[[!redirects (∞,1)-topos theory]] +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### $(\infty,1)$-Category theory +--{: .hide} [[!include quasi-category theory contents]] =-- #### $(\infty,1)$-Topos Theory +--{: .hide} [[!include (infinity,1)-topos - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The theory of [[(∞,1)-toposes]], generalizing [[topos theory]] from [[category theory]] to [[(∞,1)-category theory]]: "[[geometric homotopy theory]]". ## Related concepts * [[topos theory]] * [[2-topos theory]] * **$(\infty,1)$-topos theory** * [[higher topos theory]] * [[Awodey's conjecture]] ## References > For more see the references at *[[(infinity,1)-topos]]*: Lecture notes: * [[Charles Rezk]]: _Lectures on Higher Topos Theory_, Leeds (2019) [[pdf](https://rezk.web.illinois.edu/leeds-lectures-2019.pdf), [[RezkHigherToposTheory2019.pdf:file]]] An quick introduction is in parts 3-4 of * {#Joyal15} [[André Joyal]]: _[[A crash course in topos theory -- The big picture]]_, lecture series at [Topos à l'IHES](https://indico.math.cnrs.fr/event/747/), Paris (November 2015) [videos: part [1](https://youtu.be/Ro8KoFFdtS4), [2](https://youtu.be/4vxzK4LjFlY), [3](https://youtu.be/u_cl0qiJxWY), [4](https://youtu.be/xzyxgJBNW8w)] For origins of the notion of $(\infty,1)$-topos itself see the references at *[[(∞,1)-topos]]*. Early frameworks for [[Grothendieck topos|Grothendieck]] (as opposed to "elementary") $(\infty,1)$-topoi are due [[Charles Rezk]] via [[model categories]] * {#Rezk10} [[Charles Rezk]], _Toposes and homotopy toposes_, 2010 ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf)) and due to [[Bertrand Toen|Toën]]--[[Gabriele Vezzosi|Vezzosi]] in two versions (preprints 2002), via [[simplicially enriched category|simplically enriched categories]] and via [[Segal categories]]: * [[Bertrand Toën]], [[Gabriele Vezzosi]]: *Homotopical algebraic geometry. I. Topos theory*, Adv. Math. **193** 2 (2005) 257-372 [[doi:10.1016/j.aim.2004.05.004](https://doi.org/10.1016/j.aim.2004.05.004), [arXiv:math.AT/0207028](http://arxiv.org/abs/math/0207028)] * [[Bertrand Toën]] , [[Gabriele Vezzosi]], _Segal topoi and stacks over Segal categories_, [arXiv:math.AG/0212330](http://arXiv.org/abs/math/0212330) A general abstract conception of $(\infty,1)$-topos theory in terms of [[(∞,1)-category theory]] was given in * [[Jacob Lurie]], _[[Higher Topos Theory]]_ . The analog of the [[Sketches of an Elephant|Johnstone 2002]] for $(\infty,1)$-topos theory is still to be written.