^^Category theory.
Elementary Theory of the Category of
Sets.
Lambda calculus
wp/Category_theory
wp/Category_(mathematics)
Category theory is
all about organizing and layering structures.
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.
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 |
SSID (Wi-Fi) Service Set Identifier (Wi-Fi)
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/
- categories are the right intuition for talking about composing things with
domains and ranges.
cdsmith.wordpress.com/why-do-monads-matter/