Peircers,
Adding links, correcting typos ...
For ease of reference, here is a collection of links to posts where
basic concepts are introduced, together with a few pertinent texts:
Category Theory
===============
☞http://web.archive.org/web/20140322014010/http://permalink.gmane.org/gmane.science.philosophy.peirce/12131
Just by way of getting everyone on the same page with
respect to mathematical categories, here's a primer.
☞http://mathworld.wolfram.com/Category.html
Transcribing this into slightly more standard notation:
A mathematical category C consists of three things:
1. A collection of “objects”, usually denoted Obj(C).
2. For each pair of objects X, Y in C, a collection
of “arrows” or “morphisms” from X to Y, usually
denoted either Arr(X,Y) or Hom(X,Y).
If the arrow f is in Arr(X,Y), we write f : X → Y.
3. A binary operation (∘) called “composition”
defined on “composable” pairs of arrows.
The arrows in Arr(C) must meet the following conditions:
Existence of Composites. If f : X → Y and g : Y → Z
then there is a unique morphism f∘g : X → Z called the
“composition of f followed by g”. (Note however that it
is probably more common to write this as g∘f and to call
it the “composition of g on f”.)
Associative Axiom. Given f : W → X, g : X → Y, h : Y → Z,
we have (f∘g)∘h = f∘(g∘h).
Identity Axiom. For each object X in C, there exists
an “identity morphism” 1_X such that for any f : X → Y
we have 1_X ∘ f = f = f ∘ 1_Y.
Sign Relations
==============
☞http://web.archive.org/web/20140322014602/http://permalink.gmane.org/gmane.science.philosophy.peirce/12141
☞http://intersci.ss.uci.edu/wiki/index.php/Sign_relation
Sign Relations : Definition
===========================
☞http://intersci.ss.uci.edu/wiki/index.php/Sign_relation#Definition
Sign Relations : More Quotes
============================
☞http://forum.wolframscience.com/archive/topic/647.html
Sign Relations : Commentary
===========================
☞http://forum.wolframscience.com/archive/topic/675.html
<quote>
Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no
more refers to human thought than does the definition of a line as the place which a particle
occupies, part by part, during a lapse of time. Namely, a sign is something, A, which brings
something, B, its ''interpretant'' sign determined or created by it, into the same sort of
correspondence with something, C, its ''object'', as that in which itself stands to C.
It is from this definition, together with a definition of “formal”, that I deduce mathematically the
principles of logic. I also make a historical review of all the definitions and conceptions of
logic, and show, not merely that my definition is no novelty, but that my non-psychological
conception of logic has ''virtually'' been quite generally held, though not generally recognized.
(C.S. Peirce, NEM 4, 20–21).
</quote>
Triadic Relations
=================
☞http://web.archive.org/web/20140317002001/http://permalink.gmane.org/gmane.science.philosophy.peirce/12186
☞http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation
Background Material on Relations in General
===========================================
Relations
=========
☞http://intersci.ss.uci.edu/wiki/index.php/Relation_%28mathematics%29
Relation Theory
===============
☞http://intersci.ss.uci.edu/wiki/index.php/Relation_theory
Relation Composition
====================
☞http://intersci.ss.uci.edu/wiki/index.php/Relation_composition
Relation Reduction
==================
☞http://web.archive.org/web/20140322014802/http://permalink.gmane.org/gmane.science.philosophy.peirce/12220
☞http://intersci.ss.uci.edu/wiki/index.php/Relation_reduction
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