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href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> </ul> </li> </ul> <h2 id="toposes">Toposes</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/site">site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">pretopology</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a></p> </li> </ul> <h2 id="internal_logic">Internal Logic</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a></p> </li> </ul> </li> </ul> <h2 id="topos_morphisms">Topos morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>/<a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+section">global sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+geometric+morphism">connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+geometric+morphism">totally connected geometric morphism</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+geometric+morphism">localic geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a></p> </li> </ul> </li> </ul> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+topos">localic topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos/gros topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> </li> </ul> <h2 id="cohomology_and_homotopy">Cohomology and homotopy</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-site">2-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></p> </li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Barr%27s+theorem">Barr's theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/topos+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="category_theory">Category Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#original_texts'>Original texts</a></li> <li><a href='#Introductions'>Introductions</a></li> <li><a href='#Textbooks'>Monographs</a></li> <li><a href='#course_notes'>Course notes</a></li> <li><a href='#history'>History</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Topos theory</em> is the part of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> that studies <a class="existingWikiWord" href="/nlab/show/categories">categories</a> which are <a class="existingWikiWord" href="/nlab/show/topos">topos</a>es. This includes in particular <a class="existingWikiWord" href="/nlab/show/Grothendieck+toposes">Grothendieck toposes</a>, i.e. <a class="existingWikiWord" href="/nlab/show/categories+of+sheaves">categories of sheaves</a>.</p> <p>There are always two ways to think of topos theory: as being</p> <ul> <li> <p>about <a class="existingWikiWord" href="/nlab/show/logic">logic</a></p> </li> <li> <p>about <a class="existingWikiWord" href="/nlab/show/higher+geometry">geometry</a>.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>topos theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Giraud%27s+theorem">Giraud's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+topos+theory">fundamental theorem of topos theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos+theory">2-topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microlocal+sheaf+theory">microlocal sheaf theory</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="original_texts">Original texts</h3> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>:</p> <ul> <li id="Godement58"><a class="existingWikiWord" href="/nlab/show/Roger+Godement">Roger Godement</a>, <em>Topologie algébrique et theorie des faisceaux</em>, Actualités Sci. Ind. <strong>1252</strong>, Hermann, Paris (1958) [<a href="https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement">webpage</a>, <a class="existingWikiWord" href="/nlab/files/Godement-TopologieAlgebrique.pdf" title="pdf">pdf</a>]</li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>:</p> <ul> <li id="SGA4"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Artin">Michael Artin</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, <a class="existingWikiWord" href="/nlab/show/Jean-Louis+Verdier">Jean-Louis Verdier</a> (eds.), <em>Théorie des Topos et Cohomologie Etale des Schémas - SGA 4</em> , LNM <strong>269</strong> Springer Heidelberg 1972.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Monique+Hakim">Monique Hakim</a>, <em>Topos annelés et schémas relatifs</em>, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972) (<a href="https://www.springer.com/gp/book/9783540055730">doi:10.1007/978-3-662-59155-0</a>)</p> </li> </ul> <h3 id="Introductions">Introductions</h3> <p>Brief expositions:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <a class="existingWikiWord" href="/nlab/show/Colin+McLarty">Colin McLarty</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Wells">Charles Wells</a>, <em>Variable set theory</em>, prepared for <em>Scientific American</em> but unpublished (~1985) [<a href="https://www.math.mcgill.ca/barr/papers/vst.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BarrMcLartyWells-VariableSetTheory.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Background in topos theory</em>, chapter I in: <em>Classifying Spaces and Classifying Topoi</em>, Lecture Notes in Mathematics <strong>1616</strong>, Springer (1995) [<a href="https://doi.org/10.1007/BFb0094441">doi:10.1007/BFb0094441</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Luc+Illusie">Luc Illusie</a>, <em>What is… a Topos?</em>, Notices of the AMS <strong>51</strong> 9 (2004) [<a href="https://www.ams.org/notices/200409/what-is-illusie.pdf">pdf</a>, full volume:<a href="https://www.ams.org/notices/200409/200409FullIssue.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tom+Leinster">Tom Leinster</a>, <em><a class="existingWikiWord" href="/nlab/show/Leinster2010">An informal introduction to topos theory</a></em> (2010) [<a href="https://arxiv.org/abs/1012.5647">arXiv:1012.5647</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <em>Topos Theory in a Nutshell</em> (2021) [<a href="http://math.ucr.edu/home/baez/topos.html">web</a>]</p> </li> <li> <p>MathProofsable, Category Theory - Toposes <a href="https://www.youtube.com/watch?v=gKYpvyQPhZo&list=PL4FD0wu2mjWM3ZSxXBj4LRNsNKWZYaT7k">video playlist</a></p> </li> </ul> <p>Lecture notes:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>A survey of topos theory</em>, lecture notes (1978) [<a href="http://www.math.mq.edu.au/~street/ToposSurvey.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Street-SurveyToposTheory.pdf" title="pdf">pdf</a>]</p> </li> <li id="Wyler"> <p><a class="existingWikiWord" href="/nlab/show/Oswald+Wyler">Oswald Wyler</a>, <em>Lecture Notes on Topoi and Quasitopoi</em>, World Scientific (1991) [<a href="https://doi.org/10.1142/1047">doi:10.1142/1047</a>]</p> </li> <li id="Joyal15"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em><a class="existingWikiWord" href="/nlab/show/A+crash+course+in+topos+theory+--+The+big+picture">A crash course in topos theory – The big picture</a></em>, lecture series at <a href="https://indico.math.cnrs.fr/event/747/">Topos à l’IHES</a>, Paris (November 2015)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em>Geometric aspects of topos theory in relation with logical doctrines</em>, talk at <em><a class="existingWikiWord" href="/nlab/show/New+Spaces+for+Mathematics+and+Physics">New Spaces for Mathematics and Physics</a></em>, IHP Paris 2015 (<a href="https://www.youtube.com/watch?v=kaZpOEOAUzE">video recording</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <a class="existingWikiWord" href="/nlab/show/Jaap+van+Oosten">Jaap van Oosten</a>, <em>Topos Theory</em>, Master class notes (2007) [<a href="http://www.staff.science.uu.nl/~ooste110/syllabi/toposmoeder.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jaap+van+Oosten">Jaap van Oosten</a>, <em>Topos Theory</em> (2018) [<a href="https://webspace.science.uu.nl/~ooste110/syllabi/topostheory.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/vanOosten-ToposTheory.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, <em>Some glances at topos theory</em>, lecture notes, Como (2018) [<a href="https://tcsc.lakecomoschool.org/files/2018/01/Borceux.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Borceux-ToposTheory.pdf" title="pdf">pdf</a>, <a href="https://www.youtube.com/watch?v=s_fN9euuVAY&list=PLh_3Q6ZRqWs0LBptMGClJ8OArR0fBT6Pp&index=11">video playlist</a>]</p> </li> <li id="Schapira23"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, <em>An Introduction to Categories and Sheaves</em>, lecture notes (2023) [<a href="https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/CatShv.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Schapira-Sheaves.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>A monograph that aims to be more expository, focusing on <a class="existingWikiWord" href="/nlab/show/presheaf+toposes">presheaf toposes</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marie+La+Palme+Reyes">Marie La Palme Reyes</a>, <a class="existingWikiWord" href="/nlab/show/Gonzalo+E.+Reyes">Gonzalo E. Reyes</a>, <a class="existingWikiWord" href="/nlab/show/Houman+Zolfaghari">Houman Zolfaghari</a>, <em>Generic figures and their glueings. A constructive approach to functor categories</em>, Polimetrica (2008) [<a href="https://marieetgonzalo.files.wordpress.com/2004/06/generic-figures.pdf">pdf</a>, ISBN:8876990046]</li> </ul> <h3 id="Textbooks">Monographs</h3> <ul> <li id="Johnstone77"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Topos theory</em>, London Math. Soc. Monographs <strong>10</strong>, Acad. Press 1977, xxiii+367 pp. (Available as Dover Reprint, Mineola 2014)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Wells">Charles Wells</a>, <em><a class="existingWikiWord" href="/nlab/show/Toposes%2C+Triples%2C+and+Theories">Toposes, Triples, and Theories</a></em> , Springer Heidelberg 1985. (Available as <a href="http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html">TAC reprint no.12</a> 2005)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Colin+McLarty">Colin McLarty</a>, <em><a class="existingWikiWord" href="/nlab/show/Elementary+Categories%2C+Elementary+Toposes">Elementary Categories, Elementary Toposes</a></em>, Oxford University Press 1992 (<a href="https://global.oup.com/academic/product/elementary-categories-elementary-toposes-9780198514732?cc=ae&lang=en&">ISBN:9780198514732</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em><a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a></em>, Springer (1992) [<a href="https://link.springer.com/book/10.1007/978-1-4612-0927-0">doi:10.1007/978-1-4612-0927-0</a>]</p> </li> <li id="Borceux94"> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a> 3 - Categories of Sheaves</em>, Cambridge UP (1994) [<a href="https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/handbook-categorical-algebra-volume-3?format=PB">ISBN:9780521061247</a>, <a href="https://doi.org/10.1017/CBO9780511525872">doi:10.1017/CBO9780511525872</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Elephant">Sketches of an elephant: a topos theory compendium</a></em> (2002)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexandru+Dimca">Alexandru Dimca</a>, <em>Sheaves in Topology</em>, Universitext, Springer (2004) [<a href="https://doi.org/10.1007/978-3-642-18868-8">doi:10.1007/978-3-642-18868-8</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Masaki+Kashiwara">Masaki Kashiwara</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, <em><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></em>, Grundlehren der Mathematischen Wissenschaften <strong>332</strong> Springer (2006) [<a href="https://link.springer.com/book/10.1007/3-540-27950-4">doi:10.1007/3-540-27950-4</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf">pdf</a>]</p> </li> <li id="Warner12"> <p><a class="existingWikiWord" href="/nlab/show/Garth+Warner">Garth Warner</a>, <em>Homotopical Topos Theory</em>, EPrint Collection, University of Washington (2012) [<a href="http://hdl.handle.net/1773/19722">hdl:1773/19722</a>, <a href="https://sites.math.washington.edu//~warner/HTT_Warner.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Waner-HomotopicalTopos.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>Introducing even category theory from the scratch while still managing to cover some ground, the following textbook is the <em>royal road to topos theory</em> for people with some background in <a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a>:</p> <ul> <li id="Goldblatt84"><a class="existingWikiWord" href="/nlab/show/Robert+Goldblatt">Robert Goldblatt</a>, <em>Topoi: The Categorial Analysis of Logic</em>, 2nd ed. North-Holland Amsterdam 1984. (<a href="https://store.doverpublications.com/0486450260.html">Dover reprint</a> New York 2006; free online at <a href="http://projecteuclid.org/euclid.bia/1403013939">Project Euclid</a>.)</li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Olivia+Caramello">Olivia Caramello</a>, <em>Theories, Sites, Toposes</em>, Oxford UP 2017.</li> </ul> <h3 id="course_notes">Course notes</h3> <p>A survey is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>A survey of topos theory</em> (notes for students, 1978) <a href="http://www.math.mq.edu.au/~street/ToposSurvey.pdf">pdf</a></li> </ul> <p>A nice and concise introduction is available in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, <em>Some glances at topos theory</em> , lecture notes Como 2018. (<a href="http://tcsc.lakecomoschool.org/files/2018/06/Como2018.pdf">pdf</a>, <a href="https://www.youtube.com/watch?v=s_fN9euuVAY&list=PLh_3Q6ZRqWs0LBptMGClJ8OArR0fBT6Pp&index=11">video playlist</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <a class="existingWikiWord" href="/nlab/show/Jaap+van+Oosten">Jaap van Oosten</a>, <em>Topos theory</em> Master Class notes (2007) (<a href="http://www.staff.science.uu.nl/~ooste110/syllabi/toposmoeder.pdf">pdf</a>)</p> </li> </ul> <h3 id="history">History</h3> <ul> <li id="Lawvere00"> <p><a class="existingWikiWord" href="/nlab/show/F.+William+Lawvere">F. William Lawvere</a>, <em>Comments on the development of topos theory</em>, pp.715-734 in Pier (ed.), <em>Development of Mathematics 1950 - 2000</em> , Birkhäuser Basel 2000. (<a href="http://www.tac.mta.ca/tac/reprints/articles/24/tr24abs.html">tac reprint</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Colin+McLarty">Colin McLarty</a>, <em>The Uses and Abuses of the History of Topos Theory</em> , Brit. J. Phil. Sci., 41 (1990) (<a href="http://www.jstor.org/stable/687825">JSTOR</a>) <a href="http://bjps.oxfordjournals.org/content/41/3/351.full.pdf">PDF</a></p> </li> </ul> <p>A historical analysis of Grothendieck’s 1973 Buffalo lecture series on toposes and their precedents is in</p> <ul> <li id="McLarty18"> <p><a class="existingWikiWord" href="/nlab/show/Colin+McLarty">Colin McLarty</a>, <em>Grothendieck’s 1973 topos lectures</em>, Séminaire Lectures grothendieckiennes, 3 May (2018) (<a href="https://www.youtube.com/watch?v=5AR55ZsHmKI">YouTube video</a>)</p> </li> <li> <p><em><a href="http://www.wra1th.plus.com/gcw/math/PSSL/index.html">The peripatetic seminar on sheaves and logic 1976-1999</a></em></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 29, 2024 at 10:25:46. 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