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.

L'idea e':

- una famiglia di strutture di uguale struttura,

assieme alle mappe che conservano la struttura - si puo' considerare un'ulteriore structure-of-structures.

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

- many paradigm cases are indeed structured-families-of-structures
- we develop ways of describing what happens inside a category.

In this new setting, we revisit

- the very familiar ideas of forming new structures within a family by taking products or taking quotients, etc.

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

trade

- claims about what is going on inside structures
- for claims about the homomorphisms or other maps between them.

there is pressure to get entangled with the set-theoretic ideas

- adopting the apparatus of
*sets-for-applied-use*, we get what it takes to construct pairs, quotients, etc. in a systematic way that can be applied across the board. - going the step further into pure set theory, we get a single unifying setting for our investigations.

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

theory ZFC

Un gruppo in astratto e' la sua tavola di moltiplicazione.