1 Word has structure.
On one case of the structure, refer to the following papers.
2 Now the simplest structure example is presented.
Word has starting point of meaning. The meaning is called ms.
Word has time for shifting meaning. The time is called ts.
Word has ending point of meaning. The meaning is called me.
3 By category, word’s ms, ts and me are expressed below.
ts : ms → me
Here ts is replaced by arrow f. ms and me are replaced by objects a and b. The relation is expressed by diagram below.
f : a → b
4 Here word X and word Y are presented as category that consists of topological space or group.
Word X is expressed by f : a → b.
Word Y is expressed by g : b → c.
Composition X and Y is expressed by composite f ∘g.
f ∘g : a → c
5 Here f ∘g is putted to h.
The diagram that consists of objects a, b, c and arrow f, g, h is presented.
6 Now category’s morphism is called functor.
Functor that is commutatively transformed to another functor is called natural transformation.
Natural transformation is called component.
On component, refer to the following paper’s concept <passage>.
Natural transformation means functor’s morphism.
7 Product of two categories B and C is defined by the following.
B ×C’s arrow <b, c> →<b’, c’> is <f, g>. Here, f : b → b’ and g : c → c’
<f’, g’>∘<f, g> = <f’∘f, g’∘g>
8 C×2 consists of C×0, C×1 and arrows combined C×0, C×1.
Here when any natural transformation has unique functor, it is called universal natural transformation.
10 By upper rough sketch, <Cube Theory>’ concepts are transplanted to category.
11 Word is reconstructed by topological space or group.
12 Structure of word that is presented by KARCEVSKIJ has the chance affirmatively verified in accordance with the concepts of MACLANE.
Tokyo April 6, 2007
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