logicmatters Category Theory, A Gentle Introduction. by Peter Smith
Theorems in basic category theory are very straightforward.
Iff `if and only if'.
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esiste la struttura astratta di gruppo, e i gruppi concreti, considerati anch'essi strutture, strutture concrete.
Esistono famiglie di gruppi legati tra loro da appropriate mappe (omomorfismi) che preservano la struttura di gruppo.
Analogamente per: posets, topological spaces.
The gadgets of basic category theory do fit together rather beautifully in multiple ways. These intricate interconnections mean, however, that there isn't a single best route into the theory. Different treatments can quite appropriately take topics in very di erent orders, all illuminating in their various ways.
first talking about categories
In this new setting, we revisit
a nullary function takes no arguments. f() = 2
a unary function takes one argument. f(x) = 2x
a binary function takes two arguments f(x,y) = 2xy
there is pressure to get entangled with the set-theoretic ideas
Hence we might well suppose that category theory can, essentially, be thought of as a way of talking about set-theoretic constructions, all living in the world of sets.
That's why Saunders Mac Lane in his canonical "Categories for the Working Mathematician" can say, simply, a category will be
'any interpretation of the category axioms within set theory'.
However, there is an alternative line of thought about category theory
which apparently goes in quite the opposite direction. Mathematicians are right, the rival story goes, in their ordinary supposition that there are fundamentally different kinds of mathematical structure built up of diferent kinds of objects and maps between them. Moreover these different kinds of structure stand on their own feet, so to speak, without needing reduction to sets. Indeed, the world of pure sets is then just one big structure living in a wider democratic universe of structures. And category theory allows us to talk about the interrelations of these structures (and the place of the world of sets in the wider universe), while breaking free from set-theoretic imperialism.
"sets-for-applied-use" VS the set-theorist's universe of sets as
described in the canonical
Un gruppo in astratto e' la sua tavola di moltiplicazione.