\[
  (1) \quad \mathrm{Hom}(x, y) \in \mathrm{Ens}
  \qquad\qquad
  \mathrm{Hom}(x, y) \simeq \pi_0(\mathrm{Homhot}(x, y))
\]

\[
  (2) \quad \mathrm{Homhot}(x, y) \in (\mathrm{Hot})
  \qquad\qquad
  \mathrm{Homhot}(x, y) \simeq \int_I \underline{\mathrm{Homhot}}(x, y)
\]

\[
  (3) \quad \boxed{\underline{\mathrm{Homhot}}(x, y) \in \mathrm{Hot}(I)}
\]

\[
  (1') \quad \mathrm{Hom}(f_!(X), Y) \simeq \mathrm{Hom}(X, f^{*}Y)
\]

\[
  (2') \quad \mathrm{Homhot}(f_!(X), Y) \simeq \mathrm{Homhot}(X, f^{*}(Y))
\]

\[
  (3') \quad \boxed{\underline{\mathrm{Homhot}}(f_!X, Y) \simeq f_{*}\,
  \underline{\mathrm{Homhot}}(X, f^{*}(Y))}
\]

\[
  (1'') \quad \mathrm{Hom}(f^{*}(Y), X) \simeq \mathrm{Hom}(Y, f_{*}(X))
\]

\[
  (2'') \quad \mathrm{Homhot}(f^{*}(Y), X) \simeq \mathrm{Homhot}(Y, f_{*}(X))
\]

\[
  (3'') \quad \boxed{\underline{\mathrm{Homhot}}(Y, f_{*}(X)) \simeq f_{*}\,
  \underline{\mathrm{Homhot}}(f^{*}Y, X)}
\]

\[
  \pi_0(x, y) = \mathrm{Hom}_{\mathcal H}(x, y) \qquad
  \pi_1(x, y ; f) = \struck{\ill{}}
\]

\[
  x \xrightarrow{\ f\ } y \qquad
  \mathcal H(\Delta_1) \to \mathcal H(\Delta_0) \times \mathcal H(\Delta_0),
  \quad \varphi \longmapsto
\]

\[
  J \hookrightarrow I \qquad X_J \qquad \mathcal H(J) \qquad
  \mathcal H(I, J) \qquad \mathcal H(I) \to \mathcal H(J)
\]

\[
  X_i \to X_J, \qquad Y_i \to Y_J
\]

\[
  \pi_1(x, y ; f, g)
\]

\[
  \mathcal H(\Delta_1) \qquad \longrightarrow\longrightarrow\longrightarrow
  \qquad M \ W \qquad \mathrm{Hot}
\]

\[
    \mathrm{Hom}(\xi, \eta) = \Gamma\, \underline{\mathrm{Hom}}(\xi, \eta)
  \]

\[
    \Gamma \text{ ou } \pi_0 : \mathcal H \to \mathrm{Ens}
  \]

\[
    \mathrm{Hom}(x \otimes y, z) \simeq \mathrm{Hom}(x,
    \underline{\mathrm{Hom}}(y, z))
  \]

\[
  (1) \qquad \boxed{A \longmapsto \mathcal H(A)}
\]

\[
  (2) \qquad
  \begin{array}{c} f : A \to B \\ \text{foncteur entre $U$-catégories}
  \end{array}
  \quad \text{implique} \quad
  \mathcal H(B) \xrightarrow{\ f^{*}_{\mathcal H}\ } \mathcal H(A)
\]

\[
  (3) \qquad \mathcal H(\Delta_0) = \mathrm{Hot}_{\mathcal H}
\]

\[
  (4) \qquad \mathcal H(A) \xrightarrow{\ \alpha_A\ }
  \underline{\mathrm{Hom}}(A, \struck{\mathrm{Hot}}\ \mathrm{Hot}_{\mathcal H}),
  \qquad
  \mathcal X \longmapsto \big(a \longmapsto (i_a)^{*}_{\mathcal H}(\mathcal X)\big)
\]

\[
  i_a : \Delta_0 \longrightarrow A \quad \text{foncteur de valeur } a
\]

\[
  \underline{\mathrm{Hom}}(B, A) \longrightarrow
  \underline{\mathrm{Hom}}(\mathcal H(A), \mathcal H(B))
\]

\[
  \mathcal H(A) \longrightarrow \underline{\mathrm{Hom}}(\underline{\mathrm{Hom}}(B, A),
  \mathcal H(B))
\]

\[
  (6) \qquad \boxed{\mathcal H(A) \xrightarrow{\ \underline{L}_A\ }
  \mathcal H(A^{\wedge})}
\]

\[
  (7) \qquad \mathcal H(A) \xrightarrow{\ \underline{L}_A\ }
  \mathcal H(A^{\wedge}) \xrightarrow{\ (\varepsilon_A)^{**}_{\mathcal H}\ }
  \mathcal H(A),
  \qquad \text{composé} = \mathrm{id}_{\mathcal H(A)}
\]

\[
  \varepsilon_A : A \hookrightarrow A^{\wedge}
\]

\[
  f^{*}_{\mathcal H} : \mathcal H(B) \to \mathcal H(A)
\]

\[
  f^{\mathcal H}_{!} : \mathcal H(B) \to \mathcal H(A)
\]

\[
  \mathcal H(A) \longrightarrow \mathrm{Hot}_{\mathcal H}
\]

\[
  \int_A \mathcal X = \int_B f_!(\mathcal X)
\]

\[
  \prod_A = (\pi_A)_{*} : \mathcal H(A) \longrightarrow \mathrm{Hot}_{\mathcal H}
\]

\[
  \struck{\ast}\ \prod_A \mathcal X \simeq \prod_B f_{*}(\mathcal X).
\]

\[
  f^{*}_{\mathcal H^{\circ}} = (f^{*}_{\mathcal H})^{\circ} :
  \mathcal H^{\circ}(B) \to \mathcal H^{\circ}(A),
\]

\[
  \begin{cases}
    f^{\mathcal H^{\circ}}_{!} = (f^{\mathcal H}_{*})^{\circ} \\
    f^{\mathcal H^{\circ}}_{*} = (f^{\mathcal H}_{!})^{\circ}
  \end{cases}
\]

\[
  \mathcal H(A) = \underline{\mathrm{Hom}}(A, M)
\]

\[
  f^{*}_{M} : \underbrace{\underline{\mathrm{Hom}}(B, M)}_{\mathcal H(B)}
  \longrightarrow \underbrace{\underline{\mathrm{Hom}}(A, M)}_{\mathcal H(A)}
\]

\[
  \begin{cases}
    \int_A = \varinjlim_A \\
    \prod_A = \varprojlim_A
  \end{cases}
\]

\[
  W_A \subset \mathrm{Fl}\,\underline{\mathrm{Hom}}(A, M)
\]

\[
  \mathcal H(A) = W_A^{-1}\, \underline{\mathrm{Hom}}(A, M)
\]

\[
  \underset{(W)}{M} \xrightarrow{\ f\ } \underset{(W')}{M'}
\]

\[
  \mathcal H_W(A) \Longrightarrow \mathcal H_{W'}(A)
\]

\[
  \mathcal H_W(A) \to \mathcal H_{W'}(A)
\]

\[
  \text{(i)} \quad f^{-1}(W') = W
\]

\[
  \text{(ii)} \quad fg(X') \to X' \in W' \qquad \forall\, X' \in \mathrm{Ob}\, M'
\]

\[
  \underline{\mathrm{Hom}}(A, M)
  \underset{g^{M}}{\overset{f^{M}}{\rightleftarrows}}
  \underline{\mathrm{Hom}}(A, M')
\]

\[
  \mathcal H_W(A)
  \underset{g^{A}_{W}}{\overset{f^{A}_{W}}{\rightleftarrows}}
  \mathcal H_{W'}(A)
\]

\[
  \begin{array}{c}
    (M_1, W_1) \longrightarrow (M, W) \\
    (M_2, W_2) \nearrow
  \end{array}
\]

\[
  \underline{\mathrm{Hom}}(A, M) = \text{complexes de }
  \underline{\mathrm{Hom}}(A, \mathcal A)
\]

\[
  W_A = \text{quasi-isom.\ d'icelle.}
\]

\[
  \underline{\mathrm{Hom}}(A, \mathcal A) = \text{faisceaux de $k$-modules
  sur topos } A^{\circ\wedge} \times_{\mathrm{top}} \mathcal T
\]

\[
  f^{*} : \underline{\mathrm{Hom}}(B, M) \to \underline{\mathrm{Hom}}(A, M)
\]

\[
  g^{*} f'_{!} \xleftarrow{\ \sim\ } f_{!}\, g'^{*}
  \quad \text{si $f$ cofibrant [ou $g$ fibrant ??]}
\]

\[
  g^{*} f'_{*} \xrightarrow{\ \sim\ } f_{*}\, g'^{*}
  \quad \text{si $f$ fibrant [ou $g$ cofibrant ??]}
\]

\[
  f^{*}_{\mathcal H} : \mathcal H(B) \to \mathcal H(A) \quad \text{foncteur
  localisation}
\]

\[
  f^{\mathcal H}_{*},\ f^{\mathcal H}_{!} : \mathcal H(A) \to \mathcal H(B)
  \quad \text{sont pl.\ fid.}
\]

\[
  f^{*} f_{!} \simeq \mathrm{id}, \qquad f^{*} f_{*} \simeq \mathrm{id}\,)
\]

\[
  f^{*}_{\mathcal H} : \mathcal H(B) \to \mathcal H(A) \quad \text{pl.\ fid.,
  i.e.}
\]

\[
  f^{\mathcal H}_{!},\ f^{\mathcal H}_{*} \quad \text{foncteurs loc.}
  \quad \struck{\text{iso}} \quad \text{et} \quad
  f_! f^{*} \simeq \mathrm{id} \simeq f_{*} f^{*}
\]

\[
  \int_{A_0} i^{*}(\xi) \xrightarrow{\ \sim\ } \int_A \xi
  \qquad \text{pour tout } \xi \in \mathcal{H}(A)
\]

\[
  ?\qquad i^{*}_{\mathcal{H}} = j^{\mathcal{H}}_{!}\ {}^{(*)},
\]

\[
  p^{\mathcal{H}}_{0!}\, i^{*}_{\mathcal{H}} \xrightarrow{\ \sim\ }
  p^{\mathcal{H}}_{!}
\]

\[
  p^{\mathcal{H}}_{0!}\, i^{\mathcal{H}}_{!} \simeq p^{\mathcal{H}}_{!}
\]

\[
  \int_A \xi \xleftarrow{\ \sim\ } \xi(e)
\]

\[
  \mathcal{H}(A) \longrightarrow
  \mathcal{H}(A') \times_{\mathcal{H}(A_0)} \mathcal{H}(A'')
\]

\[
  \mathrm{Hom}(\xi, \eta) \longrightarrow
  \mathrm{Hom}(\xi', \eta') \times_{\mathrm{Hom}(\xi_0, \eta_0)}
  \mathrm{Hom}(\xi'', \eta'')
\]