The functor $\textbf{Hom}(-,W):\textbf{Vect}^{op}_k\rightarrow \textbf{Vect}_k$

Question

I've recently started to study category through Leinster's Basic Category Theory.

In example 1.2.12 it's said that $\textbf{Hom}(V,W)$ is a vector space.

Then,

Now fix a vector space $W$. Any linear map f : $V \rightarrow V′$ induces a linear map $f^*:\textbf{Hom}(V',W)\rightarrow \textbf{Hom}(V,W)$ such that for $q\in \textbf{Hom}(V',W)$ we have $f^*(q)= q f$

He concludes by saying that

This defines a functor $\textbf{Hom}(-,W):\textbf{Vect}^{op}_k\rightarrow \textbf{Vect}_k$

My problem is that I don't understand why is the op necessary there. Why wouldn't it work without it?

Answer 1

The op is necessary there because one would like to work with covariant functors $F: \mathcal{C} \rightarrow \mathcal{D}$ between categories which preserve the order of composition: i.e., given a map $f: C_1 \rightarrow C_2$ in $\mathcal{C}$ then we would like $F(f): F(C_1) \rightarrow F(C_2)$ in $\mathcal{D}$. The trouble with this is that in this case, $Hom(-,W)$ to a fixed vector space $W$ is contravariant in the first argument, as you explain above.

We need the op in order to have a covariant functor, in this case from $Vect^{op}_k \rightarrow Vect_k$.

Answer 2

(Well, just to complement the answer above, maybe this can be of utility. I also always confuse this, so maybe writing down in detail it will make it clearer. )

Suppose $Hom(-,W)$ is a functor defined on $Vect_k \to Vect_k$.

Then it will take the objects of $Vect_k$, the vector space $V$, to objects of $Vect_k$ of the kind $Hom(V,W)$, that is, the vector space of linear maps $V \to W$.

And it will operate on morphisms too. The functor $Hom(-,W)$ will take $f:V \to V'$ and will give us

$$Hom(f,W): Hom(V,W) \to Hom(V',W)$$

by definition of functor (that is, any functor has to respect $F(f): F(V) \to F(V')$).

But that is not right, because we want it operate as the composite-maker $f^*$:

$$f^*:\textbf{Hom}(V',W)\rightarrow \textbf{Hom}(V,W) $$ $$q \mapsto qf$$

for every morphism $q: V' \to W$ and map $ f: V \to V'$.

To fit this we instead take $Vect_k^{op}$ and work with contravariant functors. Then the functor $Hom(-,W)$ will take the morphism $f^{op}: V' \to V$ to the map:

$$Hom(f^{op},W): Hom(V',W) \to Hom(V,W).$$

And we are fine, because it will operate as $f^*$.

We could also define the covariant functor a $Hom(W,-)$ that operates as:

$$Hom(W,-): Vect_k \to Vect_k.$$

It will take, for every $q: W \to V$, a $f: V \to V'$ and will send to:

$$Hom(W,f): Hom(W, V) \to Hom(W, V')$$ $$q \mapsto fq.$$