^^Category Theory, A Gentle Introduction. by Peter Smith

logicmatters  Category Theory, A Gentle Introduction. by Peter Smith

Theorems in basic category theory are very straightforward.

Notation

Iff  `if and only if'.

□     end-of-proof marker

∆     end-of-definition marker

Es standard. Groups, poset, topological spaces

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.

1.3 Ascending to the categorial heights

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.

the simplest plan

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.

2 The structured-family-of-structures  of groups

wp/Arity#Examples

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

2.3 New groups from old

a key categorial theme

trade

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

2.5 Groups and sets

the familiar arithmetization of analysis >>>

2.6 Category  VS   set-theoretic imperialism

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

1. 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.
2. 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.

3 Category definition.

Talk

"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.