ex__def__direct_and_inverse_limits.tex

\documentclass{classes/tikzcd}

\begin{document}
  \begin{tikzcd}
    \BbbK \ar[r, "\iota_1^2"] & \BbbK^2 \ar[r, "\iota_2^3"] & \BbbK^3 \ar[r, "\iota_3^4"] & \cdots \ar[r, "\iota_{n-1}^{n-2}"] & \BbbK^{n-1} \ar[r, "\iota_n^{n-1}"] & \BbbK^n \ar[r, "\iota_{n+1}^n"] & \cdots
  \end{tikzcd}

  \begin{tikzcd}
    \BbbK^n \ar[rr, "\iota_n^m"] \ar[dr, "\iota_n^\infty" swap] \ar[ddr, bend right, "\alpha_n" swap] &                                      & \BbbK^m \ar[dl, "\iota_m^\infty"] \ar[ddl, bend left, "\alpha_m"] \\
                                                                                                      & \BbbK_0^\infty \ar[d, dashed, "l_A"] &                                                                   \\
                                                                                                      & A                                    &
  \end{tikzcd}

  \begin{tikzcd}
    \cdots \ar[r, "\pi_n^{n+1}"] & \BbbK^n \ar[r, "\pi_{n-1}^n"] & \BbbK^{n-1} \ar[r, "\pi_{n-2}^{n-1}"] & \cdots \ar[r, "\pi_3^4"] & \BbbK^3 \ar[r, "\pi_2^3"] & \BbbK^2 \ar[r, "\pi_1^2"] & \BbbK
  \end{tikzcd}

  \begin{tikzcd}
                                    & A \ar[d, dashed, "l_A"] \ar[ddl, bend right, "\alpha_n" swap] \ar[ddr, bend left, "\alpha_m"] &         \\
                                    & \BbbK^\infty \ar[dl, "\pi_n^\infty" swap] \ar[dr, "\pi_m^\infty"]                             &         \\
    \BbbK^n \ar[rr, "\pi_m^n" swap] &                                                                                               & \BbbK^m
  \end{tikzcd}

  \begin{tikzcd}
    \cdots \ar[r, "\iota_n^{n+1}"] & A_n \ar[r, "\iota_{n-1}^n"] & A_{n-1} \ar[r, "\iota_{n-1}^n"] & \cdots \ar[r, "\iota_3^4"] & A_3 \ar[r, "\iota_2^3"] & A_2 \ar[r, "\iota_1^2"] & A_1
  \end{tikzcd}
\end{document}