\[
u : E \longrightarrow E', \qquad v : F \longrightarrow F'
\]

\[
\Pi(u,v) : \Pi(E,F) \longrightarrow \Pi(E',F')
\]

\[
\Pi(u,v)_{*}(\varphi)(X',Y') = \varphi(u^{*}X', v^{*}Y')
\qquad : \Pi(E,F) \longrightarrow \Pi(E',F')
\]

\[
\Pi(u,v)^{*}(\varphi)(X,Y) \overset{?}{=}
\]

\[
\Pi(E,F) \longrightarrow \Pi(E,F') \longrightarrow \Pi(E',F')
\]

\[
\Pi(\mathrm{id}_{E}, v)_{*} : \Pi(E,F) \longrightarrow \Pi(E,F')
\]

\[
\Pi(\widetilde{C}, F) \longrightarrow \Pi(\widehat{C}, F') ,
\qquad\text{et}
\]

\[
\operatorname{Hom}(C, F) \longrightarrow \operatorname{Hom}(C, F')
\]

\[
\Pi(E,F) \longrightarrow E
\]

\[
\Pi(E,F) \longrightarrow F
\]

\[
\Pi(E,F) \longrightarrow E \times_{\mathrm{top}} F
\]

\[
\Pi(E,F) = \Pi(\widetilde{C}, \widetilde{D}) = \struck{\ill{}}\;
\Pi(\widetilde{C}, \widehat{D}) \cap \Pi(\widehat{C}, \widetilde{D})
\]

\begin{align}
\operatorname{Hom}_{\mathrm{top}}(X, \widehat{C}) &\cong \operatorname{Hom}'(C, X)^{\circ} \nonumber\\
\operatorname{Hom}_{\mathrm{top}}(X, \widehat{D}) &\cong \operatorname{Hom}'(D, X)^{\circ} \nonumber
\end{align}

\begin{align}
\operatorname{Hom}_{\mathrm{top}}(X, \widehat{C}) \times \operatorname{Hom}_{\mathrm{top}}(X, \widehat{D})
&\cong \bigl[\operatorname{Hom}'(C,X) \times \operatorname{Hom}'(D,X)\bigr]^{\circ} \nonumber\\
&\cong \bigl(\operatorname{Hom}'(C \times D, X)\bigr)^{\circ} \nonumber\\
&\cong \operatorname{Hom}_{\mathrm{top}}(X, \widehat{C \times D}) \nonumber
\end{align}

\[
\prod_{i} \widehat{C_{i}} \;\approx\; \widehat{{\textstyle\prod}' C_{i}}
\]

\[
\prod_{i} \operatorname{Hom}_{\mathrm{top}}(X, E'_{i}) \;\cong\;
\operatorname{Hom}_{\mathrm{top}}(X, F).
\]

\[
\varprojlim_{i} \operatorname{Hom}_{\mathrm{top}}(X, \mathcal{F}_{i})
\;\simeq\;
\varprojlim_{i} \operatorname{Hom}'(\mathcal{F}_{i}, X)
\;\simeq\; \ill{}
\]

\[
\varprojlim_{i}^{\mathrm{top}} E_{i} \longrightarrow
\prod_{i \in \operatorname{Ob} I} E_{i} \rightrightarrows
\prod_{\alpha \in \mathrm{Fl}_{1} I} E_{b(\alpha)} \rightrightarrows
\prod_{(\alpha,\beta) \in \mathrm{Fl}_{2}(I)} E_{b(\beta)}
\]

\[
\struck{\text{Prop.}} \quad (*) \qquad
\boxed{\;\operatorname{Top}(X \times Y) \longrightarrow \operatorname{Top}(X)
\overset{2}{\times}_{\mathrm{top}} \operatorname{Top}(Y)\;}
\]

\[
\bigcup_{i} U_{i} \times V_{i} = U \times V \implies
\operatorname*{Sup}_{i} U_{i} \times V_{i} = U \times V
\]

\[
\begin{cases}
\mathring{U}_{x} = U'_{x} \subset \bigcup_{i \in I_{x,y}} U_{i}, \\
V_{i} \supset V_{x,y} \text{ pour } i \in I_{x,y}
\end{cases}
\]

\[
S \overset{\text{déf}}{=} \operatorname*{Sup}_{i \in I}(U_{i} \times V_{i})
\;\geqslant\; \operatorname*{Sup}_{i \in I_{x,y}}(U_{i} \times V_{i})
\;\geqslant\; \Bigl(\operatorname*{Sup}_{i \in I_{x,y}} U_{i}\Bigr) \times V_{x,y}
\;\geqslant\; U'_{x} \times V_{x,y}
\]

\[
S \geqslant \operatorname*{Sup}_{y} \mathring{U}'_{x} \times V_{x,y}
= \mathring{U}'_{x} \times \operatorname*{Sup}_{y} V_{x,y}
= \mathring{U}'_{x} \times V
\]

\[
S \geqslant \operatorname*{Sup}_{x} (\mathring{U}'_{x} \times V)
= \Bigl(\operatorname*{Sup}_{x} \mathring{U}'_{x}\Bigr) \times V
= U \times V
\]

\[
T \times \{y\} \subset \bigcup_{i \in I_{y}} U_{i} \times V_{i},
\qquad\text{et pour } V_{y} = \bigcap_{i \in I_{y}} V_{i},
\quad \struck{\ill{}}
\]

\[
S \geqslant \operatorname*{Sup}_{i \in I_{y}}(U_{i} \times V_{i})
\geqslant \operatorname*{Sup}_{i \in I_{y}} U_{i} \times V_{y}
\geqslant \Bigl(\operatorname*{Sup}_{i \in I_{y}} U_{i}\Bigr) \times V_{y}
\]

\[
S \geqslant \operatorname*{Sup}(U'_{i}, U_{i}\ i \in I_{y}) \times V_{y}
\geqslant (U' \cup U'') \times V_{y}
\]

\[
S \geqslant \operatorname*{Sup}_{y}\bigl((U' \cup U'') \times V_{y}\bigr)
= (U' \cup U'') \times \operatorname*{Sup}_{y} V_{y}
= (U' \cup U'') \times V
\]

\[
\operatorname{Top}\Bigl(\prod_{i \in I} X_{i}\Bigr) \longrightarrow
\prod_{i}^{(2)}{}_{(\mathrm{top})} \operatorname{Top}(X_{i})
\]

\begin{align}
\prod_{i}^{\mathrm{top}} \operatorname{Top}(X_{i})
&\cong \varprojlim_{\alpha}{}^{\mathrm{top}}
   \Bigl(\prod_{i \in J_{\alpha}} \operatorname{Top}(X_{i})\Bigr) \nonumber\\
&\cong \varprojlim_{\alpha}{}^{\mathrm{top}}
   \Bigl(\operatorname{Top}\bigl({\textstyle\prod_{i \in J_{\alpha}}} X_{i}\bigr)\Bigr)
   \qquad\Bigl[{\textstyle\prod_{i \in J_{\alpha}}} X_{i} = Z_{\alpha}\Bigr] \nonumber\\
&\approx \Bigl(\underbrace{\varinjlim_{\alpha} \mathcal{O}_{Z_{\alpha}}}_{\mathcal{O}'_{Z}}\Bigr)^{\sim} \nonumber
\end{align}

\[
\{z\} \times \prod_{i \in J_{\beta(z)} - J_{\alpha}} X_{i} \subset
\bigcup_{\lambda \in \Lambda_{z}} V'_{\lambda} ,
\]

\[
U_{z} \times \prod_{i \in J_{\beta(z)} - J_{\alpha}} X_{i} \subset
\bigcup_{\lambda \in \Lambda_{z}} V'_{\lambda}
\]

\[
S \overset{\text{déf}}{=} \operatorname*{Sup}_{\lambda \in \Lambda}(V_{\lambda})
\geqslant \operatorname*{Sup}_{\lambda \in \Lambda_{z}} V_{\lambda}
\geqslant U_{z} \times Z'_{\alpha} .
\]

\[
S \geqslant \operatorname*{Sup}_{z}\bigl(U_{z} \times Z'_{\alpha}\bigr)
= \Bigl(\operatorname*{Sup}_{z \in Z} U_{z}\Bigr) \times Z'_{\alpha}
= W \times Z'_{\alpha} ,
\]

\[
\prod_{i \in I} \operatorname{Top}(X_{i}) =
\Bigl(\underbrace{\varinjlim_{\alpha} \mathcal{O}_{Z_{\alpha}}}_{\mathcal{O}'_{Z}}\Bigr)^{\sim} ,
\]

\[
\struck{\ill{}\ W(z) =} \qquad
V_{z} = \{\, z' \in Z \mid z'_{i} = z_{i} \text{ si } i \in J - J(z) \,\} .
\]

\[
f_{0} : C \longrightarrow C'
\]

\[
g_{ji} : C_{i} \longrightarrow C_{j}
\]

\[
\mathcal{X} = \varprojlim_{i, \widehat{f}_{ij}}{}^{\mathrm{top}} \widehat{C_{i}}
\;\simeq\; \Bigl(\operatorname*{colim}_{i, g_{ij}} C_{i}\Bigr)^{\wedge}
\]

\[
C_{i} \xrightarrow{\ \alpha_{i}\ } C = \operatorname*{colim} C_{j} ,
\]

\[
\mathrm{pr}_{i*}(F) = F \circ \alpha_{i} ,
\qquad\text{et pour } G \in \widehat{C_{i}} \ \ill{}
\]

\[
\mathrm{pr}_{i}^{*}(G) =
\]

\[
\operatorname{Hom}'(\widetilde{C}, E)
\qquad
\operatorname{Hom}(C, E)
\]

\begin{align}
\alpha(u) &= u \circ \varepsilon_{C} \nonumber\\
\beta(u_{0}) &= \Bigl[F \mapsto \varinjlim_{C/F} u_{0}(X)\Bigr] \nonumber
\end{align}

\[
i_{u} : \beta\alpha(u) \longrightarrow u
\]

\[
i_{u}(F) : \bigl(\beta(\alpha(u))\bigr)(F)
= \varinjlim_{C/F} u(\varepsilon_{C}(X)) \longrightarrow u(F)
\]

\[
\boxed{\;\beta\alpha \xrightarrow{\ i\ } \mathrm{id}_{\operatorname{Hom}(\widetilde{C},E)}\;}
\]

\[
j_{u_{0}} : \struck{\ill{}}\; u_{0} \longrightarrow \alpha\beta(u_{0})
\]

\[
j_{u_{0}}(Z) : \quad u_{0}(Z) \longrightarrow
\bigl(\alpha(\beta(u_{0}))\bigr)(Z)
= \beta(u_{0})(\varepsilon_{C}(Z))
= \varinjlim_{C/\varepsilon_{C}(Z)} u_{0}(X)
\]

\[
\boxed{\;j : \mathrm{id}_{\operatorname{Hom}(C,E)} \longrightarrow \alpha\beta\;}
\]

\[
u_{0}(Z) \longrightarrow \varinjlim_{C/\varepsilon_{C}(Z)} u_{0}(X)
\]

\[
\varprojlim_{C/\varepsilon_{C}(Z)} \operatorname{Hom}(u_{0}(X), \xi)
\longrightarrow \operatorname{Hom}(u_{0}(Z), \xi)
\quad\text{bijective.}
\]

\[
u'_{0} : E \longrightarrow \widehat{C}
\]

\[
u'_{0}(\xi)(Y) = \operatorname{Hom}(u_{0}(Y), \xi)
\]

\[
\varprojlim_{C/\varepsilon_{C}(Z)} P(X) \longrightarrow P(Z)
\qquad\text{un isom.}
\]

\[
\operatorname{Hom}'(\widetilde{C}, E) \longrightarrow \operatorname{Hom}(C,E)
\]

\[
u : \widetilde{C} \longrightarrow E
\]

\[
u \circ \varepsilon_{C} = u_{0} , \qquad u_{0} : C \longrightarrow E
\]

\[
F = \varinjlim_{C/F} \varepsilon_{C}(X)
\]

\[
u(F) = \varinjlim_{C/F} u_{0}(X) .
\]

\[
a : \operatorname{Hom}'(\widetilde{C}, E) \longrightarrow \operatorname{Hom}(C,E)
\]

\[
\beta : \operatorname{Hom}(C,E) \longrightarrow \operatorname{Hom}(C,E)
\]

\[
\beta(u_{0}) = F \mapsto \varinjlim_{C/F} u_{0}(X)
\]

\[
\beta \circ \alpha \simeq i
\qquad (\text{inclusion de } \operatorname{Hom}'(\widetilde{C},E)
\hookrightarrow \operatorname{Hom}(\widetilde{C},E))
\]

\begin{align}
\operatorname{Hom}'(\widetilde{C}, E) &\xrightarrow{\ \alpha'\ } \operatorname{Hom}'(\widehat{C}, E) \nonumber\\
\operatorname{Hom}'(\widehat{C}, E) &\xrightarrow{\ \beta'\ } \operatorname{Hom}(\widetilde{C}, E) \nonumber
\end{align}

\[
\alpha'(u) = u \circ a
\]

\[
\beta'(v) = v \circ j_{a} = v \mid \widetilde{C}
\qquad (\text{où } j : \widetilde{C} \to \widehat{C} \text{ est l'inclusion})
\]

\[
\beta'\alpha'(u) = u \circ a \mid \widetilde{C} \simeq u
\]

\[
\struck{\ill{}}\quad u \in \mathrm{Fl}\,\widehat{C},\ a(u) \text{ inversible}
\implies F(u) \text{ inversible.}
\]

\[
\mathcal{U}\Bigl(\varinjlim_{\alpha}{}^{\widetilde{C}} X_{\alpha}\Bigr)
= \mathcal{U}\, a \Bigl(\varinjlim_{\alpha}{}^{\widehat{C}} X_{\alpha}\Bigr)
= (\mathcal{U}a)\Bigl(\varinjlim_{\alpha} X_{\alpha}\Bigr)
= \varinjlim_{\alpha} (\mathcal{U} \circ a)(X_{\alpha})
= \varinjlim_{\alpha} \mathcal{U}(X_{\alpha})
\]

\[
\operatorname{Hom}'(\widetilde{C}, E) \xrightarrow{\ \simeq\ }
\text{sous-catégorie pleine de } \operatorname{Hom}(\widehat{C}, E)
\]

\[
\operatorname{Hom}'(\widetilde{C}, E) \xrightarrow[\ \simeq\ ]{\ \alpha_{0}\ }
\struck{\ill{}} \text{ sous-catégorie pleine de } \operatorname{Hom}(C, \struck{\widetilde{E}}\,E)
\]

\[
\alpha_{X} : \mathcal{T}(X) \longrightarrow \underline{\omega}(X), \qquad
\beta_{X} : \mathcal{T}(X) \longleftarrow \underline{\omega}(X)
\]

\[
y \underset{T}{\leq} x \iff x \in \overline{\{y\}}
\]

\[
\struck{\mathrm{Fer}}\ \mathrm{Ouv}(T) = \{\, U \subset X \mid (x \in U,\ x \leq y)
\Rightarrow y \in U \,\}
\]

\[
\mathrm{Fer}(T) = \{\, Z \subset X \mid (x \in Z,\ y \leq x) \Rightarrow y \in Z \,\}
\]

\[
T \leq \beta_{X}\alpha_{X}(T)
\]

\[
\beta\alpha(X) \longrightarrow X
\]

\[
\beta\alpha \xrightarrow{\ \rho\ } \mathrm{id}_{(\mathrm{Esp})}
\]

\[
\alpha_{X}\beta_{X}(\omega) = \omega
\]

\[
X \xrightarrow[\ \sigma\ ]{\ \sim\ } \alpha\beta(X)
\]

\[
\mathrm{id}_{(\text{Préord})} \xrightarrow[\ \sigma\ ]{\ \sim\ } \alpha\beta
\]

\[
(*) \qquad \operatorname{Hom}_{\mathrm{Esp}}(\beta I, X) \simeq
\operatorname{Hom}_{\text{Préord}}(I, \alpha X)
\]

\[
\beta : (\text{Préord}) \longrightarrow (\mathrm{Esp})
\]

\[
O_{x} = \{ y \in X \mid y \underset{T}{\leq} x \}.
\]

\[
\mathrm{Top}(X) \simeq \widehat{\underline{X}}
\]

\[
\mathrm{Top}(X) \simeq \widehat{\underline{X}}
\]

\[
\mathrm{Points}(\widehat{\underline{X}}) \simeq \mathrm{Pro}(\underline{X})
\qquad (\ill{}\ \mathrm{ind}),
\]

\[
x \longmapsto \overline{\{x\}} = \{ y \in X \mid y \geq x \}
\]

\[
(\text{topos spatiaux essentiels}) \approx (\text{Préord})
\ \struck{\text{ensembles ordonn\'es}}\ \approx (\mathrm{Esp.\ ess.})
\]

\[
\gamma(X \times Y) = \alpha(X) \wedge \beta(Y) = 1_{\sigma}
\]

\begin{align*}
\gamma(W \cap W') &= \sup_{U \times V \subset W \cap W'} \alpha(U) \wedge \beta(V)
= \sup_{\substack{U \times V \subset W \\ U' \times V' \subset W'}}
\alpha(U \cap U') \wedge \beta(V \cap V') \\
&= \sup_{U \times V \subset W} \alpha(U)\beta(V) \wedge
\sup_{U' \times V' \subset W'} \alpha(U')\beta(V') \\
&= \gamma(W) \wedge \gamma(W')
\end{align*}

\[
u \longmapsto \alpha, \beta \qquad
\operatorname{Hom}'(\underline{O}_{X \times Y}, \sigma) \longrightarrow
\operatorname{Hom}'(O_{X}, \sigma) \times \operatorname{Hom}'(O_{Y}, \sigma)
\]

\[
\gamma(W) = \sup_{\substack{U \in O_{X},\ V \in O_{Y} \\ U \times V \subset W}}
\alpha(U) \wedge \beta(V)
\]

\[
F : (\mathcal{G}, \mathcal{E}) \longmapsto
\underline{\operatorname{End}}(\mathcal{G}),\ \mathcal{E},\
\mathcal{G} \otimes \mathcal{E},\ \varphi_{\mathcal{G},\mathcal{E}}
\]

\[
\underline{C} \simeq \coprod_{g,e} \underline{C}_{g,e} \qquad
\underline{C}' \simeq \coprod_{a,e,f} \underline{C}'_{a,e,f}
\]

\[
F_{g,e} : \underline{C}_{g,e} \longrightarrow \underline{C}'_{g^{2},e,\ill{}}
\]

\[
\varphi, \varphi' : A \otimes \underline{\operatorname{End}}(\mathcal{E})
\rightrightarrows \underline{\operatorname{End}}(\mathcal{F})
\]

\[
\mathrm{TORS}^{1}_{U}(X, \Pi) \simeq H^{1}(X, \Pi)
\]

\[
T^{n} = \mathrm{TORS}^{n}_{U}(X ; \Pi) \rightleftarrows
\uncertain{H\!I\!I^{n}}(X, \Pi) \implies \pi_{0}(T^{n}) \simeq H^{n}(X, \Pi)
\]

\[
\mathrm{TORS}^{1}_{U}(X, \Pi) \xrightarrow{\ \sim\ } \uncertain{H\!I^{1}}(X, \Pi)
\]

\[
\lambda \simeq \underline{\operatorname{End}}(u), \qquad w = u \otimes v
\]

\[
\varphi \circ (\mathrm{id} \otimes \underline{\operatorname{End}}(v)) \simeq
\underline{\operatorname{End}}(w) \circ \varphi,
\]

\[
\underline{\operatorname{End}}(w)(\mathrm{id}_{g} \otimes X) =
\mathrm{id}_{g} \otimes \underline{\operatorname{End}}(v)(X) \qquad
\text{pour } X \in \Gamma\,\underline{\operatorname{End}}(\mathcal{E})
\]

\[
w(\mathrm{id}_{g} \otimes X)w^{-1} = \mathrm{id}_{g} \otimes vXv^{-1}
\]

\[
\operatorname{Hom}_{\mathrm{Sp\ sob}}(X, i(Z)) \simeq
\operatorname{Hom}_{\mathrm{Sp}}(\widetilde{X}, Z)
\]

\[
\text{« famille d'objets de } C \text{ sous-jacente »} \qquad
T(C) \xrightarrow{\ \varphi^{T}_{C}\ } C^{I}
\]

\[
u : C \longrightarrow C'
\]

\[
\varphi^{T} : \mathcal{T} \longrightarrow \mathcal{T}_{I}
\]

\[
\underline{\operatorname{Hom}}_{L}(T, T') =
\underline{\operatorname{Hom}}_{L_{0}}(\mathcal{T}_{L_{0}}, \mathcal{T}'_{L_{0}})
\]

\[
\bigl[\ \operatorname{resp.}\ \underline{\operatorname{Hom}}_{L}(T, T') =
\underline{\operatorname{Hom}}_{L_{0}}(\mathcal{T}_{L_{0}},
\mathcal{T}'_{L_{0}}) \ \bigr]
\]

\[
\underline{\operatorname{Hom}}_{L}(I ; T, T') = \text{catégorie des hom. de }
\mathcal{T}_{L_{0}} \text{ dans } \mathcal{T}'_{L_{0}}
\]

\[
\operatorname{Hom}_{\mathrm{Sp}}(X, Y) \longrightarrow
\operatorname{Hom}\mathrm{top}(\mathrm{Top}(X), \mathrm{Top}(Y))
\qquad (\text{isom classes})
\]

\[
I \longrightarrow \mathrm{Ob}\,\Gamma^{T}_{\varphi} \qquad
(\text{resp. } I \to \mathrm{Ob}\,\Gamma^{T}_{\varphi_{0}})
\]

\[
\Gamma^{T} \longrightarrow \Gamma_{0}^{T}
\]

\[
T(C) \simeq \underline{\operatorname{Hom}}_{\varprojlim}(\beta^{T}, C) \qquad
C \in \mathrm{Ob}(\mathrm{Cat}\ \varprojlim)
\]

\[
\bigl(\ T(C) \simeq \underline{\operatorname{Hom}}_{\varprojlim}(\beta_{0}, C)
\qquad C \in \mathrm{Ob}(\mathrm{Cat}\ \varprojlim)\ \bigr)
\]

\[
\underline{\operatorname{Hom}}(T, \mathcal{F}) \simeq \mathcal{F}(\beta)
\]

\[
\bigl(\ \underline{\operatorname{Hom}}_{0}(T, \mathcal{F}) \simeq
\mathcal{F}(\beta_{0})\ \bigr)
\]

\[
\underline{\operatorname{Hom}}(T, T') \simeq T'(\beta)
\]

\[
\bigl(\ \underline{\operatorname{Hom}}_{0}(T, T') \simeq T'(\beta_{0})\ \bigr)
\]

\[
\beta = \underline{\operatorname{Hom}}(T, i) = \Gamma^{T}
\]

\[
\bigl(\ \beta_{0} = \underline{\operatorname{Hom}}_{0}(T, i_{0}) =
\Gamma_{0}^{T}\ \bigr)
\]

\[
\mathrm{Cat}\bigl(X \text{ ordered by } x \leq y \text{ iff }
x \in \overline{\{y\}}\bigr) \longrightarrow \mathrm{Fib}(\mathrm{Top}(X))
\]

\[
\downarrow
\]

\[
\underline{\operatorname{Hom}}(\mathrm{Top}(Y), \mathrm{Top}(X))
\]

\[
u_{0} : \mathbb{Z}[t] \longrightarrow \mathbb{Z}[t,t^{-1}] \times \mathbb{Z}
\qquad \Bigl( \simeq \mathbb{Z}[x,y] \big/
\bigl(x(xy-1),\, y(xy-1)\bigr) \Bigr)
\]

\[
\mathcal{C}^{*} \times \{e\} \longrightarrow \mathcal{C}
\]

\[
f \longmapsto f_{*}
\]

\[
\mathrm{Ouv}(X) \longrightarrow \text{sous-objets de } e_{E}
\quad (\text{ouverts de } E),
\]

\[
\varinjlim\nolimits_{\lambda}(R, C^{\circ})^{\circ} \simeq
\varinjlim\nolimits_{\lambda}(R^{\circ}, C)
\]

\[
T(\mathrm{Ens}) \simeq \operatorname{Hom}_{\lambda}(R, (\mathrm{Ens})), \qquad
R_{T} \Longrightarrow \operatorname{Hom}(T(\mathrm{Ens}), \mathrm{Ens})
\]

\[
E \xrightarrow[\ \varprojlim \text{finies}\ ]{\ \varprojlim\ }
\operatorname{Hom}(C^{\circ}, \mathrm{Ens})
\qquad \Longleftrightarrow \qquad
C^{\circ} \longrightarrow \operatorname{Hom}_{\varprojlim\,\text{finies}}
(E, (\mathrm{Ens})) = \underline{\mathrm{Pt}}(E)^{\circ}
\]

\[
\operatorname{Hom}_{\mathrm{top}}(\widehat{C}, E) \simeq
\operatorname{Hom}(C, \underline{\mathrm{Pt}}(E))
\]

\[
\operatorname{Hom}_{\mathrm{top}}(\widehat{C}, \widehat{C}') \simeq
\operatorname{Hom}(C, \mathrm{Pro}\,C')
\qquad \bigl(\operatorname{Hom}(C, C')\ \text{pl.\ fid.}\bigr)
\]

\[
\operatorname{Hom}_{\mathrm{top}}(E, \widehat{C}) \simeq
\operatorname{Hom}_{\mathrm{ex.\,g.}}(C, E)^{\circ}
\]

\[
T(\widehat{C}\,) \longrightarrow \operatorname{Hom}_{\lambda}(C^{\circ}, S)
\qquad \bigl( S \overset{\text{déf}}{=} T(\mathrm{Ens}) \bigr)
\]

\[
\beta = (b_{\mathrm{Ens}}) : S \longrightarrow (\mathrm{Ens})^{I},
\qquad \bigl( \beta = (\beta_{i})_{i \in I},\ \beta_{i} : S \to (\mathrm{Ens})
\bigr)
\]

\[
\psi \circ \xi : C^{\circ} \longrightarrow S \xrightarrow{\ \psi\ }
(\mathrm{Ens})
\]

\[
T(C) \times R \longrightarrow C
\]

\[
T(C) \longrightarrow \operatorname{Hom}_{\lambda}(R, C)
\]

\[
E \rightleftarrows B_{G} \quad \text{correspond à} \quad e \rightleftarrows S
\]

\[
\begin{cases}
u^{*}(Y) = Y_{X} & \\
u_{!}(Y) = Y & (\text{oubli de plus structure}) \\
u_{*}(Y) & \text{plus compliqué}
\end{cases}
\]

\[
\operatorname{Hom}_{\underrightarrow{\lambda}}(\Sigma, \mathcal{T})
\xrightarrow{\ \sim\ } \operatorname{Hom}_{\lambda'}(S, \mathcal{T})
\]

\[
\operatorname{Hom}_{\underleftarrow{\lambda}}(R, \mathcal{T}^{\circ})
\qquad \parallel
\]

\[
\bigl( \forall\, x \in O,\ (y \leq x \text{ and }
x \in O \implies y \in O) \bigr).
\]

\[
\widehat{f}_{!} = \varphi^{*}, \qquad
\widehat{f}^{\,*} = \varphi_{*}, \qquad
\widehat{f}_{*} = \varphi^{!}
\]

\[
\operatorname{Hom}_{\widehat{C}'}(f_{!}(F), F') \simeq
\operatorname{Hom}_{\widehat{C}}(F, f^{*}F')
\]

\[
\operatorname{Hom}_{\widehat{C}'}(f_{!}(X), F') = f^{*}(F')(X) = F'(f(x))
= \operatorname{Hom}_{\widehat{C}'}(f(x), F')
\]

\[
\operatorname{Hom}_{\widehat{C}'}(F', f_{*}F) \simeq
\operatorname{Hom}_{\widehat{C}}(f^{*}F', F)
\]

\[
\operatorname{Hom}_{\widehat{C}'}(F', f_{*}(x)) =
\operatorname{Hom}_{\widehat{C}}(f^{*}F', x)
\]

\[
f_{*}(x)(x') = \operatorname{Hom}_{\widehat{C}}
(x' \circ f^{\circ}, x), \qquad
f_{*}(x) = \bigl( x' \longmapsto \operatorname{Hom}_{\widehat{C}}
(x' \circ f^{\circ}, x) \bigr)
\]

\[
x' \circ f^{\circ} = \bigl( y \longmapsto \operatorname{Hom}(f(y), x') \bigr)
\ \parallel\ \operatorname{Hom}(y, g\,x')
\]

\[
f_{*}(x) = \bigl( x' \longmapsto \operatorname{Hom}_{C}(g(x'), x) \bigr)
\ \wr\ \operatorname{Hom}_{C}(x', h(x))
\]

\[
\varphi(j^{*}(E)) = (E \times X) \times_{X} (e_{E}, \varepsilon) \simeq E
\]

\[
\mathrm{Top}(\mathbb{Z}, G) \longrightarrow \mathrm{Top}(\mathbb{Z}, H)
\]

\[
\mathrm{Top}(\mathbb{Z}, G)_{/\mathbb{Z}'} \longrightarrow
\mathrm{Top}(\mathbb{Z}, H), \qquad E \longmapsto E \times_{\mathbb{Z}'}(X, e)
\]

\[
\operatorname{Hom}_{\widehat{C}}(R, F) = \varprojlim\nolimits_{C/R} F(X)
\simeq \operatorname{Ker}\Bigl( F(X_{\alpha}) \rightrightarrows
\prod\nolimits_{\alpha, \beta} F(X_{\alpha} \times_{X} X_{\beta}) \Bigr)
\]

\[
\operatorname{Hom}(X, F) \longrightarrow \operatorname{Hom}(R, F)
\quad \text{mono. (biun.)} \quad \text{for } \forall\, R \subset X,\
R \in J(X)
\]

\[
\Bigl[ \operatorname{Hom}(X, F) \longrightarrow \operatorname{Ker}
\Bigl( \prod F(X_{\alpha}) \rightrightarrows F(X_{\alpha} \times_{X} X_{\beta})
\Bigr) \Bigr] \quad \text{mono. / biun.} \quad
\forall\, (X_{\alpha} \to X) \in \mathrm{Cov}(X)
\]

\[
\mathrm{id} \longrightarrow L, \qquad F \xrightarrow{\ \ell(F)\ } LF
\]

\[
LF(X) = \varinjlim\nolimits_{R \in T(X)} \operatorname{Hom}(R, F)
\]