3 函子与自然变换

3.1 函子

3.1.1 协变函子和反变函子

设 \(\mathscr{A}\) 和 \(\mathscr{B}\) 为范畴，若 \(F: \mathscr{A} \rightarrow \mathscr{B}\) 为函子，则称 \(\mathscr{A}\) 为 \(F\) 的定义域，\(\mathscr{B}\) 为 \(F\) 的值域
1. 如果函数 \(F: \operatorname{Mor} \mathscr{A} \rightarrow \mathrm{Mor} \mathscr{B}\) 满足
  1. 若 \(e\) 为 \(\mathscr{A}\) 的幺态射，则 \(F(e)\) 为 \(\mathscr{B}\) 的幺态射
  2. 若 \(f \in \mathscr{A}[A, B]\)，且 \(g \in \mathscr{A}[B, C]\)，则 \(F(g \circ f)=F(g) \circ F(f)\)
  则称 \(F\) 为一个从 \(\mathscr{A}\) 到 \(\mathscr{B}\) 的协变函子，记为 \(F: \mathscr{A} \rightarrow \mathscr{B}\) 或 \(\mathscr{A} \xrightarrow{F} \mathscr{B}\)
2. 如果函数 \(F: \operatorname{Mor} \mathscr{A} \rightarrow \mathrm{Mor} \mathscr{B}\) 满足
  1. 若 \(e\) 为 \(\mathscr{A}\) 的幺态射，则 \(F(e)\) 为 \(\mathscr{B}\) 的幺态射
  2. 若 \(f \in \mathscr{A}[A, B]\)，且 \(g \in \mathscr{A}[B, C]\)，则 \(F(g \circ f)=F(f) \circ F(g)\)
  则称 \(F\) 为一个从 \(\mathscr{A}\) 到 \(\mathscr{B}\) 的反变函子，记为 \(F: \mathscr{A} \rightarrow \mathscr{B}\) 或 \(\mathscr{A} \xrightarrow{F} \mathscr{B}\)
协变函子和反变函子统称为函子；称映射到自身的函子为自函子
设 \(F: \mathscr{A} \rightarrow \mathscr{B}\) 为函子
1. 如果 \(F\) 为协变函子，则存在唯一的函数 \(\theta_{F}: \operatorname{Ob} \mathscr{A} \rightarrow \operatorname{Ob} \mathscr{B}\) 满足
  1. 若 \(A \in \operatorname{Ob} \mathscr{A}\)，则 \(F\left(\mathbf{I}_{A}\right)=\mathbf{I}_{\theta_{F}(A)}\)
  2. 若 \(f \in \mathscr{A}[A, B]\)，则 \(F(f)=\mathscr{B}\left[\theta_{F}(A), \theta_{F}(B)\right]\)
2. 如果 \(F\) 为反变函子，则存在唯一函数 \(\theta_{F}: \operatorname{Ob} \mathscr{A} \rightarrow \operatorname{Ob} \mathscr{B}\) 满足
  1. 若 \(A \in \operatorname{Ob} \mathscr{A}\)，则 \(F\left(\mathbf{I}_{A}\right)=\mathbf{I}_{\theta_{F}(A)}\)
  2. 若 \(f \in \mathscr{A}[A, B]\)，则 \(F(f)=\mathscr{B}\left[\theta_{F}(B), \theta_{F}(A)\right]\)
对任意 \(A \in \mathscr{A}\)，将 \(\theta_{F}(A)\) 记为 \(F(A)\)
设 \(F: \mathscr{A} \rightarrow \mathscr{B}, B: \mathscr{B} \rightarrow \mathscr{C}\) 和 \(S: \mathscr{C} \rightarrow \mathscr{D}\) 均为函子
1. \(F \circ I_{\mathscr{A}}=F=I_{\mathscr{B}} \circ F\)
2. \(S \circ(G \circ F)=(S \circ G) \circ F \circ\)
典型函子：若 \(\mathscr{A}, \mathscr{B}, \mathscr{C}\) 为范畴
1. 对任意 \(f \in \operatorname{Mor} \mathscr{C}\)，令 \(I_{\mathscr{C}}(f)=f\)，可得到协变函子 \(I_{\mathscr{C}}: \mathscr{C} \rightarrow \mathscr{C}\)，称为恒等函子
2. 设 \(B \in \operatorname{Ob} \mathscr{B}\)，对任意 \(f \in \operatorname{Mor} \mathscr{A}\)，令 \(\operatorname{Const}_{B}(f)=\mathbf{I}_{B}\)，可得到函子 \(\operatorname{Const}_{B}: \mathscr{A} \rightarrow \mathscr{B}\)，称为常函子
3. 设 \(\mathscr{A} \subseteq \mathscr{B}\)，对任意 \(f \in \operatorname{Mor} \mathscr{A}\)，令 \(\operatorname{in}_{\mathscr{A}}(f)=f\)，可得到协变函子 \(\operatorname{in}_{\mathscr{A}}: \mathscr{A} \rightarrow \mathscr{B}\)，称为包含函子，简写为 \(\mathscr{A} \hookrightarrow \mathscr{B}\)
4. 设函子 \(F: \mathscr{A} \rightarrow \mathscr{B}\)，对任意 \(f \in \operatorname{Mor} \mathscr{C}\)，令 \(F^{\mathrm{op}}\left(f^{\mathrm{op}}\right)=(F(f))^{\mathrm{op}}\)，可得到函子 \(F^{\mathrm{op}}: \mathscr{A}^{\mathrm{op}} \rightarrow \mathscr{B}^{\mathrm{op}}\)，称为 \(F\) 的对偶函子
5. 设函数 \(f: X \rightarrow Y\)，定义 \(P(f): 2^{X} \rightarrow 2^{Y}\) 为 \(P(f)(B)=f(B),\ B \subseteq X\)，可得到协变函子 \(P: \mathbf{Set} \rightarrow \mathbf{Set}\)，称为幂等函子
6. 设 \(F: \mathscr{A} \rightarrow \mathscr{B}, G: \mathscr{B} \rightarrow \mathscr{C}\) 为函子，令 \((G \circ F)(f)=G(F(f))\)，可得到函子 \(G \circ F: \mathscr{A} \rightarrow \mathscr{C}\)，称为 \(G\) 与 \(F\) 的复合函子

3.1.2 多元函子

设 \(1 \leqslant i_{1}<i_{2}<\cdots<i_{r} \leqslant n\) 且 \(\mathscr{A}_{1}, \mathscr{A}_{2}, \cdots, \mathscr{A}_{n}\) 和 \(\mathscr{B}\) 均为范畴，若函数 \(F: \operatorname{Mor} \mathscr{A}_{1} \times \operatorname{Mor} \mathscr{A}_{2} \times \cdots \times \operatorname{Mor} \mathscr{A}_{n} \rightarrow \operatorname{Mor} \mathscr{B}\) 满足
1. 若 \(e_{1}, e_{2}, \cdots, e_{n}\) 分别为 \(\mathscr{A}_{1}, \mathscr{A}_{2}, \cdots, \mathscr{A}_{n}\) 的幺态射，则 \(F\left(e_{1}, e_{2}, \ldots, e_{n}\right)\) 为 \(\mathscr{B}\) 的幺态射
2. 若 \(A_{i}, A_{i}^{\prime}, A_{i}^{\prime \prime} \in \operatorname{Ob} \mathscr{A}_{i}\) 和 \(f_{i}, g_{i} \in \operatorname{Mor} \mathscr{A}_{i}(1 \leqslant i \leqslant n)\) 使得
  1. 当 \(i \notin\left\{i_{1}, i_{2}, \cdots, i_{r}\right\}\) 时恒有 \(f_{i} \in \mathscr{A}_{i}\left[A_{i}, A_{i}^{\prime}\right]\) 且 \(g_{i} \in \mathscr{A}_{i}\left[A_{i}^{\prime}, A_{i}^{\prime \prime}\right]\)
  2. 当 \(i \in\left\{i_{1}, i_{2}, \cdots, i_{r}\right\}\) 时恒有 \(f_{i} \in \mathscr{A}_{i}\left[A_{i}^{\prime \prime}, A_{i}^{\prime}\right]\) 且 \(g_{i} \in \mathscr{A}_{i}\left[A_{i}^{\prime}, A_{i}\right]\)
  则必有 \(F\left(\left\langle g_{1}, g_{2}, \cdots, g_{n}\right\rangle \circ\left\langle f_{1}, f_{2}, \cdots, f_{n}\right\rangle\right)=F\left(\widetilde{g}_{1}, \widetilde{g}_{2}, \cdots, \widetilde{g}_{n}\right) \circ F(\widetilde{f}_{1}, \widetilde{f}_{2}, \cdots, \widetilde{f}_{n})\)，其中对任意 \(1 \leqslant i \leqslant n\) 都有
  
  \[ \begin{aligned} & \widetilde{f}_{i}=\left\{\begin{aligned} & f_{i}, & i \notin\left\{i_{1}, i_{2}, \cdots, i_{r}\right\} \\ & g_{i}, & \textsf{ 否则 } \end{aligned}\right.\\ & \widetilde{g}_{i}=\left\{\begin{aligned} &g_{i}, & i \notin\left\{i_{1}, i_{2}, \cdots, i_{n}\right\} \\ &f_{i}, & \textsf{ 否则 } \end{aligned}\right. \end{aligned} \]
则称 \(F\) 为一个从 \(\mathscr{A}_{1}, \mathscr{A}_{2}, \cdots, \mathscr{A}_{n}\) 到 \(\mathscr{B}\) 的 \(n\) 元函子，记为 \(F: \mathscr{A}_{1} \times \mathscr{A}_{2} \times \cdots \times \mathscr{A}_{n} \rightarrow \mathscr{B}\) 或 \(\mathscr{A}_{1} \times \mathscr{A}_{2} \times \cdots \times \mathscr{A}_{n} \xrightarrow{F} \mathscr{B}\)，并称 \(i_{1}, i_{2}, \cdots, i_{r}\) 为 \(F\) 的反变指标
1. 若 \(r=0\)，则称 \(F\) 为 \(n\) 元协变函子
2. 若 \(r=n\)，则称 \(F\) 为 \(n\) 元反变函子
3. 若 \(0<r<n\)，则称 \(F\) 为 \(n\) 元混合型函子
4. 若 \(i \in\left\{i_{1}, i_{2}, \cdots, i_{r}\right\}\)，则称 \(F\) 的第 \(i\) 个自然变元是反变的
5. 若 \(1 \leqslant i \leqslant n\) 且 \(i \notin\left\{i_{1}, i_{2}, \cdots, i_{r}\right\}\)，则称 \(F\) 的第 \(i\) 个自然变元为协变的
若 \(n\) 元函子 \(F: \mathscr{A}_{1} \times \mathscr{A}_{2} \times \cdots \times \mathscr{A}_{n} \rightarrow \mathscr{B}\) 以 \(i_{1}, i_{2}, \cdots, i_{r}\) 为反变指标，则必有函数 \(\theta_{F}: \operatorname{Ob} \mathscr{A}_{1} \times \cdots \times \operatorname{Ob} \mathscr{A}_{n} \rightarrow \operatorname{Ob} \mathscr{B}\)
1. 若 \(A_{1} \in \operatorname{Ob} \mathscr{A}_{1}, \cdots, A_{n} \in \operatorname{Ob} \mathscr{A}_{n}\)，则 \(F\left(\mathbf{I}_{A_{1}}, \mathbf{I}_{A_{2}}, \cdots, \mathbf{I}_{A_{n}}\right)=\mathbf{I}_{\theta_{F}\left(A_{1}, A_{2}, \cdots, A_{n}\right)}\)
2. 若 \(f_{1} \in M\) or \(\mathscr{A}_{1}, \cdots, f_{n} \in M\) or \(\mathscr{A}_{n}\) 使得
  1. 如果 \(i \notin\left\{i_{1}, i_{2}, \cdots, i_{r}\right\}\)，则 \(f_{i} \in \mathscr{A}_{i}\left[A_{i}, A_{i}^{\prime}\right]\)
  2. 如果 \(i \in\left\{i_{1}, i_{2}, \cdots, i_{r}\right\}\)，则 \(f_{i} \in \mathscr{A}_{i}\left[A_{i}^{\prime}, A_{i}\right]\)
则 \(F(f_{1}, f_{2}, \cdots, f_{n}) \in \mathscr{B}[\theta_{F}(\widetilde{A}_{1}, \widetilde{A}_{2}, \cdots, \widetilde{A}_{n}), \theta_{F}(\widetilde{A}_{1}^{\prime}, \widetilde{A}_{2}^{\prime}, \cdots \widetilde{A}_{n}^{\prime})]\)，其中对任意 \(1 \leqslant i \leqslant n\) 都有

\[ \begin{aligned} & \widetilde{A}_{i}=\left\{\begin{aligned} & A_{i}, & i \notin\left\{i_{1}, i_{2}, \cdots, i_{r}\right\} \\ & A_{i}^{\prime}, & \textsf{ 否则 } \end{aligned}\right. \\ & \widetilde{A_{i}^{\prime}}=\left\{\begin{aligned} & \dot{A_{i}^{\prime}}, & \notin\left\{i_{1}, i_{2}, \cdots, i_{r}\right\} \\ & A_{i}, & \textsf{ 否则 } \end{aligned}\right. \end{aligned} \]

对任意 \(A_1 \in \mathscr{A}_{1}, A_2 \in \mathscr{A}_{2}, \cdots, A_n \in \mathscr{A}_{n}\)，将 \(\theta_{F}(A_1, A_2, \cdots, A_n)\) 记为 \(F(A_1, A_2, \cdots, A_n)\)
设 \(F: \mathscr{A}_{1} \times \mathscr{A}_{2} \times \cdots \times \mathscr{A}_{n} \rightarrow \mathscr{B}\) 为 \(n\) 元函子，\(G: \mathscr{B} \rightarrow \mathscr{C}\) 为函子，令 \((G \circ F)\left(f_{1}, f_{2}, \cdots, f_{n}\right)=G\left(F\left(f_{1}, f_{2}, \cdots, f_{n}\right)\right)\)
1. \(G \circ F: \mathscr{A}_{1} \times \mathscr{A}_{2} \times \cdots \times \mathscr{A}_{n} \rightarrow \mathscr{C}\) 为 \(n\) 元函子
2. 若 \(G\) 为协变函子且 \(1 \leqslant i \leqslant n\)，则 \(G \circ F\) 的第 \(i\) 个自变元为协变的，当且仅当 \(F\) 的第 \(i\) 个自变元为协变的
3. 若 \(G\) 为反变函子且 \(1 \leqslant i \leqslant n\)，则 \(G \circ F\) 的第 \(i\) 个自变元为协变的，当且仅当 \(F\) 的第 \(i\) 个自变元为反变的
典型多元函子
1. 设 \(\mathscr{A}_{1}, \mathscr{A}_{2}, \cdots, \mathscr{A}_{n}\) 均为范畴且 \(1 \leqslant i \leqslant n\)，令
  
  \[ \prod_{i \leqslant n}\left(f_{1}, f_{2}, \cdots, f_{n}\right)=f_{i},\ f_{i} \in \operatorname{Mor} \mathscr{A}_{i} \]
  
  可得到 \(n\) 元协变函子\(\Pi_{i}: \mathscr{A}_{1} \times \mathscr{A}_{2} \times \cdots \times \mathscr{A}_{n} \rightarrow \mathscr{A}_{i}\)，称为投影函子
2. 若对任意函数 \(f\) 和 \(g\)，令 \(F(f, g)=f \times g\)，可得到二元协变函子 \(F: \mathbf{Set} \times \mathbf{Set} \rightarrow \mathbf{Set}\)，称为乘积函子
设 \(\mathscr{C}\) 为一个局部小范畴
1. 若 \(A, B \in \operatorname{Ob} \mathscr{C}\)，则令 \(\operatorname{hom}_{\mathscr{C}}(A, B)=[A, B]\)
2. 若 \(f \in[A, B]\) 且 \(g \in[C, D]\)，则定义函数 \(\operatorname{hom}_{\mathscr{C}}(f, g):[B, C] \rightarrow[A, D]\) 为对任意 \(\varphi \in[B, C]\) 有 \(\operatorname{hom}_{\mathscr{C}}(f, g)(\varphi)=g \circ \varphi \circ f\)
不引起歧义时，常把 \(\mathrm{hom}_{\mathscr{C}}\) 简写为 \(\mathrm{hom}\)
1. 设 \(\mathscr{C}\) 为一个局部小范畴
  1. \(\operatorname{hom}: \mathscr{C} \times \mathscr{C} \rightarrow \mathbf{Set}\) 为二元混合型函子
  2. \(\operatorname{hom}\) 关于第一个自变元为反变的，关于第二个自变元为协变的
2. 若 \(A, B \in \operatorname{Ob} \mathscr{C}\)，分别定义 \(h_{A}\) 与 \(h^{B}\) 为
  
  \[ \begin{aligned} h_{A}(B) & =\operatorname{hom}(A, B) & & A \in \operatorname{Ob} \mathscr{C} \\ h_{A}(g) & =\operatorname{hom}\left(\mathbf{I}_{A}, g\right) & & g \in \operatorname{Mor} \mathscr{C} \\[0.5em] h^{B}(A) & =\operatorname{hom}(A, B) & & A \in \operatorname{Ob} \mathscr{C} \\ h^{B}(f) & =\operatorname{hom}\left(f, \mathbf{I}_{B}\right) & & f \in \operatorname{Mor\mathscr {C}} \end{aligned} \]
  1. \(h_{A}: \mathscr{C} \rightarrow \mathbf{Set}\) 为协变函子，\(h^{B}: \mathscr{C} \rightarrow \mathbf{Set}\) 为反变函子
  2. 若 \(f \in[A, B]\)，则 \(h_{A}(f)\left(\mathbf{I}_{A}\right)=f=h^{B}(f)\left(\mathbf{I}_{B}\right)\)
  3. 若 \(f, g \in [C, D]\)，则 \(f=g \leftrightarrow \forall A \in \operatorname{Ob} \mathscr{C}: h_{A}(f)=h_{A}(g) \leftrightarrow \forall B \in \operatorname{Ob} \mathscr{C}: h^{B}(f)=h^{B}(g)\)
  4. 设 \(f \in[C, D]\)
    1. \(f\) 为单态射当且仅当对每个 \(A \in \operatorname{Ob} \mathscr{C}\)，皆有 \(h_{A}(f)\) 为单射
    2. \(f\) 为外态射当且仅当对每个 \(B \in \operatorname{Ob} \mathscr{C}\)，皆有 \(h^{B}(f)\) 为单射
  5. 设 \(A \in \operatorname{Ob} \mathscr{C}\)
    1. \(A\) 为终止对象当且仅当对每个 \(B \in \operatorname{Ob} \mathscr{C}\) 都有 \(h_{B}(A)\) 为单元素集合
    2. \(A\) 为初始对象当且仅当对每个 \(B \in \operatorname{Ob} \mathscr{C}\) 都有 \(h^{B}(A)\) 为空集
3. 设 \(\mathscr{C}\) 为局部小范畴且 \(\langle\mathcal{D}, \Gamma\rangle\) 为 \(\mathscr{C}\) 中一个图
  1. 若 \(\langle L, \tau\rangle\) 为 \(\langle\mathcal{D}, \Gamma\rangle\) 的锥，则 \(\varprojlim\langle\mathcal{D}, \Gamma\rangle=\langle L, \tau\rangle\) 当且仅当对每个 \(A \in \operatorname{Ob} \mathscr{C}\) 有 \(\varprojlim \left\langle\mathcal{D}, h_{A} \circ \Gamma\right\rangle=\left\langle h_{A}(L), h_{A} \circ \tau\right\rangle\)
  2. 若 \(\langle\tau, L\rangle\) 为 \(\langle\mathcal{D}, \Gamma\rangle\) 的余锥，则 \(\varinjlim \langle\mathcal{D}, \Gamma\rangle=\langle\tau, L\rangle\) 当且仅当对每个 \(B \in \operatorname{Ob} \mathscr{C}\) 有 \(\varinjlim\left\langle\mathcal{D}, h^{B} \circ \Gamma\right\rangle=\left\langle h^{B} \circ \tau, h^{B}(L)\right\rangle\)
4. 设 \(\mathscr{A}, \mathscr{B}\) 为范畴且 \(\mathscr{C}\) 为局部小范畴，\(F: \mathscr{A} \rightarrow \mathscr{C}\) 和 \(G: \mathscr{A} \rightarrow \mathscr{C}\) 是函子．令
  
  \[ \begin{aligned} \operatorname{hom}(F, G)(A, B)&=\operatorname{hom}(F(A), G(B)) & A \in \operatorname{Ob} \mathscr{A} \textsf{ 且 } B \in \operatorname{Ob} \mathscr{B} \\ \operatorname{hom}(F, G)(f, g)&=\operatorname{hom}(F(f), G(g)) & f \in \operatorname{Mor} \mathscr{A} \textsf{ 且 } g \in \operatorname{Mor} \mathscr{B} \end{aligned} \]
  
  则可得到二元函子 \(\operatorname{hom}(F, G): \mathscr{A} \times \mathscr{B} \rightarrow \mathbf{Set}\) 是 \(\operatorname{hom}\) 的推广．此时令
  
  \[ \begin{aligned} h_{F}&=\operatorname{hom}\left(F, I_{\mathscr{C}}\right) \\ h^{G}&=\operatorname{hom}\left(I_{\mathscr{C}}, G\right) \end{aligned} \]
  
  则 \(h_{F}: \mathscr{A} \times \mathscr{C} \rightarrow \mathbf{Set}\) 与 \(h^{G}: \mathscr{C} \times \mathscr{B} \rightarrow \mathbf{Set}\) 均为二元函子

3.1.3 函子的性质

设 \(\mathscr{A}\) 和 \(\mathscr{B}\) 均为范畴，\(F: \mathscr{A} \rightarrow \mathscr{B}\) 为协变函子
1. 设 \(F: \mathscr{A} \rightarrow \mathscr{B}\) 为协变函子，\(P\) 为关于态射的某种性质
  1. \(F\) 保持 \(P\)：若 \(f \in \operatorname{Mor} \mathscr{A}\) 具有 \(P\)，则 \(F(f)\) 也具有 \(P\)
  2. \(F\) 反射 \(P\)：若 \(f \in \operatorname{Mor} \mathscr{A}\) 使 \(F(f)\) 具有 \(P\)，则 \(f\) 也具有 \(P\)
  易知协变函子 \(F\) 保持幺态射，\(\text{section}\)，\(\text{retraction}\)，同构态射以及范畴中图的交换性
2. 设 \(A, B \in \operatorname{Ob} \mathscr{A}\)，定义函数 \(F(A, B): \mathscr{A}[A, B] \rightarrow \mathscr{B}[F(A), F(B)]\) 为 \(\forall f \in \mathscr{A}[A, B]: F(A, B)(f)=F(f)\)
  1. 若每个 \(F(A, B)\) 均为单射，则称 \(F\) 为忠信的
  2. 若每个 \(F(A, B)\) 均为满射，则称 \(F\) 为完全的
  3. 若函数 \(F: \operatorname{Mor} \mathscr{A} \rightarrow M\) or \(\mathscr{B}\) 为单射，则称 \(F\) 为嵌入函子
  4. 若对每个 \(B \in \operatorname{Ob} \mathscr{B}\) 皆有 \(A \in \operatorname{Ob} \mathscr{A}\) 使 \(F(A) \simeq B\)，则称 \(F\) 为稠密的
设协变函子 \(F: \mathscr{A} \rightarrow \mathscr{B}\) 为忠信的
1. 如果 \(f, g \in \mathscr{A}[A, B]\) 使 \(F(f)= F(g)\)，则 \(f=g\)
2. \(F\) 反射单态射、外态射、双态射、常态射、余常态射及范畴 \(\mathscr{B}\) 中图的交换性
3. 若 \(F\) 还是完全的，则 \(F\) 反射 \(\text{section}\)，\(\text{retraction}\) 和同构态射
设 \(\mathscr{C}\) 为范畴且 \(P, Q \in \operatorname{Ob} \mathscr{C}\)，若 \(h_{P}\) 保持外态射，则称 \(P\) 为可投影的；若 \(h^{Q}\) 保持外态射，则称 \(Q\) 为可注入的
1. \(P\) 为可投影的当且仅当对每个外态射 \(f \in[B, C]\) 及任意的 \(g \in[P, C]\)，皆有 \(h \in[P, B]\) 使 \(f \circ h=g\)
2. \(Q\) 为可注入的当且仅当对每个外态射 \(f \in[C, B]\) 及任意的 \(g \in[C, Q]\)，皆有 \(h \in[B, Q]\) 使 \(h \circ f=g\)
设 \(\mathscr{C}\) 为范畴且 \(S, C \in \operatorname{Ob} \mathscr{C}\)，若 \(h_{S}\) 为忠信函子，则称 \(S\) 为 \(\mathscr{C}\) 的分离子；若 \(h^{C}\) 为忠信函子，则称 \(C\) 为 \(\mathscr{C}\) 的余分离子
1. \(S\) 为分离子当且仅当若 \(f, g \in[A, B]\) 使 \(f \neq g\)，则有 \(h \in[S, A]\) 使 \(f \circ h \neq g \circ h\)
2. \(C\) 为余分离子当且仅当若 \(f, g \in[A, B]\) 使 \(f \neq g\)，则有 \(h \in[B, C]\) 使 \(h \circ f \neq h \circ g\)

3.2 自然变换

3.2.1 自然变换的性质

设 \(\mathscr{A}\) 和 \(\mathscr{B}\) 均为范畴，\(F: \mathscr{A} \rightarrow \mathscr{B}\) 和 \(G: \mathscr{A} \rightarrow \mathscr{B}\) 均为协变函子．若函数 \(\alpha: \operatorname{Ob} \mathscr{A} \rightarrow \operatorname{Mor} \mathscr{B}\) 满足
1. 若 \(A \in \operatorname{Ob} \mathscr{A}\)，则 \(\alpha(A) \in \mathscr{B}[F(A), G(A)]\)
2. 若 \(A, A^{\prime} \in \operatorname{Ob} \mathscr{A}\) 且 \(f \in \mathscr{A}\left[A, A^{\prime}\right]\)，则 \(G(f) \circ \alpha(A)=\alpha\left(A^{\prime}\right) \circ F(f)\)
则称 \(\alpha\) 为一个从 \(F\) 到 \(G\) 的自然变换，记为 \(\alpha: F \rightarrow G\) 或 \(F \xrightarrow{\alpha} G\)．若 \(A \in \operatorname{Ob} \mathscr{A}\) 蕴含 \(\alpha(A)\) 为同构态射，则称 \(\alpha\) 为自然同构
1. 若存在自然同构 \(\alpha: F \rightarrow G\)，则称 \(F\) 自然同构于 \(G\) 或 \(F\) 与 \(G\) 自然同构，记为 \(F \cong G\)
2. 若 \(F: \mathscr{A} \rightarrow \mathscr{B}\) 和 \(G: \mathscr{A} \rightarrow \mathscr{B}\) 均为协变函子，则自然变换 \(\alpha: F \rightarrow G\) 为自然同构当且仅当有唯一的自然变换 \(\beta: G \rightarrow F\) 使得
  
  \[ \begin{aligned} & \alpha(A) \circ \beta(A)=\mathbf{I}_{G(A)} \\ & \beta(A) \circ \alpha(A)=\mathbf{I}_{F(A)} \end{aligned} \quad A \in \operatorname{Ob} \mathscr{A} \]
3. 设 \(F: \mathscr{A} \rightarrow \mathscr{B}, G: \mathscr{A} \rightarrow \mathscr{B}\) 和 \(H: \mathscr{A} \rightarrow \mathscr{B}\) 均为协变函子且 \(\alpha: F \rightarrow G\) 和 \(\beta: G \rightarrow H\) 为自然变换．若令
  
  \[ \forall A \in \operatorname{Ob} \mathscr{A}: (\beta \circ \alpha)(A)=\beta(A) \circ \alpha(A) \]
  
  则 \(\beta \circ \alpha: F \rightarrow H\) 为自然变换，称为 \(\alpha\) 与 \(\beta\) 的复合
4. 设 \(\mathscr{A}\) 和 \(\mathscr{B}\) 为范畴，\(F: \mathscr{A} \rightarrow \mathscr{B}\) 和 \(G: \mathscr{A} \rightarrow \mathscr{B}\) 为协变函子且 \(\alpha: F \rightarrow G\) 为自然变换．若自然变换 \(\beta: G \rightarrow F\) 使得
  
  \[ \alpha \circ \beta=\mathbf{I}_{G} \wedge \beta \circ \alpha=\mathbf{I}_{F} \]
  
  则称 \(\beta\) 为 \(\alpha\) 的逆自然变换，简称 \(\alpha\) 的逆．如果有自然变换 \(\beta: G \rightarrow F\) 为 \(\alpha\) 的逆，则称 \(\alpha\) 有逆
设 \(F, G, H, K\) 均为从 \(\mathscr{A}\) 到 \(\mathscr{B}\) 的协变函子，\(\alpha: F \to G, \beta: G \to H, \gamma: H \to K\) 为自然变换
1. \(\gamma \circ(\beta \circ \alpha)=(\gamma \circ \beta) \circ \alpha\)
2. \(\alpha \circ \mathbf{I}_{F}=\alpha=\mathbf{I}_{G} \circ \alpha\)
3. \(\alpha\) 有逆当且仅当 \(\alpha\) 为自然同构
  1. 若 \(\alpha\) 有逆，则 \(\alpha\) 的逆必唯一，常用 \(\alpha^{-1}\) 表示
  2. 若 \(\alpha\) 有逆，则 \(\alpha^{-1}\) 也有逆，且 \(\left(\alpha^{-1}\right)^{-1}=\alpha\)
  3. 若 \(\alpha\) 和 \(\beta\) 都有逆，则 \(\beta \circ \alpha\) 有逆，且 \((\beta \circ \alpha)^{-1}=\alpha^{-1} \circ \beta^{-1}\)
若 \(\mathscr{A}\) 和 \(\mathscr{B}\) 均为小范畴
1. 以从 \(\mathscr{A}\) 到 \(\mathscr{B}\) 的协变函子为对象
2. 以协变函子之间的自然变换为态射
构成的范畴称为从 \(\mathscr{A}\) 到 \(\mathscr{B}\) 的协变函子范畴，记为 \(\mathscr{B}^{\mathscr{A}}\) 或 \(\operatorname{Funct}(\mathscr{A}, \mathscr{B})\)
1. 称 \(\operatorname{Funct}(\mathscr{A}, \mathscr{A})\) 为 \(\mathscr{A}\) 上的自函子范畴
2. 称以全体小范畴之间的协变函子为对象构成一个范畴为小范畴的协变函子范畴，记为 \(\text{Funct}\)
任意协变函子 \(F: \mathscr{A} \rightarrow \mathscr{B}\) 都可诱导出图同态 \(F^{*}: \mathbf{G}(\mathscr{A}) \rightarrow \mathbf{G}(\mathscr{B})\)

\[ \begin{aligned} F^{*}(A)&=F(A) & A \in \operatorname{Ob} \mathscr{A} \\ F^{*}(f)&=F(f) & f \in \operatorname{Mor} \mathscr{A} \end{aligned} \]

从而获得范畴 \(\mathscr{B}\) 中一个图 \(\left\langle\mathbf{G}(\mathscr{A}), F^{*}\right\rangle\)，常用 \(\mathcal{D}_{F}\) 表示
1. 设 \(F: \mathscr{A} \rightarrow \mathscr{B}\) 为协变函子
  1. \(\langle L, \lambda\rangle\) 为 \(\mathcal{D}_{F}\) 的锥当且仅当 \(\lambda:\) Const \(_{L} \rightarrow F\) 为自然变换，称 \(\varprojlim \mathcal{D}_{F}\) 为 \(F\) 的极限，记为 \(\varprojlim F\)
  2. \(\langle\tau, K\rangle\) 为 \(\mathcal{D}_{F}\) 的余锥当且仅当 \(\tau: F \rightarrow\) Const \(_{K}\) 为自然变换，称 \(\varinjlim \mathcal{D}_{F}\) 为 \(F\) 的余极限，记为 \(\varinjlim F\)
2. 设 \(F: \mathscr{A} \rightarrow \mathscr{B}\) 为协变函子
  1. 若 \(\mathscr{B}\) 有积和等子，则 \(\varprojlim F\) 存在
  2. 若 \(\mathscr{B}\) 有余积和余等子，则 \(\varinjlim F\) 存在
3. 设 \(F: \mathscr{A} \rightarrow \mathscr{B}\) 为协变函子且 \(\mathscr{B}\) 为局部小范畴
  1. \(\varprojlim F=\langle L, \lambda\rangle\) 当且仅当对每个 \(B \in \operatorname{Ob} \mathscr{B}\) 皆有 \(\varprojlim \left(h_{B} \circ F\right)=\left\langle h_{B}(L), h_{B} \tau\right\rangle\)
  2. \(\varinjlim F=\langle\tau, K\rangle\) 当且仅当对每个 \(B \in \operatorname{Ob} \mathscr{B}\) 皆有 \(\varinjlim \left(h^{B} \circ F\right)=\left\langle h^{B} \tau, h^{B}(K)\right\rangle\)
4. 设 \(F: \mathscr{A} \rightarrow \mathscr{B}\) 为协变函子，\(\mathscr{B}\) 为局部小范畴且 \(L \in \operatorname{Ob} \mathscr{B}\)
  1. 若 \(\lambda:\) Const \(_{L} \rightarrow F\) 为自然变换，则 \(\varprojlim F=\langle L, \lambda\rangle\) 当且仅当 \(B \in \operatorname{Ob} \mathscr{B}\) 蕴含 \(\varprojlim \left(h_{B} \circ F\right)=\left\langle h_{B}(L), h_{B} \lambda\right\rangle\)
  2. 若 \(\tau: F \rightarrow\) Const \(_{L}\) 为自然变换，则 \(\varinjlim F=\langle\tau, L\rangle\) 当且仅当 \(B \in \operatorname{Ob} \mathscr{B}\) 蕴含 \(\varinjlim \left(h^{B} \circ F\right)=\left\langle h^{B} \tau, h^{B}(L)\right\rangle\)
若 \(F: \mathscr{A} \rightarrow \mathscr{B}\) 为协变函子，则下列条件等价
1. \(F\) 为忠信的、完全的和稠密的
2. 存在协变函子 \(G: \mathscr{B} \rightarrow \mathscr{A}\) 使得 \(F \circ G \cong I_{\mathscr{B}}\) 且 \(G \circ F=\mathbf{I}_{\mathscr{A}}\)
3. 存在协变函子 \(G: \mathscr{B} \rightarrow \mathscr{A}\) 及自然同构 \(\eta: \mathbf{I}_{\mathscr{A}} \rightarrow G \circ F\) 和 \(\varepsilon: F \circ G \rightarrow \mathbf{I}_{\mathscr{B}}\) 使得 \(F \eta=(\varepsilon F)^{-1}\) 且 \(G \varepsilon=(\eta G)^{-1}\)

3.2.2 自然变换的积

设 \(\mathscr{A}, \mathscr{B}, \mathscr{C}\) 均为范畴，协变函子 \(F, G, H, K\) 及自然变换 \(\alpha\) 和 \(\beta\) 如图所示
1. 若定义函数 \(\beta * \alpha: \operatorname{Ob} \mathscr{A} \rightarrow \operatorname{Mor} \mathscr{C}\) 为
  
  \[ \forall A \in \operatorname{Ob} \mathscr{A}: (\beta * \alpha)(A)=\beta(G(A)) \circ H(\alpha(A)) \]
  
  则 \(\beta * \alpha: H \circ F \rightarrow K \circ G\) 为自然变换，称为 \(\alpha\) 与 \(\beta\) 的 \(*-\)积
2. 若 \(A \in \operatorname{Ob} \mathscr{A}\)，则 \((\beta * \alpha)(A)=K(\alpha(A)) \circ \beta(F(A))\)
设范畴、协变函子和自然变换的关系如图所示
1. 结合律：\((\delta * \eta) * \varepsilon=\delta *(\eta * \varepsilon)\)
2. 若对任意 \(B \in \operatorname{Ob} \mathscr{B}, A \in \operatorname{Ob} \mathscr{A}\)，令 \(H \eta(B)=H(\eta(B)), \eta S(A)=\eta(S(A))\)，则 \(H \eta=\mathbf{I}_{H} * \eta\) 且 \(\eta S=\eta * \mathbf{I}_{S}\)
设范畴、协变函子和自然变换的关系如图所示

则 \((\nu \circ \mu) *(\eta \circ \varepsilon)=(\nu * \eta) \circ(\mu * \varepsilon)\)
设范畴、协变函子和自然变换的关系如图所示
1. \((G \circ F) \alpha=G(F \alpha)\)
2. \(\alpha(K \circ L)=(\alpha K) L\)
3. \(\mathbf{I}_{U} K=\mathbf{I}_{U \circ K}\) 且 \(F \mathbf{I}_{U}=\mathbf{I}_{F \circ U}\)
4. \(F(\alpha K)=(F \alpha) K\)
5. \(F(\beta \circ \alpha)=F \beta \circ F \alpha\)
6. \((\beta \circ \alpha) K=\beta K \circ \alpha K\)
7. \(F(\beta \circ \alpha) K=F \beta K \circ F \alpha K\)
8. \(\tau V \circ F \alpha=H \alpha \circ \tau U\)