Xjyuk

I think I don't understand the categorical duality principle. For instance, if you prove that a certain category has kernels, by duality principle it has cokernels. But why this doesn't apply with Grothendieck axioms for abelian categories? For example, if a category has arbitrary direct sums, why the dual statement doesn't hold in general?

For instance, if you prove that a certain category has kernels, by duality principle it has cokernels.

You have to be careful what certain means: If for a property $mathsf P$ of categories you prove $mathsf P(mathscr C)Rightarrow mathscr Ctext has kernels$ for any $mathscr C$, then this also gives you $mathsf P(mathscr C^textop)Rightarrow left(mathscr C^textoptext has kernelsLeftrightarrow mathscr Ctext has cokernelsright)$. However, $mathsf P((-)^textop)$ is a different property than $mathsf P$ in general, and only if you know that $mathsf P(mathscr C^textop)Leftrightarrow mathsf P(mathscr C)$ you get the dual statement for categories satisfying $mathsf P$ for free.

Examples:

• If $mathsf P(mathscr C):=mathscrCtext is abelian$, then $mathsf P(mathscr C^textop)Leftrightarrow mathsf P(mathscr C)$ because being abelian is a self-dual property. Therefore, with any statement about abelian categories you get another, 'dual', statement about abelian categories for free.




• If $mathsf P(mathscr C):=mathscrCtext is Grothendieck$, then $mathsf P((-)^textop)notLeftrightarrow mathsf P$: In fact, the only Grothendieck abelian categories with Grothendieck dual are the trivial categories (those where every object is a zero object). Hence, statements about Grothendieck abelian categories do not come in dual pairs! Instead, with every statement about a Grothendieck abelian category, you get a dual statement about co-Grothendieck abelian categories, but as these aren't very popular (the only concrete example I can think of is $textabelian groups^textopcong textcompact abelian topological groups$), that's usually not very useful. It's important to notice this striking asymmetry in the focus on Grothendieck abelian categories in homological algebra.

If you have some dual notions like kernels and cokernels or more generally limits and colimits then duality states that if $mathcalC$ has kernels, the opposite category $mathcalC^textop$ has cokernels. That is because a colimit is the same as a limit when all involved arrows are reversed. In general you do not get the dual notions in the same category. You can construct an easy example for that by considering a poset as a category.