Hom}}_C(i,c)$ are all singleton sets.

Dually, an object $t$ is called terminal if for every object $c\in C$ there exists a unique arrow $c\to t$, i.e., the hom-sets ${\mathrm{Hom}}_C(c,t)$ are all singleton sets.

1. Show that an initial (respectively, terminal) object, if it exists, is unique up to unique isomorphism.
2. Show that in $\mathbf{S}\mathbf{e}\mathbf{t}$, the empty set is initial and the singleton set is terminal.
3. Show that in $\mathbf{A}\mathbf{b}$, the trivial group $0$ is both initial and terminal. (Such an object is called a null or zero object.)
4. Show that $\mathbf{F}\mathbf{i}\mathbf{e}\mathbf{l}\mathbf{d}$ contains neither an initial nor a terminal object.

An arrow $f:a\to b$ in a category $C$ is called monic (or a monomorphism) if it is left-cancellable; i.e., if whenever $g_1,g_2:c\to a$ are arrows such that $f\circ g_1=f\circ g_2$, then $g_1=g_2$.

Dually, the arrow $f$ is called epic (or epi, or an epimorphism) if it is right-cancellable; i.e., if whenever $g_1,g_2:b\to c$ are arrows such that $g_1\circ f=g_2\circ f$, then $g_1=g_2$.

1. In $\mathbf{S}\mathbf{e}\mathbf{t}$, show that a set map $f:X\to Y$ is monic (respectively, epic) if and only if it is injective (respectively, surjective).
2. Show that, in a general category $C$, if an arrow $f:a\to b$ is an isomorphism (i.e., invertible), then $f$ is both monic and epic.
3. Show that in $\mathbf{R}\mathbf{i}\mathbf{n}\mathbf{g}$, the ring inclusion $i:\mathbf{Z}\to \mathbf{Q}$ is both monic and epic, even though the map $i$ is not surjective.

There are two functors vying for the name "power set functor".

For the first, define $\mathcal{P}:\mathbf{S}\mathbf{e}\mathbf{t}\to \mathbf{S}\mathbf{e}\mathbf{t}$ as follows:

• On objects: for each set $X$, define $\mathcal{P}(X)$ to be the power set of $X$, i.e., the set of all subsets of $X$.
• On arrows: for each set map $f:X\to Y$, define a set map $\mathcal{P}(f):\mathcal{P}(X)\to \mathcal{P}(Y)$ by sending each subset $S\subseteq X$ to its image $f(S)\subseteq Y$.

For the second, define ${\mathcal{P}}^{\prime }:{\mathbf{S}\mathbf{e}\mathbf{t}}^{\text{op}}\to \mathbf{S}\mathbf{e}\mathbf{t}$ as follows^[1]:

• On objects: for each set $X$, define ${\mathcal{P}}^{\prime }(X)$ to once again be the power set of $X$.
• On arrows: for each arrow $f^{\text{op}}:X\to Y$ (corresponding to a set map $f:Y\to X$), define a set map ${\mathcal{P}}^{\prime }(f^{\text{op}}):{\mathcal{P}}^{\prime }(X)\to {\mathcal{P}}^{\prime }(Y)$ by sending each subset $S\subseteq X$ to its preimage $f^{-1}(X)\subset Y$. (Recall that $f^{-1}(X)=\{y\in Y\mid f(y)\in X\}$.)

Verify that $\mathcal{P}$ and ${\mathcal{P}}^{\prime }$ are indeed both functors. The first is sometimes called the covariant power set functor and the second the contravariant power set functor.

Prove there does not exist a functor $\mathbf{G}\mathbf{r}\mathbf{p}\to \mathbf{A}\mathbf{b}$ with object function sending each group $G$ to its center.