\[
\pi_0(\Gamma \times C) = \pi_0(\Gamma) \times \pi_0(C)
= \Gamma \times \pi_0(C) \xrightarrow{\ p\ } \pi_0(C)
\]

\[
\pi_1(\Gamma \times C) = \pi_1(\Gamma) \times \pi_1(C) = \pi_1(C),
\qquad
k_{\Gamma \times C} = p^{*}(k_C).
\]

\[
\varphi : \pi_0(C) \times \pi_0(C) \longrightarrow \pi_1(C).
\]

\[
\varphi(e_i, e_j) =
\begin{cases}
0 & \text{si } i \neq j \\
n_i & \text{si } i = j
\end{cases}
\]

\[
\overline{\varphi} : \pi_0(\Gamma \times C) \times \pi_0(\Gamma \times C)
\longrightarrow \pi_1(\Gamma \times C)
\]

\[
\pi_0(\Gamma \times^{\varphi} C) \approx \pi_0(\Gamma \times C),
\qquad
\pi_1(\Gamma \times^{\varphi} C) \cong \pi_1(C).
\]

\[
\Lambda(u) \otimes \Lambda(w) \simeq \Lambda(v) ;
\]

\[
\widetilde{P}(X) = \mathrm{Rep}(\mathcal{X} ; P, \alpha_X)
= \mathrm{Rep}(X, P, \alpha_X)
\]

\[
\varphi : G \longrightarrow \mathrm{Autext}(F)
\]

\[
\pi_0(P(F)) \in \mathrm{Autext}\,F ; \qquad \pi_1(P(F)) = Z(F),
\ \text{centre de } F.
\]

\[
0 \longrightarrow \mathrm{Ext}^2(M,N) \longrightarrow \pi_0(\mathcal{P})
\longrightarrow \mathrm{Hom}(M, {}_2N) \longrightarrow 0
\]

\[
\mathrm{Hom}\bigl((L,L'), (U,U')\bigr)
= \bigl[\mathrm{Hom}(L,U) \times \mathrm{Hom}(L',U')\bigr] / N
\quad \text{(diagonale)}
\]

\[
\mathrm{Ob}\,C = G, \qquad
\mathrm{Fl}\,C = G \times M \rightrightarrows G, \qquad
s = b = \mathrm{pr}_1,
\]

\[
(g,m) \cdot (g,m') = (g, m+m'), \qquad
g \otimes g' = gg',
\]

\[
(g,m) \otimes (g',m') = (gg', m+m'), \qquad
a_{g,g',g''} = \mathrm{id}_{gg'g''}.
\]

\[
\alpha'(x) = \alpha(x)\,\varphi(x),
\]

\[
\mathrm{Hom}(\alpha, \alpha') = \{\, m \in M \mid \alpha'(x) =
\alpha(x)(gm - m) \,\}
\]

\[
L(x,g) \otimes L(gx, g') \simeq L(x, g'g).
\]

\[
P_u \otimes P_v \otimes P_w \longrightarrow P_{uvw}
\]

\[
P \otimes Q \to S \otimes R^{-1}, \qquad
Q \otimes R \to P^{-1} \otimes S,
\]

\[
R \otimes S^{-1} \to Q^{-1} \otimes P^{-1}, \qquad
S^{-1} \otimes P \to R^{-1} \otimes Q^{-1}.
\]

\[
\mathrm{Fl}_3\,F, \qquad
M_{\Gamma_2} \to \mathrm{Fl}_2(\Gamma) = \mathrm{Fl}\,\Gamma
\times_{\mathrm{Ob}} \mathrm{Fl}\,\Gamma, \qquad
M_{\Gamma_1} \to \mathrm{Fl}_1(\Gamma) = \mathrm{Fl}\,F, \qquad
M_{\Gamma_0} \to \mathrm{Fl}_0(\Gamma) = \mathrm{Ob}\,\Gamma
\]

\[
H^2(\Gamma, M), \qquad H^1(\Gamma, M), \qquad H^0(\Gamma, M),
\]

\[
L_{.} \otimes L_{.} \longrightarrow M, \qquad
L_{.} \otimes L_{.} \otimes M^{-1} \simeq\ \ill{}, \qquad
L_{.} \otimes \cdots \otimes L_{.} \otimes S^{-1} \simeq 1.
\]

\[
P \otimes Q \to R, \qquad
R \otimes Q^{-1} \to P, \qquad
P^{-1} \otimes R \to Q.
\]

\[
L_u L_v \sim L_{uv}, \qquad
L_{uv} L_w \sim L_{uvw}, \qquad
L_u L_v L_w \sim L_{uvw} ;
\]

\[
L_v L_w \sim L_{vw}, \qquad
L_u L_{vw} \sim L_{uvw}, \qquad
L_u L_v L_w \sim L_{uvw}.
\]

\[
\bigl(\cdots((P_{u_1} \otimes P_{u_2}) \otimes P_{u_3}) \cdots\bigr)
\longrightarrow P_{u_1 u_2 \cdots u_n},
\]

\[
\mathrm{Fl}_1(\Gamma) \longrightarrow \pi_0(\widetilde{P}), \qquad
\mathrm{Fl}_2(\Gamma) \longrightarrow \ill{}
\]

\[
\mathrm{Fl}(\Gamma) \xrightarrow{\ \alpha\ } \pi_0(P) \longrightarrow
\mathrm{Aut}(\pi_1(P)),
\]

\[
\lambda(u) : P_u \longrightarrow P'_u
\]

\[
f(uv) = f(u).(\alpha(u).f(v)),
\]

\[
\mu_u = \lambda_u \otimes (m(y) - \alpha(u).m(x)) \qquad (?)
\]

\[
(\alpha) \qquad \mathrm{Hom}.(A,B)(e) \simeq \mathrm{Hom}(A,B)
\]

\[
(\beta) \qquad \varphi_{e} = \mathrm{id}_{\hat{A}}.
\]

\[
(a) \qquad \mathrm{Hom}.(A,B) \times \mathrm{Hom}.(B,C) \longrightarrow
\mathrm{Hom}.(A,C)
\]

\[
(b) \qquad \varphi_{K \times L} \Longleftarrow \varphi_{K} \circ \varphi_{L}
\]

\[
\mathrm{Hom}(A, f_{*}(B)) \simeq \mathrm{Hom}(f^{*}(A), B)
\]

\[
\Gamma(Y, A) \simeq \Gamma(X, f^{*}(A)).
\]

\[
X_{00}, \qquad
X_{20} \times X_{02} \longleftarrow Q \longleftarrow X_{22},
\]

\[
Q \longleftarrow P \longleftarrow P \times \Omega Q \longleftarrow
\Omega Q \longleftarrow \Omega P \longleftarrow \Omega P \times \Omega^{2} Q,
\]

\[
X_{20} \times X_{11} \times X_{02} \longleftarrow Q \longrightarrow\ ?,
\qquad
X_{00} \longleftarrow X_{11} \longleftarrow Q.
\]

\[
X_{20} \longleftarrow Q \longleftarrow \uncertain{X_{12}} \longleftarrow
\Omega X_{20},
\]

\[
X_{20} \times X_{02} \longleftarrow Q \longleftarrow X_{22} \longleftarrow
\uncertain{\Omega X_{10} \times \Omega X_{02}},
\]

\[
X_{02} \longleftarrow Q \longleftarrow X_{21} \longleftarrow \ill{}
\]

\[
e_{(L \otimes M)} = \{\, x \otimes y \mid x \in e_{L},\ y \in e_{M} \,\} ;
\]

\[
\Omega\Gamma_{f} \to \Omega\Gamma_{gf} \to \Omega A \to e, \qquad
\Omega\Gamma_{g} \to \Omega D \to \Gamma(f) \to e,
\]

\[
\Omega C \to \Gamma(gf) \to \Gamma(g) \to e, \qquad
\Omega D \to \Gamma(hgf) \to \Gamma(hg) \to \Gamma(h) \to e,
\]

\[
e \leftarrow \Sigma\Gamma_{f} \leftarrow \Sigma\Gamma_{gf} \leftarrow
\Sigma\Gamma_{hgf} \to \Sigma A, \qquad
\Sigma\Gamma_{g} \to \Sigma\Gamma_{hg} \to \Sigma B \to
\Sigma^{2}\Gamma_{f},
\]

\[
\Sigma\Gamma_{h} \to \Sigma C \to \Sigma^{2}\Gamma_{gf}, \qquad \Sigma D.
\]

\[
\Omega S \to F \to X \to S
\]

\[
\mathrm{Hom}(\varphi, \psi) \in \hat{C}, \qquad
\mathrm{Hom}(\varphi, \psi)(S) =
\mathrm{Hom}(\uncertain{i_{S} \circ \varphi},\ \uncertain{i_{S} \circ \psi}).
\]

\[
\mathrm{Aut}(1_{C}) \rightrightarrows \mathrm{Aut}(X)
\]

\[
\phi_{a,b} : L_{a} \otimes L_{b} \simeq L_{ab},
\]

\[
(L_{a} \otimes L_{b}) \otimes L_{c} \simeq L_{a} \otimes (L_{b} \otimes
L_{c}),
\]

\[
L_{abc} \simeq L_{abc},
\]

\[
f(a,b,c) \in \pi_{1}(C).
\]

\[
f : \pi_{0} \times \pi_{0} \times \pi_{0} \longrightarrow \pi_{1}.
\]

\[
k(C) \in H^{3}(\pi_{0}(C), \pi_{1}(C)).
\]

\[
s(C) = s : \pi_{0} \longrightarrow {}_{2}(\pi_{1}),
\]

\[
\mathrm{Hom}_{\otimes \mathrm{AUC}}(S^{-1}C, G) \longrightarrow
\mathrm{Hom}^{S}_{\otimes \mathrm{AUC}}(C, G)
\]

\[
X \otimes (Y \oplus Z) \simeq X \otimes Y \oplus X \otimes Z
\]

\[
\otimes : C \times C \longrightarrow C, \qquad
(X, Y) \longmapsto X \otimes Y
\]

\[
a_{X,Y,Z} : X \otimes (Y \otimes Z) \xrightarrow{\ \sim\ }
(X \otimes Y) \otimes Z, \qquad X, Y, Z \in \mathrm{Ob}\,C
\]

\[
c_{X,Y} : X \otimes Y \xrightarrow{\ \sim\ } Y \otimes X, \qquad
X, Y \in \mathrm{Ob}\,C,
\]

\[
c_{Y,X} \circ c_{X,Y} = \mathrm{id}_{X \otimes Y}.
\]

\[
g_{X} : X \xrightarrow{\ \sim\ } 1 \otimes X, \qquad
d_{X} : X \xrightarrow{\ \sim\ } X \otimes 1, \qquad
X \in \mathrm{Ob}\,C
\]

\[
X' \otimes X \simeq X \otimes X'' \simeq 1.
\]

\[
\check{F}_{X,Y} : FX \otimes FY \longrightarrow F(X \otimes Y), \qquad
X, Y \in \mathrm{Ob}\,C.
\]

\[
\gamma_{X} : u \longmapsto u \otimes \mathrm{id}_{X} \ = \
\mathrm{Aut}(1) \xrightarrow{\ \sim\ } \mathrm{Aut}(X),
\]

\[
\delta_{X} : u \longmapsto \mathrm{id}_{X} \otimes u \ = \
\mathrm{Aut}(1) \xrightarrow{\ \sim\ } \mathrm{Aut}(X).
\]

\[
su = \delta_{X}^{-1} \gamma_{X}(u)
\]

\[
\varepsilon_{0} : M \xrightarrow{\ \sim\ } \Pi_{0}(P), \qquad
\varepsilon_{1} : N \xrightarrow{\ \sim\ } \Pi_{1}(P)
\]

\[
\varepsilon_{0} : M \xrightarrow{\ \sim\ } \Pi_{0}(P), \qquad
\varepsilon_{1} : N \xrightarrow{\ \sim\ } \Pi_{1}(P).
\]

\[
L_{.}(M) : L_{3}(M) \xrightarrow{\ d_{3}\ } L_{2}(M)
\xrightarrow{\ d_{2}\ } L_{1}(M) \xrightarrow{\ d_{1}\ } L_{0}(M)
\longrightarrow M
\]

\[
{}'L_{.}(M) : {}'L_{3}(M) \xrightarrow{\ 'd_{3}\ } {}'L_{2}(M)
\xrightarrow{\ 'd_{2}\ } {}'L_{1}(M) \xrightarrow{\ 'd_{1}\ } {}'L_{0}(M)
\longrightarrow M
\]

\[
L_{0}(M) = {}'L_{0}(M) = \mathbf{Z}[M]
\]

\[
L_{1}(M) = {}'L_{1}(M) = \mathbf{Z}[M \times M]
\]

\[
L_{2}(M) = {}'L_{2}(M) = \mathbf{Z}[M \times M \times M] +
\mathbf{Z}[M \times M]
\]

\[
L_{3}(M) = {}'L_{3}(M) + \mathbf{Z}[M]
\]

\[
{}'L_{3}(M) = \mathbf{Z}[M \times M \times M \times M] +
\mathbf{Z}[M \times M \times M] + \mathbf{Z}[M \times M]
\]

\[
d_{1}[x,y] = {}'d_{1}[x,y] = [y] - [x+y] + [x]
\]

\[
d_{2}[x,y] = {}'d_{2}[x,y] = [x,y] - [y,x]
\]

\[
d_{2}[x,y,z] = {}'d_{2}[x,y,z] = [y,z] - [x+y,z] + [x,y+z] - [x,y]
\]

\[
d_{3}[x,y,z,t] = {}'d_{3}[x,y,z,t] = [y,z,t] - [x+y,z,t] + [x,y+z,t] -
[x,y,z+t] + [x,y,z]
\]

\[
d_{3}[x,y,z] = {}'d_{3}[x,y,z] = [x,y,z] - [x,z,y] + [z,x,y] - [y,z] +
[x+y,z] - [x,z]
\]

\[
d_{3}[x,y] = [x,y] + [y,x] = {}'d_{3}[x,y]
\]

\[
d_{3}[x] = [x,x],
\]

\[
\lambda : (D, \check{D}) \circ (T, \check{T}) \xrightarrow{\ \sim\ }
(I_{P}, \check{I}_{P}),
\]

\[
(K, \check{K}) : A \longrightarrow B, \qquad
K(u) = \mathrm{id} \ \text{for all} \ u \in S
\]

\[
(A'_{1}, B'_{1}, u) \ R_{A,B} \ (A'_{2}, B'_{2}, u)
\]

\[
u' : A'_{1} \otimes C'_{1} \xrightarrow{\ \sim\ } A'_{2} \otimes C'_{2},
\qquad
v' : B'_{1} \otimes C'_{1} \xrightarrow{\ \sim\ } B'_{2} \otimes C'_{2}
\]

\[
[B'', C'', v] \circ [A', B', u] = [A' \otimes B'',\ B' \otimes C'',\
\omega] : A \longrightarrow C
\]

\[
A \otimes E \ (\text{in } P) = A \otimes E \ (\text{in } A)
\]

\[
[A', B', u] \otimes [E', F', v] = [A' \otimes E',\ B' \otimes F',\ W]
\]

\[
\bigl([A', A', a \otimes \mathrm{id}],\ [A', A', c \otimes \mathrm{id}],\
(1_{P} = TA'_{0},\ g_{A} = [A'_{0} \otimes A', A', t_{A}],\
d_{A} = [A'_{0} \otimes A', A', r_{A}])\bigr)
\]

\[
g_{A} : A \longrightarrow 1_{P} \otimes A, \qquad
d_{A} : A \longrightarrow A \otimes 1_{P}
\]

\[
DA = A, \qquad Du = [A', A', u \otimes \mathrm{id}_{TA'}], \qquad
\check{D}_{A,B} = \mathrm{id}_{A \otimes B}
\]

\[
\lambda : (D, \check{D}) \circ (T, \check{T}) \xrightarrow{\ \sim\ }
(I_{P}, \check{I}_{P})
\]

\[
DTA' = TA'
\xrightarrow{\ \lambda_{A'} = [A'_{0},\, A',\, c_{TA', TA'_{0}}]\ }
I_{P} A' = TA'_{0}, \qquad A' \in \mathrm{Ob}\,A'
\]

\[
\mathrm{Hom}^{\otimes \mathrm{ACU}}(C \times C', Q) \longrightarrow
\mathrm{Hom}^{\otimes \mathrm{ACU}}(C, Q) \times
\mathrm{Hom}^{\otimes \mathrm{ACU}}(C', Q).
\]

\[
\Pi_{0}(P) \simeq K^{0}(R), \qquad \Pi_{1}(P) \simeq K^{1}(R)
\]