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\begin{document}

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\section*{Sheaves in Geometry and Logic}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

This entry collects hyperlinks related to the textbook

\begin{itemize}%
\item [[Saunders Mac Lane]], [[Ieke Moerdijk]],

\emph{Sheaves in Geometry and Logic -- A first introduction to topos theory}

Springer Verlag, 1992

(\href{https://link.springer.com/book/10.1007/978-1-4612-0927-0}{doi:10.1007/978-1-4612-0927-0})



\end{itemize}
on [[sheaf and topos theory]] and its application in [[categorical logic]].

For a similar link lists see also

\begin{itemize}%
\item \emph{[[Categories and Sheaves]]}


\item \emph{[[Sketches of an Elephant]]}



\end{itemize}
\hypertarget{content}{}\section*{{Content}}\label{content}

\noindent\hyperlink{categorical_preliminaries}{Categorical Preliminaries}\dotfill \pageref*{categorical_preliminaries} \linebreak
\noindent\hyperlink{i_categories_of_functors}{I Categories of Functors}\dotfill \pageref*{i_categories_of_functors} \linebreak
\noindent\hyperlink{ii_sheaves_of_sets}{II Sheaves of Sets}\dotfill \pageref*{ii_sheaves_of_sets} \linebreak
\noindent\hyperlink{iii_grothendieck_topologies_and_sheaves}{III Grothendieck Topologies and Sheaves}\dotfill \pageref*{iii_grothendieck_topologies_and_sheaves} \linebreak
\noindent\hyperlink{iv_first_properties_of_elementary_topoi}{IV First Properties of Elementary Topoi}\dotfill \pageref*{iv_first_properties_of_elementary_topoi} \linebreak
\noindent\hyperlink{1_definition_of_a_topos}{1. Definition of a topos}\dotfill \pageref*{1_definition_of_a_topos} \linebreak
\noindent\hyperlink{2_the_construction_of_exponentials}{2. The construction of exponentials}\dotfill \pageref*{2_the_construction_of_exponentials} \linebreak
\noindent\hyperlink{3_direct_image}{3. Direct image}\dotfill \pageref*{3_direct_image} \linebreak
\noindent\hyperlink{4_monads_and_becks_theorem}{4. Monads and Beck's theorem}\dotfill \pageref*{4_monads_and_becks_theorem} \linebreak
\noindent\hyperlink{5_the_construction_of_colimits}{5. The construction of colimits}\dotfill \pageref*{5_the_construction_of_colimits} \linebreak
\noindent\hyperlink{6_factorization_and_images}{6. Factorization and images}\dotfill \pageref*{6_factorization_and_images} \linebreak
\noindent\hyperlink{7_the_slice_category_as_a_topos}{7. The slice category as a topos}\dotfill \pageref*{7_the_slice_category_as_a_topos} \linebreak
\noindent\hyperlink{8_lattice_and_heyting_algebra_objects_in_a_topos}{8. Lattice and Heyting algebra objects in a topos}\dotfill \pageref*{8_lattice_and_heyting_algebra_objects_in_a_topos} \linebreak
\noindent\hyperlink{9_the_beckchevalley_condition}{9. The Beck-Chevalley condition}\dotfill \pageref*{9_the_beckchevalley_condition} \linebreak
\noindent\hyperlink{10_injective_objects}{10. Injective objects}\dotfill \pageref*{10_injective_objects} \linebreak
\noindent\hyperlink{v_basic_constructions_of_topoi}{V Basic Constructions of Topoi}\dotfill \pageref*{v_basic_constructions_of_topoi} \linebreak
\noindent\hyperlink{1_lawveretierney_topologies}{1. Lawvere-Tierney topologies}\dotfill \pageref*{1_lawveretierney_topologies} \linebreak
\noindent\hyperlink{2_sheaves}{2. Sheaves}\dotfill \pageref*{2_sheaves} \linebreak
\noindent\hyperlink{3_the_associated_sheaf_functor}{3. The associated sheaf functor}\dotfill \pageref*{3_the_associated_sheaf_functor} \linebreak
\noindent\hyperlink{4_lawveretierney_subsumes_grothendieck}{4. Lawvere-Tierney subsumes Grothendieck}\dotfill \pageref*{4_lawveretierney_subsumes_grothendieck} \linebreak
\noindent\hyperlink{5_internal_versus_external}{5. Internal versus external}\dotfill \pageref*{5_internal_versus_external} \linebreak
\noindent\hyperlink{6_group_actions}{6. Group actions}\dotfill \pageref*{6_group_actions} \linebreak
\noindent\hyperlink{7_category_actions}{7. Category actions}\dotfill \pageref*{7_category_actions} \linebreak
\noindent\hyperlink{8_the_topos_of_coalgebras}{8. The topos of coalgebras}\dotfill \pageref*{8_the_topos_of_coalgebras} \linebreak
\noindent\hyperlink{9_the_filterquotient_construction}{9. The filter-quotient construction}\dotfill \pageref*{9_the_filterquotient_construction} \linebreak
\noindent\hyperlink{vi_topoi_and_logic}{VI Topoi and Logic}\dotfill \pageref*{vi_topoi_and_logic} \linebreak
\noindent\hyperlink{vii_geometric_morphisms}{VII Geometric Morphisms}\dotfill \pageref*{vii_geometric_morphisms} \linebreak
\noindent\hyperlink{vii_1_geometric_morphisms_and_basic_examples}{VII 1. Geometric Morphisms and Basic Examples}\dotfill \pageref*{vii_1_geometric_morphisms_and_basic_examples} \linebreak
\noindent\hyperlink{vii_2_tensor_products}{VII 2. Tensor products}\dotfill \pageref*{vii_2_tensor_products} \linebreak
\noindent\hyperlink{vii_3_group_actions}{VII 3. Group actions}\dotfill \pageref*{vii_3_group_actions} \linebreak
\noindent\hyperlink{vii_4_embeddings_and_surjections}{VII 4. Embeddings and surjections}\dotfill \pageref*{vii_4_embeddings_and_surjections} \linebreak
\noindent\hyperlink{vii_5_points}{VII 5. Points}\dotfill \pageref*{vii_5_points} \linebreak
\noindent\hyperlink{vii_6_filtering_functors}{VII 6. Filtering functors}\dotfill \pageref*{vii_6_filtering_functors} \linebreak
\noindent\hyperlink{vii_7_morphisms_into_grothendieck_topoi}{VII 7. Morphisms into Grothendieck Topoi}\dotfill \pageref*{vii_7_morphisms_into_grothendieck_topoi} \linebreak
\noindent\hyperlink{vii_8_filtering_functors_into_a_topos}{VII 8. Filtering functors into a topos}\dotfill \pageref*{vii_8_filtering_functors_into_a_topos} \linebreak
\noindent\hyperlink{vii_9_geometric_morphisms_as_filtering_functors}{VII 9. Geometric morphisms as filtering functors}\dotfill \pageref*{vii_9_geometric_morphisms_as_filtering_functors} \linebreak
\noindent\hyperlink{vii_10_morphisms_between_sites}{VII 10. Morphisms between sites}\dotfill \pageref*{vii_10_morphisms_between_sites} \linebreak
\noindent\hyperlink{viii_classifying_topoi}{VIII Classifying Topoi}\dotfill \pageref*{viii_classifying_topoi} \linebreak
\noindent\hyperlink{ix_localic_topoi}{IX Localic Topoi}\dotfill \pageref*{ix_localic_topoi} \linebreak
\noindent\hyperlink{geometric_logic_and_classifying_topoi}{Geometric Logic and Classifying Topoi}\dotfill \pageref*{geometric_logic_and_classifying_topoi} \linebreak
\noindent\hyperlink{appendix_sites_for_topoi}{Appendix: Sites for Topoi}\dotfill \pageref*{appendix_sites_for_topoi} \linebreak


\hypertarget{categorical_preliminaries}{}\subsection*{{Categorical Preliminaries}}\label{categorical_preliminaries}

\hypertarget{i_categories_of_functors}{}\subsection*{{I Categories of Functors}}\label{i_categories_of_functors}

\begin{itemize}%
\item [[presheaf]]


\item [[category of presheaves]]



\end{itemize}
\hypertarget{ii_sheaves_of_sets}{}\subsection*{{II Sheaves of Sets}}\label{ii_sheaves_of_sets}

\begin{itemize}%
\item [[category of open subsets]]


\item [[sheaf]]


\item [[global section]]/[[constant sheaf]]



\end{itemize}
\hypertarget{iii_grothendieck_topologies_and_sheaves}{}\subsection*{{III Grothendieck Topologies and Sheaves}}\label{iii_grothendieck_topologies_and_sheaves}

\begin{itemize}%
\item [[presheaf]]


\item [[site]]

\begin{itemize}%
\item [[coverage]]


\item [[basis for a topology]]


\item [[Grothendieck topology]]


\item [[sheaf]]

\begin{itemize}%
\item [[sheafification]]


\item [[direct image]]


\item [[inverse image]]


\item [[restriction and extension of sheaves]]


\item [[stalk]]



\end{itemize}

\item [[category of sheaves]]



\end{itemize}

\item [[Grothendieck topos]]



\end{itemize}
\hypertarget{iv_first_properties_of_elementary_topoi}{}\subsection*{{IV First Properties of Elementary Topoi}}\label{iv_first_properties_of_elementary_topoi}

\hypertarget{1_definition_of_a_topos}{}\subsubsection*{{1. Definition of a topos}}\label{1_definition_of_a_topos}

\begin{itemize}%
\item [[subobject classifier]]

\begin{itemize}%
\item [[subobject]]

\end{itemize}

\item [[topos]]

\begin{itemize}%
\item [[Grothendieck topos]]


\item [[well-pointed topos]]


\item [[Boolean topos]]


\item [[pretopos]]


\item [[universe in a topos]]



\end{itemize}

\item [[point of a topos]]


\item [[fundamental group of a topos]]


\item [[dependent product]]



\end{itemize}
\hypertarget{2_the_construction_of_exponentials}{}\subsubsection*{{2. The construction of exponentials}}\label{2_the_construction_of_exponentials}

\begin{itemize}%
\item [[exponential object]]

\end{itemize}
\hypertarget{3_direct_image}{}\subsubsection*{{3. Direct image}}\label{3_direct_image}

\begin{itemize}%
\item [[direct image]]

\end{itemize}
\hypertarget{4_monads_and_becks_theorem}{}\subsubsection*{{4. Monads and Beck's theorem}}\label{4_monads_and_becks_theorem}

\begin{itemize}%
\item [[monad]]


\item [[Eilenberg-Moore category]]


\item [[monadic functor]]


\item [[monadicity theorem]]



\end{itemize}
\hypertarget{5_the_construction_of_colimits}{}\subsubsection*{{5. The construction of colimits}}\label{5_the_construction_of_colimits}

\begin{itemize}%
\item [[quotient type]]

\end{itemize}
\hypertarget{6_factorization_and_images}{}\subsubsection*{{6. Factorization and images}}\label{6_factorization_and_images}

\begin{itemize}%
\item [[image]]

\end{itemize}
\hypertarget{7_the_slice_category_as_a_topos}{}\subsubsection*{{7. The slice category as a topos}}\label{7_the_slice_category_as_a_topos}

\begin{itemize}%
\item [[over-topos]]

\end{itemize}
\hypertarget{8_lattice_and_heyting_algebra_objects_in_a_topos}{}\subsubsection*{{8. Lattice and Heyting algebra objects in a topos}}\label{8_lattice_and_heyting_algebra_objects_in_a_topos}

\begin{itemize}%
\item [[lattice]]


\item [[Heyting algebra]]



\end{itemize}
\hypertarget{9_the_beckchevalley_condition}{}\subsubsection*{{9. The Beck-Chevalley condition}}\label{9_the_beckchevalley_condition}

\begin{itemize}%
\item [[Beck-Chevalley condition]]

\end{itemize}
\hypertarget{10_injective_objects}{}\subsubsection*{{10. Injective objects}}\label{10_injective_objects}

\begin{itemize}%
\item [[injective objects]]

\end{itemize}
\hypertarget{v_basic_constructions_of_topoi}{}\subsection*{{V Basic Constructions of Topoi}}\label{v_basic_constructions_of_topoi}

\hypertarget{1_lawveretierney_topologies}{}\subsubsection*{{1. Lawvere-Tierney topologies}}\label{1_lawveretierney_topologies}

\begin{itemize}%
\item [[Lawvere-Tierney topology]]

\end{itemize}
\hypertarget{2_sheaves}{}\subsubsection*{{2. Sheaves}}\label{2_sheaves}

\begin{itemize}%
\item [[dense monomorphism]]


\item [[sheaf]]



\end{itemize}
\hypertarget{3_the_associated_sheaf_functor}{}\subsubsection*{{3. The associated sheaf functor}}\label{3_the_associated_sheaf_functor}

\begin{itemize}%
\item [[sheafification]]

\end{itemize}
\hypertarget{4_lawveretierney_subsumes_grothendieck}{}\subsubsection*{{4. Lawvere-Tierney subsumes Grothendieck}}\label{4_lawveretierney_subsumes_grothendieck}

\begin{itemize}%
\item [[Grothendieck topology]]

\end{itemize}
\hypertarget{5_internal_versus_external}{}\subsubsection*{{5. Internal versus external}}\label{5_internal_versus_external}

\hypertarget{6_group_actions}{}\subsubsection*{{6. Group actions}}\label{6_group_actions}

\begin{itemize}%
\item [[group]]


\item [[action]]


\item [[permutation representation]]



\end{itemize}
\hypertarget{7_category_actions}{}\subsubsection*{{7. Category actions}}\label{7_category_actions}

\hypertarget{8_the_topos_of_coalgebras}{}\subsubsection*{{8. The topos of coalgebras}}\label{8_the_topos_of_coalgebras}

\begin{itemize}%
\item [[comonad]]


\item [[topos of coalgebras]]



\end{itemize}
\hypertarget{9_the_filterquotient_construction}{}\subsubsection*{{9. The filter-quotient construction}}\label{9_the_filterquotient_construction}

\hypertarget{vi_topoi_and_logic}{}\subsection*{{VI Topoi and Logic}}\label{vi_topoi_and_logic}

\begin{itemize}%
\item [[ETCS]]


\item [[continuum hypothesis]]


\item [[logic]]


\item [[foundations and logic]]

\begin{itemize}%
\item [[internal logic]]

\begin{itemize}%
\item [[intuitionistic logic]]


\item [[classical logic]]


\item [[paraconsistent logic]]


\item [[coherent logic]]



\end{itemize}


\end{itemize}

\item [[Boolean algebra]]


\item [[Heyting algebra]]



\end{itemize}
\hypertarget{vii_geometric_morphisms}{}\subsection*{{VII Geometric Morphisms}}\label{vii_geometric_morphisms}

\hypertarget{vii_1_geometric_morphisms_and_basic_examples}{}\subsubsection*{{VII 1. Geometric Morphisms and Basic Examples}}\label{vii_1_geometric_morphisms_and_basic_examples}

\begin{itemize}%
\item [[geometric morphism]]

\begin{itemize}%
\item [[direct image]], [[inverse image]]

\end{itemize}


\end{itemize}
Examples:

\begin{itemize}%
\item [[continuous function]]


\item [[base change geometric morphism]]


\item [[sheafification]], [[geometric embedding]]


\item [[global section geometric morphism]]



\end{itemize}
\hypertarget{vii_2_tensor_products}{}\subsubsection*{{VII 2. Tensor products}}\label{vii_2_tensor_products}

\begin{itemize}%
\item [[Kan extension]]

\end{itemize}
\hypertarget{vii_3_group_actions}{}\subsubsection*{{VII 3. Group actions}}\label{vii_3_group_actions}

\begin{itemize}%
\item [[group object]]

\end{itemize}
\hypertarget{vii_4_embeddings_and_surjections}{}\subsubsection*{{VII 4. Embeddings and surjections}}\label{vii_4_embeddings_and_surjections}

\begin{itemize}%
\item [[geometric embedding]]


\item [[geometric surjection]]


\item [[geometric surjection/embedding factorization]]



\end{itemize}
\hypertarget{vii_5_points}{}\subsubsection*{{VII 5. Points}}\label{vii_5_points}

\begin{itemize}%
\item [[point of a topos]]


\item [[stalk]]



\end{itemize}
\hypertarget{vii_6_filtering_functors}{}\subsubsection*{{VII 6. Filtering functors}}\label{vii_6_filtering_functors}

\hypertarget{vii_7_morphisms_into_grothendieck_topoi}{}\subsubsection*{{VII 7. Morphisms into Grothendieck Topoi}}\label{vii_7_morphisms_into_grothendieck_topoi}

\hypertarget{vii_8_filtering_functors_into_a_topos}{}\subsubsection*{{VII 8. Filtering functors into a topos}}\label{vii_8_filtering_functors_into_a_topos}

\hypertarget{vii_9_geometric_morphisms_as_filtering_functors}{}\subsubsection*{{VII 9. Geometric morphisms as filtering functors}}\label{vii_9_geometric_morphisms_as_filtering_functors}

\hypertarget{vii_10_morphisms_between_sites}{}\subsubsection*{{VII 10. Morphisms between sites}}\label{vii_10_morphisms_between_sites}

\begin{itemize}%
\item [[morphism of sites]]


\item [[covering lifting property]]



\end{itemize}
\hypertarget{viii_classifying_topoi}{}\subsection*{{VIII Classifying Topoi}}\label{viii_classifying_topoi}

\begin{itemize}%
\item [[classifying topos]]

\end{itemize}
\hypertarget{ix_localic_topoi}{}\subsection*{{IX Localic Topoi}}\label{ix_localic_topoi}

\begin{itemize}%
\item [[locale]]


\item [[localic topos]]



\end{itemize}
\hypertarget{geometric_logic_and_classifying_topoi}{}\subsection*{{Geometric Logic and Classifying Topoi}}\label{geometric_logic_and_classifying_topoi}

\begin{itemize}%
\item [[classifying topos]]

\end{itemize}
\hypertarget{appendix_sites_for_topoi}{}\subsection*{{Appendix: Sites for Topoi}}\label{appendix_sites_for_topoi}

\begin{itemize}%
\item [[Giraud's theorem]]

\end{itemize}
category: reference

[[!redirects Sheaves in geometry and logic]] [[!redirects sheaves in geometry and logic]] [[!redirects Mac Lane-Moerdijk]]



\end{document}