more background.
Tannaka duality, see for example this MO discussion.
Pontrjagin duality establishes an equivalence between the category of locally compact abelian groups and its opposite category. This restricts to an anti-equivalence between the category of compact abelian groups and discrete abelian groups!

Remark that in the first four examples, in some sense, geometry is connected with algebra. This not just enables you to argue about geometry with algebra, but also the other way round!
By the way, the equivalence between the category of finite sets and the non-negative integers is not surprising at all. Every category is equivalent to its skeleton, essentially by the definitions. A skeleton is some kind of classification of the objects, and a finite set is classified by its cardinality.

A:

What one person finds surprising another person won't,It’s a riveting and disorienting re-telling of Canadian history. From Chapter Six onward, Macdonald figures prominently. For example: “On 24 March 1882, the prime minister announced to Parliament that all Indians in the territory of Assiniboia would be removed, by force if necessary, from the land south of the proposed railway.” Growing up in southern Saskatchewan, I came to wonder why there were no aboriginal people around. Now I know. Reading Daschuk’s book, you realize that they were systematically lied to, starved, beaten, cheated, and forcefully relocated. The treaties, once signed, were ignored, since the true goal was to clear the land for European settlement, for the railway, and for the (often Montreal) investors who were looking for a profit.

Macdonald was the architect behind much of this terrible