\[
f \longmapsto \struck{\ill{}} \quad \struck{multibundle} \quad \text{MM-fibré} \quad M_1
\]

\[
M \longrightarrow M_1
\]

\[
M(\eta_0, \xi_{n-1}) \longrightarrow M(\xi'_0, \xi_n)
\]

\[
M\bigl(\xi'_0, \ill{}\, \eta_{n-1}, \xi'_{n-1}\bigr) \qquad M(\eta_0, \eta_{n-1})
\]

\[
M(\eta_0, \eta_1, \dots, \eta_{n-1})
\]

\[
M\bigl(\xi_0, \xi'_0, \xi_1 \cdots \eta_{n-1}, \xi_{n-1}, \xi_n\bigr)
\;\simeq\; M(\xi_0, \xi_1, \dots, \xi_n)
\]

\[
M\bigl(\xi'_0, \xi_1, \xi_2 \cdots \xi_{n-1}, \xi'_n\bigr)
\longrightarrow \prod M(\eta_i, \xi_{i+1}, \eta_{i+1})
\qquad \text{si } i \leqslant n-2
\]

\[
M\bigl(\xi'_0, \eta_0, \eta_1, \dots, \eta_{n-1}, \xi'_n\bigr)
\longrightarrow \prod M(\eta_i, \eta_{i+1})
\qquad \text{à } i \leqslant n-2
\]

\[
M(\eta_0, \eta_{n-1}) \longleftarrow M(\eta_0, \eta_1, \dots, \eta_{n-1})
\qquad \text{subm } (\uncertain{mieux})
\]

\[
P^{k} \;\underset{\text{déf}}{=}\; \operatorname{Hom}_{\mathbb{Z}}(P^{\circ}, Ab)
\;\simeq\; \operatorname{Hom}_k(P^{\circ}, Ab_k)
\]

\[
\operatorname{Kar} P \longrightarrow P^{k} \quad \text{qui induit équivalence}
\]

\[
\operatorname{Kar} P \;\approx\; \uncertain{\mathrm{Ul}}\operatorname{Proj}(P^{k})
\]

\[
\operatorname{Hom}_{k!}(P^{k}, M) \overset{\sim}{\longrightarrow}
\operatorname{Hom}_k(P, M)
\qquad \text{si $M$ $k$-additive \uncertain{stable} par $\varinjlim$}
\]

\[
\operatorname{Hom}_{k!}(P^{k\circ}, M) \overset{\sim}{\longrightarrow}
\operatorname{Hom}_k(P^{\circ}, M)
\qquad \text{si $M$ $k$-additive stable par $\varprojlim$}
\]

\[
\operatorname{Hom}_{k!}(Q^{k\circ}, M) \simeq \operatorname{Hom}_k(P, M)
\]

\[
\underbrace{\operatorname{Hom}^{!}_{k}(Q^{k\circ}, M)}_{\textstyle \mathcal{Q}_M}
\;\simeq\;
\underbrace{\operatorname{Hom}_{k!}(P^{k}, M)}_{\textstyle \mathcal{P}_M}
\qquad \text{si $M$ $k$-add.\ stable pour les deux types de lim}
\]

\[
\mathcal{P}^{k}_{M} \times Q^{k} \longrightarrow M,
\qquad
\mathcal{P}_M \times Q \longrightarrow M,
\qquad
P \times \mathcal{P}^{\dagger}_{M} \longrightarrow M,
\qquad
P^{k} \times P^{k}_{M}
\]

\[
(F *_k L')^{\circ} \;\simeq\; \struck{\ill{}}\ \operatorname{Hom}_k(L', F^{\circ}),
\qquad
\bigl(\operatorname{Hom}_k(L, F)\bigr)^{\circ} \;\simeq\; F^{\circ} * L
\]

\[
\operatorname{Hom}_k(P \otimes_k Q, M) \;\simeq\; \operatorname{Bil}_k(P, Q; M)
\qquad \text{tout $M$ $k$-additive}
\]

\[
\zeta = \bigoplus_{i \in I} a_i \otimes_k b_i ,
\qquad \ill{},\ \text{pour } \zeta' = \bigoplus_{j \in J} a'_j \otimes_k b'_j ,
\]

\[
\operatorname{Hom}(\zeta, \zeta') = \prod_{I \times J}
\operatorname{Hom}(a_i, a'_j) \otimes_k \operatorname{Hom}(b_i, b'_j)
\]

\[
P \otimes_k Q \;\hookrightarrow\; (P \otimes_k Q)^{k} \;\simeq\;
\struck{\operatorname{Bil}}\ \operatorname{Bil}_k(P^{\circ}, Q^{\circ}; (Ab_k))
\]

\[
(*) \qquad a \otimes_k b \;\longmapsto\;
\bigl((x, y) \mapsto \operatorname{Hom}_P(x, a) \otimes_k
\operatorname{Hom}_Q(y, b)\bigr)
\]

\[
\begin{aligned}
P^{k}_{M} &= \operatorname{Hom}_k\bigl(P^{\circ},
   \operatorname{Hom}_k(Q^{\circ}, Ab_k)\bigr) \\
 &\simeq \operatorname{Hom}_k(P^{\circ} \otimes_k Q^{\circ}, Ab_k) \\
 &\simeq \operatorname{Hom}_k\bigl((P \otimes_k Q)^{\circ}, Ab_k\bigr)
   = (P \otimes_k Q)^{k}
\end{aligned}
\]

\[
\begin{aligned}
P^{\circ k}_{M^{\vee}} &\simeq \operatorname{Hom}_k\bigl(
   (P^{\circ} \otimes_k Q^{\circ})^{\circ}, Ab_k\bigr) \\
 &\simeq \operatorname{Hom}_k(P \otimes_k Q, Ab_k) = (P \otimes_k Q)^{\circ k}
\end{aligned}
\]

\[
\operatorname{Bil}_{k!!}(P^{k}, Q^{k}; M) \;\simeq\;
\struck{\operatorname{Hom}}\ \operatorname{Hom}_{k!}\bigl((P \otimes_k Q)^{k}, M\bigr)
\]

\[
\bigl(\simeq \operatorname{Bil}_k(P, Q; M) \text{ si $M$ $k$-add.\ avec }
\varinjlim\bigr)
\]

\[
\begin{aligned}
\operatorname{Bil}_{k!!}(P^{k}, Q^{k}; M)
 &\simeq \operatorname{Hom}_{k!}\bigl(P^{k},
    \underbrace{\operatorname{Hom}_{k!}(Q^{k}, M)}_{\textstyle
    \operatorname{Hom}_k(Q, M)}\bigr) \\
 &\simeq \operatorname{Hom}_k\bigl(P, \operatorname{Hom}_k(Q, M)\bigr) \\
 &\simeq \operatorname{Hom}_k(P \otimes_k Q, M)
    \overset{\sim}{\longleftarrow}
    \operatorname{Hom}_{k!}\bigl((P \otimes_k Q)^{k}, M\bigr)
\end{aligned}
\]

\[
\operatorname{Bil}_{k!!}(P^{k\circ}, Q^{k\circ}; M) \;\simeq\;
\struck{\operatorname{Hom}}\ \operatorname{Hom}_{k!}\bigl(
(P \otimes_k Q)^{k\circ}, M\bigr)
\]

\[
(\circledast) \qquad
P^{k} \otimes_k M \longrightarrow \operatorname{Hom}_k(P^{\circ}, M) = P^{k}_{M}
\]

\[
L \otimes_k x \;\longmapsto\; \bigl(a \mapsto L(a) \otimes_k x\bigr)
= \widetilde{x} \circ L
\]

\[
P^{\circ} \overset{L}{\longrightarrow} Ab_k
\overset{\widetilde{x}}{\longrightarrow} M ,
\qquad \operatorname{Hom}_{k!}(Ab_k, M) \simeq M
\]

\[
P^{k} \otimes_k M \dashrightarrow \operatorname{Bil}_k(P^{k} \times M^{\circ}, Ab_k)
\qquad \text{pl.\ fid.\,?}
\]

\[
P^{k} \otimes_k Q^{k} \longrightarrow (P \otimes_k Q)^{k}
\simeq \operatorname{Bil}_k(P, Q; Ab_k)
\qquad \text{pl.\ fid.}
\]

\[
F \otimes_k G \;\longmapsto\; \bigl((a,b) \mapsto F(a) \otimes_k G(b)\bigr)
\]

\[
(L \otimes_k x) *_k L' \;\simeq\;
\underbrace{(L *_k L')}_{\textstyle \in Ab_k} \otimes_k x
\]

\[
\operatorname{Hom}_{P^{k}_{M}}(L \otimes_k x, F)
\;\simeq\; \operatorname{Hom}_M\bigl(x, \operatorname{Hom}_k(L, F)\bigr)
\;\simeq\; \operatorname{Hom}_{P^{k}}\bigl(L, \operatorname{Hom}_M(x, F)\bigr)
\]

\[
\operatorname{Hom}_M(x, F) : \quad a \longmapsto
\operatorname{Hom}_M\bigl(x, F(a)\bigr)
\]

\[
    Ab_k \longrightarrow N, \qquad M \longmapsto M \otimes_k a
    \]

\[
    Ab_k \longrightarrow N, \qquad M \longmapsto M \otimes_k a
    \]

\[
    Q^{k} \longrightarrow N, \qquad L' \longmapsto \Phi *_k L'
    \]