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<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/%28infinity%2C1%29-category'>(∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-functor'>(∞,1)-functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-presheaf'>(∞,1)-presheaf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves'>(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> <h2 id='sidebar_definitions'>Definitions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/elementary+%28infinity%2C1%29-topos'>elementary (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-site'>(∞,1)-site</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/reflective+sub-%28infinity%2C1%29-category'>reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/localization+of+an+%28infinity%2C1%29-category'>localization of an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/topological+localization'>topological localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/hypercompletion'>hypercompletion</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-sheaf'>(∞,1)-sheaf</a>/<a class='existingWikiWord' href='/nlab/show/infinity-stack'>∞-stack</a>/<a class='existingWikiWord' href='/nlab/show/derived+stack'>derived stack</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-topos'>(∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28n%2C1%29-topos'>(n,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/n-topos'>n-topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/truncation'>n-truncated object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/connected+object'>n-connected object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/topos'>(1,1)-topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/presheaf'>presheaf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/sheaf'>sheaf</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/2-topos'>(2,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/2-topos'>2-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/%282%2C1%29-presheaf'>(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-quasitopos'>(∞,1)-quasitopos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/separated+%28infinity%2C1%29-presheaf'>separated (∞,1)-presheaf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/quasitopos'>quasitopos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/separated+presheaf'>separated presheaf</a></li> </ul> </li> <li> <p><span class='newWikiWord'>(2,1)-quasitopos<a href='/nlab/new/%282%2C1%29-quasitopos'>?</a></span></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/separated+%282%2C1%29-presheaf'>separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C2%29-topos'>(∞,2)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2Cn%29-topos'>(∞,n)-topos</a></p> </li> </ul> <h2 id='sidebar_characterization'>Characterization</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/pullback-stable+colimit'>universal colimits</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos'>object classifier</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category'>groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/effective+epimorphism'>effective epimorphism</a></li> </ul> </li> </ul> <h2 id='sidebar_morphisms'>Morphisms</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-geometric+morphism'>(∞,1)-geometric morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29Topos'>(∞,1)Topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Lawvere+distribution'>Lawvere distribution</a></p> </li> </ul> <h2 id='sidebar_extra'>Extra stuff, structure and property</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/hypercomplete+%28infinity%2C1%29-topos'>hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/hypercomplete+object'>hypercomplete object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/over-%28infinity%2C1%29-topos'>over-(∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/n-localic+%28infinity%2C1%29-topos'>n-localic (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/locally+n-connected+%28n%2B1%2C1%29-topos'>locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/structured+%28infinity%2C1%29-topos'>structured (∞,1)-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/geometry+%28for+structured+%28infinity%2C1%29-toposes%29'>geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/locally+n-connected+%28n%2B1%2C1%29-topos'>locally ∞-connected (∞,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/locally+n-connected+%28n%2B1%2C1%29-topos'>∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-local+geometric+morphism'>local (∞,1)-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/concrete+%28infinity%2C1%29-sheaf'>concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/cohesive+%28infinity%2C1%29-topos'>cohesive (∞,1)-topos</a></p> </li> </ul> <h2 id='sidebar_models'>Models</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/presentations+of+%28infinity%2C1%29-sheaf+%28infinity%2C1%29-toposes'>models for ∞-stack (∞,1)-toposes</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/model+category'>model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+functors'>model structure on functors</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/model+site'>model site</a>/<a class='existingWikiWord' href='/nlab/show/sSet-site'>sSet-site</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+presheaves'>model structure on simplicial presheaves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/descent+for+simplicial+presheaves'>descent for simplicial presheaves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves'>descent for presheaves with values in strict ∞-groupoids</a></p> </li> </ul> </li> </ul> <h2 id='sidebar_constructions'>Constructions</h2> <p><strong>structures in a <a class='existingWikiWord' href='/nlab/show/cohesive+%28infinity%2C1%29-topos'>cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/shape+of+an+%28infinity%2C1%29-topos'>shape</a> / <a class='existingWikiWord' href='/nlab/show/coshape+of+an+%28infinity%2C1%29-topos'>coshape</a></p> </li> <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> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical</a>/<a class='existingWikiWord' href='/nlab/show/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric</a> homotopy groups</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Postnikov+tower+in+an+%28infinity%2C1%29-category'>Postnikov tower</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Whitehead+tower+in+an+%28infinity%2C1%29-topos'>Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/function+algebras+on+infinity-stacks'>rational homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/dimension'>dimension</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy+dimension'>homotopy dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/cohomological+dimension'>cohomological dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/covering+dimension'>covering dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Heyting+dimension'>Heyting dimension</a></p> </li> </ul> </li> </ul> <div> <p> <a href='/nlab/edit/%28infinity%2C1%29-topos+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#examples'>Examples</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>The notion of <strong>model site</strong> is a <a class='existingWikiWord' href='/nlab/show/model+category'>model category</a> model for the notion of <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-site'>(∞,1)-site</a>.</p> <p>An <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category'>(∞,1)-category</a> equipped with the structure of a <a class='existingWikiWord' href='/nlab/show/site'>site</a> on its <a class='existingWikiWord' href='/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy category</a> serves as a domain for <a class='existingWikiWord' href='/nlab/show/infinity-stack'>∞-stack</a>s. It may be modeled by a <a class='existingWikiWord' href='/nlab/show/model+category'>model category</a>.</p> <p>A <a class='existingWikiWord' href='/nlab/show/model+category'>model category</a> with a <a class='existingWikiWord' href='/nlab/show/site'>site</a> structure on its <a class='existingWikiWord' href='/nlab/show/homotopy+category'>homotopy category</a> is called a pseudo <strong>model site</strong> (<a href='http://arxiv.org/PS_cache/math/pdf/0207/0207028v4.pdf#page=43'>def 4.1.1</a> in <a href='http://arxiv.org/abs/math/0404373'>ToVeI</a>).</p> <h2 id='examples'>Examples</h2> <ul> <li>The <a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+algebras'>model structure on simplicial algebras</a> equipped with the covering of the <a class='existingWikiWord' href='/nlab/show/%C3%A9tale+%28infinity%2C1%29-site'>etale (infinity,1)-site</a>.</li> </ul> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/site'>site</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/2-site'>2-site</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-site'>(∞,1)-site</a></p> <ul> <li> <p><strong>model site</strong>, <a class='existingWikiWord' href='/nlab/show/sSet-site'>simplicial site</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/model+topos'>model topos</a></p> </li> </ul> </li> </ul> <h2 id='references'>References</h2> <ul id='ToenVezzosi'> <li><a class='existingWikiWord' href='/nlab/show/Bertrand+To%C3%ABn'>Bertrand Toën</a>, <a class='existingWikiWord' href='/nlab/show/Gabriele+Vezzosi'>Gabriele Vezzosi</a>, <em>Homotopical Algebraic Geometry I: Topos theory</em> (<a href='http://arxiv.org/abs/math/0207028'>arXiv</a>)</li> </ul> <p> </p> <p> </p> </div>  </div>  </body> </html>