^^Category theory.
Elementary Theory of the Category of
Sets.
Lambda calculus
wp/Category_theory
wp/Category_(mathematics)
Connections are
- symmetric, so if 'a' is connected to 'b', then b is connected to a.
- transitive, meaning that if a is connected to b, and b is connected to
c, then
a is connected to c; that is, all a, b, and c are connected.
join
the transitive closure of the union of the connections in A and B
A≤B
xCy ∈ A, then xCy ∈ B.
Definizione informale
- Category theory is
all about organizing and layering structures.
- categories are the right intuition for talking about composing things with
domains and ranges.
cdsmith/why-do-monads-matter
- Category theory attempts to find commonalities by turning concepts from
disparate fields into common objects called categories, which are
essentially graphs with a composition rule.
There’s a style of programming called
point-free
programming which involves coding without variables. So, instead of
writing an absolute value function as if
x > 0 then x else -x, you write it as
sqrt . square, where the .
operator is function composition. Category theory is like doing everything
in the point-free style. It can sometimes lead to beautifully short
definitions that enable a lot of insight, but it can also serve to obscure
needlessly.
A Rosetta Stone, by category theory. Physics, Topology, Logic and
Computation.
John C. Baez 2009
Category theory is a very general formalism.
A category has objects and morphisms,
which represent things and ways to go between things.
In physics
- objects are physical systems
- morphisms are processes turning a state of one physical system into a
state of another system perhaps the same one.
In quantum physics we often formalize this by taking Hilbert spaces as
objects, and linear
operators as morphisms.
At present, the deductive systems in mathematical logic look like hieroglyphs
to most physicists. Similarly, quantum eld theory is Greek to most computer
scientists, and so on. So, there is a need for a new Rosetta Stone to aid
researchers attempting to translate between fields.
The Rosetta Stone (pocket version)
| Category Theory |
Physics |
Topology |
Logic |
Computation |
| object |
system |
manifold |
proposition |
data type |
| morphism |
process |
cobordism |
proof |
program |
Links
-
maddmaths.simai.eu/la-teoria-delle-categorie-sbarca-su-forbes
- T. Leinster,
Basic category theory, Cambridge University Press (2014)
- D. Spivak and B. Fong, An
invitation to applied category theory: seven sketches in compositionality,
Cambridge University Press (2019)
- https://plato.stanford.edu/entries/category-theory/
- johndcook/Category
theory