By John R. Pierce

Covers encoding and binary digits, entropy, language and that means, effective encoding and the noisy channel, and explores ways that info conception pertains to physics, cybernetics, psychology, and paintings. "Uncommonly good...the so much pleasurable dialogue to be found." - medical American. 1980 version.

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T h e small category we obtained in this way will be considered as the diagram scheme for our diagram. If the diagram scheme consists of two objects X and Y and of three morphisms \ > 1 , and x : X —• Y, then we call this category 2. T h e diagrams of Funct(2, ? ) are in one-one correspondence to the morphisms of €. T h u s one calls Funct(2, ? ) the morphism category of ? A morieJ c x c F 26 1. PRELIMINARY NOTIONS phism in Funct(2, ? 5, we want to generalize again a notion from S to arbitrary categories.

A l l surjective set maps are retractions. I n A b the map Zsn\-> 2n e Z is a kernel of the residue class homomorphism Z —>- Z/2Z; however, it is not a section. In fact, if g : Z —>- Z were a corresponding retraction, then 2^(1) — 1 e Z. But there is no such element ^(1) in Z. LI 1 Products and Coproducts Another important notion in the category of sets is the notion of a produet of two sets A and B. T h e produet is the set of pairs A x B = {(a b)\aeA t and b e B} 30 1. PRELIMINARY NOTIONS Furthermore, there are maps p : A x 5 9 (a, 6) H-> ae A A and p : A X B3{a b)\-> B y be B We want to investigate whether this notion can again be generalized i n the desired way to morphism sets.

If each [finite, nonempty] family of objects i n ? has a produet, then we call ? a category, with \ßnite nonempty] produets. If (A {pu}) is a produet of a y } 32 1. PRELIMINARY NOTIONS family of objects {A^ i n ^ and if h : B —> A is an isomorphism, then (B, { pih}) is another produet for the A . i£l i L E M M A 2. Assume that in the category there is a produet for each pair of objects. Then is a category with finite, nonempty produets. Proof. L e t A A be a family of objects i n ? W e show that (••• (A X A ) X •*•) X A is a produet of the A .