# 0.2. category

<The Detail of This Chapter.pdf>

### 0.2.1.what is a category?

Category is a set that express a routine:

Take some objects that compose a class O,  then we construct a class M of morphisms, a morphism hom(i, j) as an element of M, is defined from i to j, i and j are any element of O, and we name that i is the domain of f, j is the codomain of f. Then we have:

For any three elements (i, j, k) of O, if we can get two morphism hom(i, j) and hom(j, k), then we define a composite operation * from hom(i, j) to hom(j, k), get a composite morphism hom(i, k), means hom(i, j) * hom(j, k) = hom(i, k)

Such composite operation satisfy associative law.

For any element j of O, there is a identity morphism ID(j), it is from j to j, and for any hom(i, j) or hom(j, k), we have hom(i, j) * ID(j)=hom(i, j) and  ID(j) * hom(j, k)=hom(j, k).

So, a category is just a set {O, M, *}.

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SET_A={O, M, *}; O={i, j}; M={ID(i)_1, ID(i)_2, hom(i, j), hom(j, i)_1, hom(j, i)_2}. Is SET_A a category?

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### 0.2.2.why we need category?

Because category is the most general routine in modern mathematics.

0.2.3. product and plus of categories