# 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, *}.

________________

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**