Category Theory LO21644

AM de Lange (
Mon, 17 May 1999 09:03:33 +0200

Replying to LO21568 --

Dear Organlearners,

>I have kept this last paragraph for Rick. Rick, I will send you
>under a separate message as file attachment a bitmap
>illustrating (i) the Axiom of Identity and (ii) the Axiom of
>Associaitivity as CAT diagrams. With your permission, please
>make it available on the LO server and give our fellow learners
>the URL. Thank you very much.

Many thanks to Rick, our host.

The URL is:
< >

I think that these two diagrams are so abstract that a little
explanation is required.

When you read, for example, the (i) identity diagram, picture
in your mind the following scheme:
[being] --(becoming)--> [being] --(becoming)--> [being]
[a] --(f)--> [b] --(g)--> [c]
where [a], [b] and [c] are objects
and --(f)--> and --(g)--> are arrows.

It can be any being-becoming pattern, for example,
[John] --(kick)--> [ball] --(move)--> [goal]
which will be in the natural language English:
John kick{s the} ball {. The ball} move{s to the} goal {posts}.
I have put all extra syntaxis which English requires into braces { }.
Note, for example the {s} in the tenses to establish concorde. (My own
mother tongue Afrikaans has done away with concorde in tenses.)
Without the braces the sentence reads:
John kicks the ball. The ball moves to the goal posts.

Bringing in the identity arrow results into
[John] --(kick)--> [ball] --(move)--> [goal]
/ ^
Think of a looping arrow --(is)--> beginning and ending on [ball].
I hope the representation with ASCII text comes out all right.
It looks like a triangle on my screen. The "English" would be
John kick{s the} ball {. The ball is ball} move{s to the} goal
This is reduced to proper English by
John kick{s the} ball {which} move{s to the} goal {posts}.
In other words, the {which} corresponds to {. The ball is ball}. This
is how we used the "identity arrow" in English to join the two
John kicks the ball. The ball moves to the goal posts.
into one compound sentence
John kicks the ball which moves to the goal posts.

Category Theory is fast replacing Set Theory as the background against
which mathematical systems operate. The example above illustrate
clearly why. Try to analyze the two sentences and their compound
sentence in terms of Set Theory. The result is far less natural than
with Category Theory.

Who would think that there is mathematics in a language?

Best wishes


At de Lange <> Snailmail: A M de Lange Gold Fields Computer Centre Faculty of Science - University of Pretoria Pretoria 0001 - Rep of South Africa

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