Talk:Category theory: Difference between revisions
imported>J. Noel Chiappa m (→English, please?: Clarify comment) |
imported>Peter Lyall Easthope (New draft of introductory paragraph.) |
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:: I'm having a bit of struggle seeing the common thread among the examples in the intro - and between them and the examples at the end of the article. Perhaps you could explain the concept in words, at slightly more length than "two mathematical concepts .. the object and the map or morphism"? Having done that, having some examples following that text might then be more illuminating. [[User:J. Noel Chiappa|J. Noel Chiappa]] 15:24, 18 May 2008 (CDT) | :: I'm having a bit of struggle seeing the common thread among the examples in the intro - and between them and the examples at the end of the article. Perhaps you could explain the concept in words, at slightly more length than "two mathematical concepts .. the object and the map or morphism"? Having done that, having some examples following that text might then be more illuminating. [[User:J. Noel Chiappa|J. Noel Chiappa]] 15:24, 18 May 2008 (CDT) | ||
:Noel, is this any better? | |||
Languages such as English have nouns and verbs. A noun identifies an | |||
object while a verb identifies an action or process. Thus the sentence | |||
"Please lift the tray." conjures an image of a tray on a table, a person | |||
who can lift it and the tray in its elevated position. | |||
In a pocket calculator, a datum is a number or pair of numbers. | |||
The calculator has a selection of operations which can be performed. | |||
Given the number 5, pressing the "square" key produces the | |||
number 25. | |||
High school mathematics introduces the concepts of set and function. | |||
Given the function | |||
<!-- <math>f(a) = a^{2}</math> --> | |||
''f''(''a'') = ''a''<sup>2</sup> | |||
we know that the solution set for | |||
<!-- <math>f(a) = 25</math> --> | |||
''f''(''a'') = 25 | |||
is | |||
<!-- <math>{-5, 5}</math> --> | |||
{-5, 5}. | |||
The mathematical abstraction drawn from these examples is based | |||
on two concepts: objects and the things which act on objects. In | |||
category theory, the thing which acts upon an object to produce | |||
another object is called a map or morphism. | |||
Morphisms can be composed. In the first example the tray can be | |||
lifted L and then rotated R. Composition simply means that two | |||
actions such as L and R can be thought of as combined into a | |||
single action R∘L. The symbol ∘ denotes composition. | |||
Morphisms are associative. Think of three motions of the tray. | |||
L: Lifting of the tray 10 cm above the table.<br/> | |||
R: Rotation of the tray 180 degrees clockwise.<br/> | |||
S: Shifting of the tray 1 m north while maintaining the elevated position.<br/> | |||
The lift and rotation can be thought of as combined into a single | |||
motion followed by the shift; this is denoted S∘(R∘L). | |||
Alternatively, the rotation and shift can be thought of as a single | |||
motion following the lift: (S∘R)∘L. Associativity | |||
simply means that | |||
S∘(R∘L) = (S∘R)∘L. | |||
An identity motion is any motion which brings the tray back | |||
to a starting position. If M denotes lowering the tray | |||
10 cm then M∘L is an identity motion. The identity rule | |||
in the formal definition of a category states that any action | |||
preceded or following by the identity is equal to the action | |||
alone. | |||
This formal definition embodies the preceding concepts in | |||
concise mathematical notation. | |||
... [[User:Peter Lyall Easthope|Peter Lyall Easthope]] 12:40, 19 May 2008 (CDT) | |||
Revision as of 11:40, 19 May 2008
English, please?
An introductory section in English that someone with 'only' beginning college math might understand is desireable. J. Noel Chiappa 07:24, 18 May 2008 (CDT)
- I've drafted something. Does it help any? Criticize away, ... Peter Lyall Easthope 14:55, 18 May 2008 (CDT)
- I'm having a bit of struggle seeing the common thread among the examples in the intro - and between them and the examples at the end of the article. Perhaps you could explain the concept in words, at slightly more length than "two mathematical concepts .. the object and the map or morphism"? Having done that, having some examples following that text might then be more illuminating. J. Noel Chiappa 15:24, 18 May 2008 (CDT)
- Noel, is this any better?
Languages such as English have nouns and verbs. A noun identifies an object while a verb identifies an action or process. Thus the sentence "Please lift the tray." conjures an image of a tray on a table, a person who can lift it and the tray in its elevated position.
In a pocket calculator, a datum is a number or pair of numbers. The calculator has a selection of operations which can be performed. Given the number 5, pressing the "square" key produces the number 25.
High school mathematics introduces the concepts of set and function. Given the function f(a) = a2 we know that the solution set for f(a) = 25 is {-5, 5}.
The mathematical abstraction drawn from these examples is based on two concepts: objects and the things which act on objects. In category theory, the thing which acts upon an object to produce another object is called a map or morphism.
Morphisms can be composed. In the first example the tray can be lifted L and then rotated R. Composition simply means that two actions such as L and R can be thought of as combined into a single action R∘L. The symbol ∘ denotes composition.
Morphisms are associative. Think of three motions of the tray.
L: Lifting of the tray 10 cm above the table.
R: Rotation of the tray 180 degrees clockwise.
S: Shifting of the tray 1 m north while maintaining the elevated position.
The lift and rotation can be thought of as combined into a single
motion followed by the shift; this is denoted S∘(R∘L).
Alternatively, the rotation and shift can be thought of as a single
motion following the lift: (S∘R)∘L. Associativity
simply means that
S∘(R∘L) = (S∘R)∘L.
An identity motion is any motion which brings the tray back to a starting position. If M denotes lowering the tray 10 cm then M∘L is an identity motion. The identity rule in the formal definition of a category states that any action preceded or following by the identity is equal to the action alone.
This formal definition embodies the preceding concepts in concise mathematical notation.
... Peter Lyall Easthope 12:40, 19 May 2008 (CDT)