\[ \operatorname{Ob} \mathcal{V}^{A} = \operatorname{Ob} \mathcal{V} . \]

\[ \operatorname{Hom}_{\mathcal{V}^{A}}(X,Y) = A(X \times Y) . \]

\[
\begin{array}{ccc}
\operatorname{Hom}_{\mathcal{V}^{A}}(X,Y) \times \operatorname{Hom}_{\mathcal{V}^{A}}(Y,Z) & \longrightarrow & \operatorname{Hom}_{\mathcal{V}^{A}}(X,Z) \\
(u, v) & \longmapsto & v \circ u
\end{array}
\]

\[ A(X \times Y) \times A(Y \times Z) \longrightarrow A(X \times Z) \]

\[ \boxed{\, v \circ u = p_{31*}\bigl( p_{32}^{*}(v)\, p_{21}^{*}(u) \bigr) . \,} \]

\[ X \xrightarrow{\;u\;} Y \xrightarrow{\;v\;} Z \xrightarrow{\;w\;} T \]

\[ w(vu) = (wv)u \]

\[
w(vu) \stackrel{\text{déf}}{=} \gamma_{*}\bigl( \beta^{*}(w)\, \alpha^{*}(vu) \bigr)
\stackrel{\text{déf de } vu}{=} \gamma_{*}\Bigl( \beta^{*}(w)\, \alpha^{*}\bigl( \nu_{*}( \mu^{*}(v)\, \lambda^{*}(u) ) \bigr) \Bigr)
\]

\[
= \gamma_{*}\Bigl( \beta^{*}(w)\, \nu'_{*}\bigl( \alpha'^{*}( \mu^{*}(v)\, \lambda^{*}(u) ) \bigr) \Bigr)
\]

\[
\alpha'^{*}\mu^{*}(v)\, \alpha'^{*}\lambda^{*}(u) = p_{YZ}^{*}(v)\, p_{X,Y}^{*}(u)
\]

\[
\nu'_{*}\bigl( \nu'^{*}\beta^{*}(w)\, p_{YZ}^{*}(v)\, p_{XY}^{*}(u) \bigr),
\qquad \nu'^{*}\beta^{*}(w) = p_{ZT}^{*}(w)
\]

\[
= p_{XT*}\bigl( p_{ZT}^{*}(w)\, p_{YZ}^{*}(v)\, p_{XY}^{*}(u) \bigr)
\]

\[
(wv)u = c_{*}\bigl( b^{*}(wv)\, a^{*}(u) \bigr)
= c_{*}\Bigl( b^{*}\bigl( n_{*}( m^{*}(w)\, l^{*}(v) ) \bigr)\, a^{*}(u) \Bigr)
\]

\[
n'_{*}\bigl( b'^{*}( m^{*}(w)\, l^{*}(v) ) \bigr),
\qquad b'^{*}\bigl( m^{*}(w)\, l^{*}(v) \bigr) = p_{ZT}^{*}(w)\, p_{YZ}^{*}(v)
\]

\[
= c_{*}\Bigl( n'_{*}\bigl( p_{ZT}^{*}(w)\, p_{YZ}^{*}(v) \bigr)\, a^{*}(u) \Bigr)
\]

\[
= n'_{*}\bigl( p_{ZT}^{*}(w)\, p_{YZ}^{*}(v)\; n'^{*} a^{*}(u) \bigr),
\qquad n'^{*}a^{*}(u) = p_{XY}^{*}(u)
\]

\[
= p_{XT*}\bigl( p_{ZT}^{*}(w)\, p_{YZ}^{*}(v)\, p_{XY}^{*}(u) \bigr)
\]

\[ \Delta_{X} \in \operatorname{Hom}_{\mathcal{V}^{A}}(X,X) = A(X \times X) \]

\[ \delta_{X} \colon X \to X \times X \qquad (\text{diag.}) \]

\[ \Delta_{X} = A(\delta_{X})_{*}(1_{A(X)}) \]

\[ X \xrightarrow{\;\Delta_{X}\;} X \xrightarrow{\;u\;} Y \]

\[ u \circ \Delta_{X} = p_{31*}\bigl( p_{32}^{*}(u)\, p_{21}^{*}(\Delta_{X}) \bigr) \]

\[
p_{21}^{*}(\Delta_{X}) = p_{21}^{*}\bigl( \delta_{X*}(1) \bigr) \stackrel{\text{échange}}{=} q_{*}\bigl( p^{*}(1) \bigr) = q_{*}(1)
\]

\[
p_{32}^{*}(u)\, p_{21}^{*}(\Delta_{X}) = p_{32}^{*}(u)\, q_{*}(1) = q_{*}\bigl( q^{*}( p_{32}^{*}(u) ) \bigr)
\]

\[ = q_{*}(u) \quad \text{car } p_{32}\, q = \mathrm{id}_{X \times Y} \]

\[ u \circ \Delta_{X} = p_{31*}\, q_{*}(u) = u \]

\[
\varphi_{X,Y} \colon \operatorname{Hom}_{\mathcal{V}}(X,Y) \longrightarrow \operatorname{Hom}_{\mathcal{V}^{A}}(X,Y) = A(X \times Y)
\]

\[ \Gamma_{f} = (\mathrm{id}_{X}, f) \colon X \to X \times Y \]

\[ A(\Gamma_{f})_{*} \colon A(X) \to A(X \times Y) \]

\[ \varphi_{X,Y}(f) = A(\Gamma_{f})_{*}(1) \]

\[ \varphi(g)\, \varphi(f) = \varphi(gf) \]

\[
p_{31*}\bigl( p_{32}^{*}(\varphi(g))\, p_{21}^{*}(\varphi(f)) \bigr)
\]

\[
p_{32}^{*}(\varphi(g)) = p_{32}^{*}\bigl( \Gamma_{g*}(1) \bigr) = \lambda_{*}\bigl( \mu^{*}(1) \bigr) = \lambda_{*}(1),
\qquad
p_{21}^{*}(\varphi(f)) = p_{21}^{*}\, \Gamma_{f*}(1) = \lambda'_{*}\bigl( \mu'^{*}(1) \bigr) = \lambda'_{*}(1)
\]

\[
\varphi(g)\, \varphi(f) = p_{31*}\bigl( \lambda_{*}(1)\, \lambda'_{*}(1) \bigr)
\]

\[
\lambda_{*}(1)\, \lambda'_{*}(1) = \lambda_{*}\bigl( \lambda^{*}( \lambda'_{*}(1) ) \bigr) \quad (\text{form.\ proj.})
\]

\[
\lambda^{*} \lambda'_{*}(1) = \Gamma_{f*}\bigl( \Gamma_{gf}^{*}(1) \bigr) = \Gamma_{f*}(1)
\]

\[
= \lambda_{*}\, \Gamma_{f*}(1) = \nu_{*}(1)
\]

\[
\varphi(g)\, \varphi(f) = p_{31*}\bigl( \nu_{*}(1) \bigr) = \Gamma_{gf*}(1) = \varphi(gf) \qquad \text{OK.}
\]

\[ u(x) = p_{2*}\bigl( u\, p_{1}^{*}(x) \bigr) \]

\[
\Delta_{X}(x) = p_{2*}\bigl( \Delta_{X}\, p_{1}^{*}(x) \bigr) = p_{2*}\bigl( \delta_{*}(1)\, p_{1}^{*}(x) \bigr)
\]

\[
\delta_{*}(1)\, p_{1}^{*}(x) = \delta_{*}\bigl( 1 \cdot \delta^{*} p_{1}^{*}(x) \bigr),
\qquad \delta^{*} p_{1}^{*}(x) = x
\]

\[
= p_{2*}\, \delta_{*}(x) = x \qquad \text{OK.}
\]

\[ X \xrightarrow{\;u\;} Y \xrightarrow{\;v\;} Z \]

\[
v(u(x)) = q_{2*}\bigl( v\, q_{1}^{*}(u(x)) \bigr)
= q_{2*}\Bigl( v\, q_{1}^{*}\bigl( p_{2*}( u\, p_{1}^{*}(x) ) \bigr) \Bigr)
\]

\[
q_{1}^{*}\, p_{2*}\bigl( u\, p_{1}^{*}(x) \bigr) = \mu_{*}\bigl( \lambda^{*}( u\, p_{1}^{*}(x) ) \bigr) = \mu_{*}\bigl( \lambda^{*}(u)\, p_{X}^{*}(x) \bigr)
\]

\[
= q_{2*}\Bigl( \mu_{*}\bigl( \mu^{*}(v)\, \lambda^{*}(u)\, p_{X}^{*}(x) \bigr) \Bigr)
\]

\[
= p_{Z*}\bigl( p_{YZ}^{*}(v)\, p_{XY}^{*}(u)\, p_{X}^{*}(x) \bigr)
\]

\[
(v \circ u)(x) = r_{2*}\bigl( (v \circ u)\, r_{1}^{*}(x) \bigr)
= r_{2*}\Bigl( p_{XZ*}\bigl( p_{YZ}^{*}(v)\, p_{XY}^{*}(u) \bigr)\, r_{1}^{*}(x) \Bigr)
\]

\[
p_{XZ*}\bigl( p_{YZ}^{*}(v)\, p_{XY}^{*}(u)\, p_{XZ}^{*}\, r_{1}^{*}(x) \bigr),
\qquad p_{XZ}^{*}\, r_{1}^{*}(x) = p_{X}^{*}(x)
\]

\[
= p_{Z*}\bigl( p_{YZ}^{*}(v)\, p_{XY}^{*}(u)\, p_{X}^{*}(x) \bigr)
\]

\[ f_{*}(x) = \varphi(f)(x) \]

\[
\varphi(f)(x) = p_{2*}\bigl( \varphi(f)\, p_{1}^{*}(x) \bigr) = p_{2*}\bigl( \Gamma_{f*}(1)\, p_{1}^{*}(x) \bigr)
\]

\[
\Gamma_{f*}(1)\, p_{1}^{*}(x) = \Gamma_{f*}\bigl( \Gamma_{f}^{*}\, p_{1}^{*}(x) \bigr),
\qquad \Gamma_{f}^{*}\, p_{1}^{*}(x) = x
\]

\[
= p_{2*}\, \Gamma_{f*}(x) = x \qquad \text{OK.}
\]

\[
\varphi(f)^{*} \in \operatorname{Hom}_{\mathcal{V}^{A}}(Y,X) = A(Y \times X),
\]

\[ s \colon \mathcal{V}^{A\circ} \to \mathcal{V}^{A\circ} \]

\[
u \mapsto {}^{t}u \colon \operatorname{Hom}_{\mathcal{V}^{A}}(X,Y) = A(X \times Y) \to \operatorname{Hom}_{\mathcal{V}^{A}}(Y,X) = A(Y \times X)
\]

\[ \varphi^{*} = s \varphi^{\circ} \colon \mathcal{V}^{\circ} \to \mathcal{V}^{A} \]

\[
\psi \varphi^{*} = \psi(s \varphi^{\circ}) = (\psi s) \varphi^{\circ} \colon \mathcal{V}^{\circ} \to k\text{-mod gr.}
\]

\[ u \in \operatorname{Hom}_{\mathcal{V}^{A}}(X,Y) = A(X \times Y) \]

\[ su \in \operatorname{Hom}_{\mathcal{V}^{A}}(Y,X) = A(Y \times X) \]

\[ \tilde{u} \colon A(X) \to A(Y) \qquad \widetilde{su} \colon A(Y) \to A(X) . \]

\[
k_{Y}\bigl( y\, u_{*}(x) \bigr) = k_{X}\bigl( u^{*}(y)\, x \bigr)
\]

\[
g_{*}\bigl( y\, u_{*}(x) \bigr) = f_{*}\bigl( u^{*}(y)\, x \bigr) .
\]

\[
g_{*}\bigl( y\, u_{*}(x) \bigr) = g_{*}\bigl( y\, p_{2*}( u\, p_{1}^{*}(x) ) \bigr) = g_{*}\, p_{2*}\bigl( p_{2}^{*}(y)\, u\, p_{1}^{*}(x) \bigr) = k\bigl( p_{2}^{*}(y)\, u\, p_{1}^{*}(x) \bigr)
\]

\[
f_{*}\bigl( u^{*}(y)\, x \bigr) = f_{*}\bigl( p_{1*}( u\, p_{2}^{*}(y) )\, x \bigr) = f_{*}\, p_{1*}\bigl( u\, p_{2}^{*}(y)\, p_{1}^{*}(x) \bigr) = k\bigl( u\, p_{2}^{*}(y)\, p_{1}^{*}(x) \bigr)
\]

\[
= k\bigl( p_{2}^{*}(y)\, u\, p_{1}^{*}(x) \bigr) \qquad (\text{mod.\ signe !})
\]

\[
x \otimes y = \delta^{*}(x \boxtimes y) \qquad [x, y \in A(X),\ \delta = \mathrm{diag}_{X} \colon X \to X \times X]
\]

\[ x \boxtimes y = p_{1}^{*}(x)\, p_{2}^{*}(y) . \]

\[ u(x) = p_{2*}\bigl( u\, p_{1}^{*}(x) \bigr) . \]

\[ f_{*}(x) = \gamma_{f}(x) , \]

\[
x_{1}, \dots, x_{n} \longmapsto u(x_{1} \boxtimes \dots \boxtimes x_{n})
\]

\[
\begin{aligned}
\varphi(X_{1}, \dots, X_{n}) &: A(X_{1}) \times \dots \times A(X_{n}) \to A(X) \\
\psi(Y_{1}, \dots, Y_{m}) &: A(Y_{1}) \times \dots \times A(Y_{m}) \to A(Y)
\end{aligned}
\]

\[
\chi(\varphi, \psi) : A(X_{1}, \dots, X_{n}) \times A(Y_{1}) \times \dots \times A(Y_{m}) \to A(Z)
\]

\[
(g \times f)_{*}\bigl(s(x \otimes y)\bigr) = (g \times f)_{*}(y \otimes x)(-1)^{xy} = g_{*}(y) \otimes f_{*}(x)\,(-1)^{xy + fx}
\]

\[
s\bigl((f \times g)_{*}(x \otimes y)\bigr) = s\bigl(f_{*}(x) \otimes g_{*}(y)\bigr)(-1)^{gx} = g_{*}(y) \otimes f_{*}(x)\,(-1)^{gx + (x+f)(y+g)}
\]

\[
(-1)^{xy + fx + gx + gx + fy + xy + fg} \qquad (-1)^{f(x+y+g)} \qquad (-1)^{fy + gx + fg}
\]

\[ \kappa_{A}\bigl(f^{*}(b)\, a\bigr) = \kappa_{B}\bigl(b\, f_{*}(a)\bigr) \]

\[ f_{*}(a)\, b = f_{*}\bigl(a\, f^{*} b\bigr) \]

\[ \mathcal{V} \longrightarrow \text{algèbres bivariantes graduées} \]

\[ f^{*} \otimes g^{*}(a \otimes b) = f^{*}(a) \otimes g^{*}(b) \]

\[ \kappa_{A \otimes B} = \kappa_{A} \otimes \kappa_{B} \]

\[ \varphi(f^{*} \otimes g^{*}) = \varphi(f^{*}) \otimes \varphi(g^{*}) \]

\[ (*) \qquad (f \times g)_{*}(a' \otimes b') = f_{*}(a') \otimes g_{*}(b')\,(-1)^{g a'} \]

\[ (-1)^{ag + bf + Bf} \]

\[ (-1)^{a'g + bf} \]

\[ (-1)^{bf} = (-1)^{B'f} \]

\[ (-1)^{bf + (AB + A'B')} \]

\[
\boxed{\ (f \times g)_{*} = \bigl(f_{*} \overset{\mathrm{ord}}{\otimes} g_{*}\bigr)\bigl(\mathrm{id}_{A} \otimes [(-1)^{B}]_{B'}^{\deg f}\bigr)\ }
\]

\[
(g \times f)_{*}\bigl(s(x \otimes y)\bigr) = s\bigl((f \times g)_{*}(x \otimes y)\bigr)
\]

\[
\begin{aligned}
s(x \otimes y) &= y \otimes x\,(-1)^{xy}, & (f \times g)_{*}(x \otimes y) &= f_{*}(x) \otimes g_{*}(y), \\
g_{*}(y) \otimes f_{*}(x)\,(-1)^{xy} &\ \Vert & g_{*}(y) \otimes f_{*}(x)\,&(-1)^{(x+f)(y+g)} .
\end{aligned}
\]

\[
\left\{
\begin{aligned}
p^{*}\bigl(f_{*}(x)\bigr) &= f'_{*}\bigl(q^{*}(x)\bigr) \\
f^{*}\bigl(p_{*}(m)\bigr) &= q_{*}\bigl(f'^{*}(m)\bigr)
\end{aligned}
\right.
\]

\[ f_{*}(x) \otimes 1 = (f_{*} \otimes 1)(x \otimes 1) \qquad \text{OK} \]

\[
f^{*}\bigl((\mathrm{id}_{A} \otimes \kappa_{R})(a \otimes r)\bigr) \overset{?}{=} (\mathrm{id}_{B} \otimes \kappa_{R})(f^{*} \otimes \mathrm{id}_{R})(a \otimes r)
\]

\[
\kappa(r)\, a \ \mapsto\ \kappa(r)\, f^{*}(a) \qquad\qquad f^{*}(a)\, r \ \mapsto\ \kappa(r)\, f^{*}(a) \qquad \text{O.K.}
\]

\[
X \xrightarrow{\ f\ } Y \xrightarrow{\ g\ } Z \qquad\qquad A \xleftarrow{\ f^{*}\ } B \xleftarrow{\ g^{*}\ } C
\]

\[ \therefore\ \lambda'^{*}\bigl(\lambda_{*}(u)\bigr) = (\Gamma_{f})_{*}\bigl(\Gamma_{gf}^{*}(u)\bigr) \]

\[
\bigl[\, f_{*}\bigl(x f^{*}(y)\bigr) = f_{*}(x)\, y \,\bigr], \quad f_{*}\bigl(f^{*}(y)\, x\bigr) = y\, f_{*}(x)
\]

\[ \kappa_{A} : A \to k \]

\[ (x, y) \longmapsto \kappa(xy) \]

\[
\varphi_{A} : A \to \check{A} \qquad \bigl(\check{A} = \mathrm{Hom}_{k\text{-mod}}(A, k)\bigr), \qquad \langle y, \varphi_{A}(x) \rangle = \kappa(yx) .
\]

\[ (***) \qquad \kappa\bigl(y\, f_{*}(x)\bigr) = \kappa\bigl(f^{*}(y)\, x\bigr) \]

\[ A \to A^{d} \xrightarrow{\ \bar{\kappa}_{A}^{(d)}\ } k , \]

\[ d - i = d' - i' \qquad \text{i.e.} \qquad i' = i + (d' - d) . \]

\[ A(I') \xrightarrow{\ \alpha^{*}\ } A(I), \qquad A(I'') \xrightarrow{\ \beta^{*}\ } A(I) \]

\[ A(I') \otimes A(I'') \xrightarrow{\ \sim\ } A(I' \sqcup I'') . \]

\[
  A(I' \sqcup I'') \xleftarrow[\ \sim\ ]{(\alpha^{*},\,\beta^{*})} A(I') \otimes A(I'') ,
\]

\[
  x \longmapsto C(x) = C.x \quad : \quad A(I) \to A(I')
\]

\[
  A(I) \xrightarrow{\ \alpha^{*}\ } A(I \sqcup I') \xrightarrow{\ y \mapsto Cy\ }
  A(I \sqcup I') \xrightarrow{\ \alpha'_{*}\ } A(I') .
\]

\[
  A(I'', I') \times A(I', I) \longrightarrow A(I'', I)
\]

\[
  v \circ u = p_{31*}\bigl(p_{21}^{*}(u)\, p_{32}^{*}(v)\bigr) .
\]

\[
  \widetilde{v \circ u} = \widetilde{v} \circ \widetilde{u} .
\]

\begin{align*}
  (\widetilde{v \circ u})(x)
  &\overset{\text{déf.\ de }\sim}{=} \struck{\ill{}}\; \beta''_{*}\bigl[(v \circ u)\, \beta^{*}(x)\bigr] \\
  &\overset{\text{déf.\ de }v \circ u}{=} \beta''_{*}\bigl[\underbrace{p_{31*}\bigl(p_{21}^{*}(u)\, p_{32}^{*}(v)\bigr) \beta^{*}(x)}_{=\ \text{formule de proj.}}\bigr] \\
  &= \beta''_{*}\, p_{31*}\bigl(p_{21}^{*}(u)\, p_{32}^{*}(v)\, \underbrace{p_{31}^{*} \beta^{*}(x)}_{p_{1}^{*}(x)}\bigr) \\
  &\overset{\text{transitivité}}{=} p_{3*}\bigl(p_{21}^{*}(u)\, p_{32}^{*}(v)\, p_{1}^{*}(x)\bigr) \\
  &\struck{\overset{\text{projection}}{=} p_{3*}\bigl(p_{21}^{*}(u)\, p_{32}^{*}(v)\bigr) x}
\end{align*}

\begin{align*}
  \widetilde{v}\bigl(\widetilde{u}(x)\bigr)
  &\overset{\text{déf.\ }\widetilde{v}}{=} \struck{\ill{}}\; \alpha''_{*}\bigl(v\, \alpha'^{*}(\widetilde{u}(x))\bigr) \\
  &\overset{\text{déf.\ }\widetilde{u}}{=} \alpha''_{*}\bigl(v\, \underbrace{\alpha'^{*}\bigl(\gamma'_{*}(u\, \gamma^{*}(x))\bigr)}_{\text{échange}}\bigr)
\end{align*}

\[
  \alpha'^{*}\gamma'_{*}\bigl(u\,\gamma^{*}(x)\bigr) = \struck{\ill{}}\;
  p_{32*}\bigl(p_{21}^{*}(u\, \gamma^{*}(x))\bigr)
  = p_{32*}\bigl(p_{21}^{*}(u)\, \underbrace{p_{21}^{*} \gamma^{*}(x)}_{p_{1}^{*}(x)}\bigr)
\]

\begin{align*}
  &= \alpha''_{*}\bigl(v\, \underbrace{p_{32*}\bigl(p_{21}^{*}(u)\, p_{1}^{*}(x)\bigr)}_{\text{formule de projection, mod signes}}\bigr) \\
  &= \underbrace{\alpha''_{*}\, p_{32*}}\bigl(p_{32}^{*}(v)\, p_{21}^{*}(u)\, p_{1}^{*}(x)\bigr) \\
  &= p_{3*}\bigl(p_{32}^{*}(v)\, p_{21}^{*}(u)\, p_{1}^{*}(x)\bigr)
\end{align*}

\[
  \Delta_{I} \in A(I \sqcup I)
\]

\[
  \widetilde{\delta}_{I}(x) = x \quad (x \in A(I)) \qquad \text{i.e.}\quad
  \widetilde{\delta}_{I} = \mathrm{id}_{A(I)} .
\]

\[
  \Delta_{I} = \delta_{*}(1) .
\]

\begin{align*}
  \widetilde{\Delta}(x)
  &\overset{\text{déf.\ de }\sim}{=} p_{2*}\bigl(\Delta\, p_{1}^{*}(x)\bigr) \\
  &= p_{2*}\bigl(\underbrace{\delta_{*}(1)\, p_{1}^{*}(x)}_{\delta_{*}(\delta^{*} p_{1}^{*}(x)) \,=\, \delta_{*}(x)}\bigr) \\
  &= p_{2*}\, \delta_{*}(x) = x
\end{align*}

\[
  \left\{
  \begin{array}{l}
    y_{1}^{2} = y_{2}^{2} = y_{3}^{2} = 0 ,\\
    \delta_{12}^{2} = \chi\, y_{1} y_{2} ,\ \delta_{23}^{2} = \chi\, y_{2} y_{3} ,\quad
    \delta_{31}^{2} = \chi\, y_{3} y_{1} ,\quad
    \delta_{12} \delta_{23} = \delta_{12} \delta_{31} = \delta_{23} \delta_{31} = \delta \\
    \delta_{12} y_{1} = \delta_{12} y_{2} = y_{1} y_{2} ,\quad
    \delta_{23} y_{2} = \delta_{23} y_{3} = y_{2} y_{3} ,\quad
    \delta_{31} y_{3} = \delta_{31} y_{1} = y_{3} y_{1} ,\\
    \delta y_{1} = \delta y_{2} = \delta y_{3} = y_{1} y_{2} y_{3} \\
    \delta^{2} = 0 \\
    \delta_{12} \delta = \delta_{23} \delta = \delta_{31} \delta = \chi\, y_{1} y_{2} y_{3}
  \end{array}
  \right.
  \]

\[
  X^{k-i} \longrightarrow X^{k}
  \qquad
  \xleftarrow{\ \ill{}\ } [1, k'] \subset [1, k]
\]

\[
  X^{I}
\]

\[
  (R, J) \quad
  \left\{
  \begin{array}{l}
    R \subset \mathfrak{P}(I) \text{ tel que } \struck{\ill{}}
    \left\{
    \begin{array}{l}
      \text{a)}\ \bigcup_{A \in R} A = I \\
      \text{b)}\ A, A' \in R,\ A \neq A' \Rightarrow A \cap A' = \emptyset \\
      \text{c)}\ \emptyset \in R
    \end{array}
    \right. \\[1ex]
    J \in R
  \end{array}
  \right.
\]

\[
  \varepsilon_{R,J}\, \varepsilon_{R',J'} =
  \left\{
  \begin{array}{l}
    0 \quad \text{si } J \cap J' \neq \emptyset \\
    \phantom{0}
  \end{array}
  \right.
\]

\[
  Z_{R,J} = \left\{ (x_{i})_{i \in I} \text{ tel que }
  \left\{
  \begin{array}{ll}
    x_{i} = a & \text{si } i \in J \\
    x_{i} = x_{j} & \text{si } \exists A \in R, \text{ t.q. } i, j \in A
  \end{array}
  \right.
  \right\}
\]

\[
  Z^{a}_{R,J} \cap Z^{a'}_{R',J'} \underset{\text{du pt de vue ens.}}{=} Z_{R'',J''}
\]

\[
  \struck{\widetilde{J} \neq \widetilde{J}'}
\]

\[
  \left|
  \begin{array}{l}
    \widetilde{J} = \widetilde{J}' \neq \emptyset \Rightarrow
    \varepsilon_{R,J}\, \varepsilon_{R',J'} = 0 \\[1ex]
    \widetilde{J} \neq \widetilde{J}' \text{ ou } \widetilde{J} = \widetilde{J}' = \emptyset
    \quad \text{alors} \quad
    \varepsilon_{R,J}\, \varepsilon_{R',J'} =
    \Bigl(\prod_{i \in \widetilde{J} \cup \widetilde{J}'} y_{i}\Bigr)
    \prod_{\substack{A'' \in R'' \\ A'' \neq \widetilde{J} \\ A'' \neq \widetilde{J}'}}
    \varepsilon_{R_{A''}}\, \varepsilon_{R'_{A''}}
  \end{array}
  \right.
\]

\[
  1 \leq \operatorname{card} R = s \leq k \quad \text{donc} \quad
  \dim Z_{R} = sn \qquad \operatorname{codim} Z_{R} = kn - sn = (k-s)n
\]

\[
  \operatorname{card} R' = s' \qquad \operatorname{codim} Z_{R'} = (k-s')n
\]

\[
  \operatorname{card} R'' = 1 \qquad \operatorname{codim} Z_{R''} = (k-1)n
\]

\[
  (k-1)n \leq (k-s)n + (k-s')n \quad \text{i.e.} \quad \struck{\ill{}}
\]

\[
  \boxed{s + s' \leq k+1}
\]

\[
  X^{p} \to X^{q} \qquad X^{p} \times X \to X^{q} \times X \qquad
  \Delta_{p} \leftarrow \Delta_{q} \qquad
  \Delta_{p} \sqcup 1 \leftarrow \Delta_{q} \sqcup 1
\]

\[
  \boxed{\text{engendrée, via } \boxtimes, \text{ par } \struck{\ill{}}\ y, \text{ et les } \delta_{I}}
\]

\[
  (\delta_{2} \boxtimes \delta_{2})
\]