A,B\mathscr{A},\mathscr{B}2F,GF,G
Gyazo


F,GF,GA\mathscr{A}AAB\mathscr{B}F(A),G(A)F(A),G(A)
Gyazo



Gyazo


Gyazo


Gyazo
αA2F(f)=G(f)αA1\alpha_{A_2}\circ F(f)= G(f)\circ \alpha_{A_1}
A\mathscr{A}2f:A1A2f:A_1\rightarrow A_2F(A1)G(A2)F(A_1)\rightarrow G(A_2)



B\mathscr{B}{F(A)αAG(A)}Aob(A)\{F(A)\xrightarrow{\alpha_A} G(A)\}_{A\in\mathscr{\mathrm{ob}(A)}}
α:FG\alpha:F\Rightarrow G


ob(A)\mathrm{ob}(\mathscr{A})
Gyazo


I={1,2,n}I=\{1,2,\dots n\}
{Oi}iI\{O_i\}_{i\in I}O1,O2,O3,,OnO_1, O_2,O_3,\dots,O_n
I=[0,1]I=[0,1]
{Ai}iI\{A_i\}_{i\in I}A0,A0.5,A0.33..,A12,A_0,A_{0.5},A_{0.33..},A_{\frac{1}{\sqrt{2}}},\cdots


ob(A)\mathrm{ob}(\mathscr{A})A\mathscr{A}
AA{F(A)αAG(A)}Aob(A)\{F(A)\xrightarrow{\alpha_A} G(A)\}_{A\in\mathscr{\mathrm{ob}(A)}}
A\mathscr{A}nnA1,A2,,AnA_1,A_2,\cdots,A_n
{F(A1)αA1G(A1),F(A2)αA2G(A2),,F(An)αAnG(An)}\{F(A_1)\xrightarrow{\alpha_{A_1}} G(A_1),F(A_2)\xrightarrow{\alpha_{A_2}} G(A_2),\cdots,F(A_n)\xrightarrow{\alpha_{A_n}} G(A_n)\}
mrsekut
Aob(A)A\in\mathrm{ob}(\mathscr{A})α\alpha



α:FG\alpha: F\rightarrow G
B\mathscr{B}{F(A)αAG(A)}AA\{F(A)\xrightarrow{\alpha_A} G(A)\}_{A\in\mathscr{A}}A\mathscr{A}AfAA\xrightarrow{f}A'
Gyazo
αBF(f)=G(f)αA\alpha_{B}\circ F(f)= G(f)\circ \alpha_{A}



ff
f:AAf:A\rightarrow A
f=1Af=1_A
F,GF,G
F(f):F(A)F(A)F(f):F(A)\rightarrow F(A)
G(f):G(A)G(A)G(f): G(A)\rightarrow G(A)
F(f)=1F(A),G(f)=1G(A)F(f)=1_{F(A)},G(f)=1_{G(A)}
Gyazo
2
Gyazo
1G(A)αA=1F(A)αA1_{G(A)}\circ \alpha_A=1_{F(A)}\circ\alpha_AαA=αA\alpha_A=\alpha_A
A\mathscr{A}2f:A1A2f:A_1\rightarrow A_2F(A1)G(A2)F(A_1)\rightarrow G(A_2)
αA\alpha_A



A\mathscr{A}F,G:ABF,G:\mathscr{A}\rightarrow\mathscr{B}
B\mathscr{B}(F(A))AA(F(A))_{A\in\mathscr{A}}
B\mathscr{B}(F(A)αAG(A))AA(F(A)\xrightarrow{\alpha_A}G(A))_{A\in\mathscr{A}}