# Concrete category

Concrete category 是用來描述該 category 的 object 與 function 可以直接對應 set 與 function。諸如 $\mathbf{Grp}$ (the category of groups), $\mathbf{Rng}$ (the category of rings), $\mathbf{Top}$ (the category of topological space), 以及所有的 small category。然而, 並非所有的 large category 都是 concrete, 而其中包含比較重要的例子有 $\mathbf{Toph}$, 其 objects 為 pointed topological space 而 morphisms 為 homotopy classes, 由 Peter Freyd 於 1970 年提出證明Continue reading

# Splitting Lemma

Splittin Lemma. Let $0 \rightarrow A_{1} \overset{f}{\rightarrow} B \overset{g}{\rightarrow} A_{2} \rightarrow 0$ be a short exact sequence of $R-$module homomorphism. The the following are equivalent.

1. (Right split): There is an $R-$module homomorphism $h : A_{2} \rightarrow B$ with $gh = 1_{A_{2}}$ ,
2. (Left split): There is an $R-$module homomorphism $k : B \rightarrow A_{1}$ with $kf = 1_{A_{1}}$ ,
3. (Direct Sum): The given sequence is isomorphic to $0 \rightarrow A_{1} \rightarrow A_{1} \oplus A_{2} \rightarrow A_{2} \rightarrow 0$, and thus $B \cong A_{1} \oplus A_{2}$. Continue reading