A"> B"> X"> e"> f"> p"> 0"> 1"> 2"> 0"> 1"> ]> Category 2

Copyright © 2009 Peter Lyall Easthope. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

### Category 2

The skeletal version of the full category of sets of cardinality < 2, having as only objects the ordinal numbers 0 and 1. More briefly, "sets of cardinality < 2". This category can also be thought of as the ordinal or cardinal number "two". It is cartesian closed and concrete. It was suggested by Fred Linton in the Categories List, 2008-02-14. Where possible, the 2 is in boldface.

Objects 0 and 1.

In this category the maps are 0→0, 0→1, 1→1. There is no map 1→0.

Products 0×0, 0×1, 1×1.

Map Objects 00, 01, 10, 11.

A B A B
Notations
Expression
in
Arithmetic
Arithmetical
Expression in
English
Logical
Expression in
English
Expression
in
Logic
0zerofalse,
bottom,
up tack

&UpTack;
1onetrue,
top,
down tack

&DownTack; (1)
1-&mathA;one minus &mathA;not &mathA;~&mathA; (outmoded),
&NotSign;&mathA;
max(&mathA;, &mathB;)the greater of &mathA; and &mathB; &mathA; or &mathB;&mathA;&LogicalOr;&mathB;
min(&mathA;,&mathB;),
&mathA;·&mathB;
the lesser of &mathA; and &mathB; &mathA; and &mathB;&mathA;&LogicalAnd;&mathB;
max(1-&mathA;,&mathB;),
&mathB;&mathA;
the greater of, one
minus &mathA;, and &mathB;
&mathA; entails &mathB; (2),
if &mathA; then &mathB;,
&mathA; implies &mathB;,
&mathA; necessitates &mathB;,
&mathA; suffices for &mathB;
(&NotSign;&mathA;)&LogicalOr;&mathB;,
&mathA;&RightwardsDoubleArrow;&mathB;,
$\genfrac{}{}{0.2ex}{}{A}{B}$, (3)
&SmallDelta;&mathA;&mathB; Kronecker delta,
one when &mathA; equals B,
zero otherwise
&mathA; equals &mathB;,
&mathA; is equivalent to &mathB;,
&mathA; is necessary and
sufficient for &mathB;
(&mathA;&RightwardsDoubleArrow;&mathB;)&LogicalAnd; (&mathB;&RightwardsDoubleArrow;&mathA;),
&mathA;&LeftRightDoubleArrow;&mathB;,
 A ─── B
 A === B
(4)
(1) DownTack, Unicode U+22A4, is not available to some browsers.
(2) Logical consequence is more involved than is evident here.
(3) The horizontal line notation is attributed to G. Gentzen.
The expression to the left is constructed with MathML.
The expression to the right is constructed with SVG.
(4) A double line can distinguish equivalence from implication.
These expressions are constructed with tables. The double line is
represented by EqualsSign because BoxDrawingsDoubleHorizontal,
U+2550, ═, is not available to some browsers.
&mathf; &mathf;&Sub1; &mathf;&Sub2; &mathp;&Sub1; &mathp;&Sub2; &mathf; &mathf;&Sub1; &mathf;&Sub2; &mathp;&Sub1; &mathp;&Sub2; &mathf; &mathf;&Sub1; &mathf;&Sub2; &mathp;&Sub1; &mathp;&Sub2; &mathf; &mathe; &mathf; &mathe;

### Product Diagrams

These are all the instances of the diagram in the definition of the product, Lawvere & Schanuel, Conceptual Mathematics: A First Introduction to Categories, pages 217 & 237. For each of the three products and for each test object there is a diagram. The pale arrows are in place of the non-existent map 1&RightwardsArrow;0. There are four intact diagrams. Some browsers will not display a SVG diagram correctly.

0 0 0 0 &LabelProduct0x0; 1 0 0 0 &LabelProduct0x0; 0 0 0 1 &LabelProduct0x1; 1 0 0 1 &LabelProduct0x1; 0 1 1 1 &LabelProduct1x1; 1 1 1 1 &LabelProduct1x1;

### Map Object Diagrams

These are all the instances of the diagram in the definition of the map object, Conceptual Mathematics, page 313. For each of four map objects and for each test object there is a diagram. The pale arrows are in place of the non-existent map 1&RightwardsArrow;0. There are seven intact diagrams. Some browsers will not display a SVG diagram correctly.

0×0 1&Sub0;&TimesNameOf.f; 0×0&Super0; 0 0&Super0; 0×1 1&Sub0;&TimesNameOf.f; 0×0&Super0; 0 0&Super0; 1×0 1&Sub1;&TimesNameOf.f; 1×0&Super1; 0 0&Super1; 1×1 1&Sub0; &TimesNameOf.f; 1×0&Super1; 0 0&Super1; 0×0 1&Sub0;&TimesNameOf.f; 0×1&Super0; 1 1&Super0; 0×1 1&Sub0;&TimesNameOf.f; 0×1&Super0; 1 1&Super0; 1×0 1&Sub1;&TimesNameOf.f; 1×1&Super1; 1 1&Super1; 1×1 1&Sub1;&TimesNameOf.f; 1×1&Super1; 1 1&Super1;

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