About category theory and direct sum

Question

What is the explicit meaning of the statement that * is compatible with direct sums ; what are these maps ; I guess they are projection and injection map but could somebody explain how duality works in this kind of maps intutively? Also how Hom(M,N) is isomorphic to Hom(M**,N**)?

Answer 1

Well as is said $*$ is a functor, let's say $Mod_R \to Mod_R$ for some commutative ring $R$. The compatibility with direct sums can be rephrased categorically as does this duality functor preserve finite coproducts (which are the same as finite products in an additive category). The projection and injection map are the maps defined for the product/coproduct respectively. For your final question, if we know that $**$ is naturally isomorphic to the identity functor, then in particular it is full and faithful, which means that we get an isomorphism of Hom sets.

Answer 2

$*$ is a contravariant functor $\mathscr{M\to M}$.

So if you have a canonical projection map $\pi_i : M_1\oplus ...\oplus M_n \to M_i$, you get $\pi_i^* : M_i^*\to (M_1\oplus ...\oplus M_n)^*$. The requirement is that this is the "injection" of $M_i^*$ into a direct sum of the $M_i^*$'s.

Dually, if you have a canonical injection $\rho_i : M_i \to M_1\oplus ...\oplus M_n$, the requirement is that $\rho_i^*$ be the canonical projection map (note that the author says "direct sum" whereas after they only mention finite direct sums; this is something that should be clarified)

For your last question, there's a requirement that $(-)^{**}$ be naturally isomorphic to $id_\mathscr{M}$, but classically if $F\simeq G$, then $\hom (F(x),F(y))\cong \hom (G(x), G(y))$. It's just the same argument as saying that if $A\simeq B, C\simeq D$, then $\hom (A,C) \simeq \hom (B, D)$, only that since the initial isomorphism are natural, so are the induced ones.

Note that this doesn't follow from the other assumptions, it is another assumption