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\begin{document}

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\section*{(infinity,1)-topos}

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

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{AsAGeometricEmbedding}{As a geometric embedding into a $(\infty,1)$-presheaf category}\dotfill \pageref*{AsAGeometricEmbedding} \linebreak
\noindent\hyperlink{GiraudAxioms}{By Giraud-Rezk-Lurie axioms}\dotfill \pageref*{GiraudAxioms} \linebreak
\noindent\hyperlink{morphisms}{Morphisms}\dotfill \pageref*{morphisms} \linebreak
\noindent\hyperlink{types_of_toposes}{Types of $(\infty,1)$-toposes}\dotfill \pageref*{types_of_toposes} \linebreak
\noindent\hyperlink{topological_localizations__sheaf_toposes}{Topological localizations / $(\infty,1)$-sheaf toposes}\dotfill \pageref*{topological_localizations__sheaf_toposes} \linebreak
\noindent\hyperlink{hypercomplete_toposes}{Hypercomplete $(\infty,1)$-toposes}\dotfill \pageref*{hypercomplete_toposes} \linebreak
\noindent\hyperlink{cubical_type_theory}{Cubical type theory}\dotfill \pageref*{cubical_type_theory} \linebreak
\noindent\hyperlink{models}{Models}\dotfill \pageref*{models} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{global_sections_geometric_morphism}{Global sections geometric morphism}\dotfill \pageref*{global_sections_geometric_morphism} \linebreak
\noindent\hyperlink{Powering}{Powering and copowering over $\infty Grpd$ -- Hochschild homology}\dotfill \pageref*{Powering} \linebreak
\noindent\hyperlink{ClosedMonoidalStructure}{Closed monoidal structure}\dotfill \pageref*{ClosedMonoidalStructure} \linebreak
\noindent\hyperlink{overtoposes}{Over-$(\infty,1)$-toposes}\dotfill \pageref*{overtoposes} \linebreak
\noindent\hyperlink{syntax_in_univalent_homotopy_type_theory}{Syntax in univalent homotopy type theory}\dotfill \pageref*{syntax_in_univalent_homotopy_type_theory} \linebreak
\noindent\hyperlink{ToposTheory}{$(\infty,1)$-Topos theory}\dotfill \pageref*{ToposTheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{giraudrezklurie_axioms}{Giraud-Rezk-Lurie axioms}\dotfill \pageref*{giraudrezklurie_axioms} \linebreak
\noindent\hyperlink{homotopy_type_theory}{Homotopy type theory}\dotfill \pageref*{homotopy_type_theory} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Recall the following familiar 1-categorical statement:

\begin{itemize}%
\item Working in the 1-[[category]] [[Set]] of [[0-category|0-categories]] amounts to doing [[set theory]]. The point of [[sheaf topos|sheaf]] [[toposes]] is to pass to \emph{parameterized} [[0-category|0-categories]], namely [[presheaf]] categories. Although these [[topos|topoi]] behave much like the 1-topos [[Set]], their objects are generalized [[spaces]] that may carry more structure. For instance, a (pre)[[sheaf]] on [[Diff]] is a [[generalized smooth space]].

\end{itemize}
The idea of $(\infty,1)$-toposes is to generalize the above situation from $1$ to $(\infty,1)$ (recall the notion of [[(n,r)-category]] and see the general discussion at [[∞-topos]]):

\begin{itemize}%
\item Working in the [[(∞,1)-category]] [[∞Grpd]] of [[infinity-groupoid|(∞,0)-categories]] amounts to doing [[homotopy theory]]. The point of [[(∞,1)-sheaves]] is to pass to \emph{parameterized} [[(∞,0)-categories]], namely [[(∞,1)-presheaf]] categories. Although these $(\infty,1)$-topoi behave much like the $(\infty,1)$-topos [[∞Grpd]], their objects are generalized [[spaces]] with higher [[homotopies]] that may carry more structure. More generally we have topoi of [[sheaves]], and $(\infty,1)$-topoi of [[(∞,1)-sheaves]]. For instance, an [[∞-Lie groupoid]] is an [[(∞,1)-sheaf]] on [[CartSp]].

\end{itemize}
\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\hypertarget{AsAGeometricEmbedding}{}\subsubsection*{{As a geometric embedding into a $(\infty,1)$-presheaf category}}\label{AsAGeometricEmbedding}

Recall that [[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]] and that the inclusion functor is necessarily an [[accessible functor]]. This characterization has the following immediate generalization to a definition in [[(∞,1)-category theory]], where the only subtlety is that accessibility needs to be explicitly required:

\begin{defn}
\label{ToposByLocalization}\hypertarget{ToposByLocalization}{}
A [[Alexander Grothendieck|Grothendieck]]--[[Charles Rezk|Rezk]]--[[Jacob Lurie|Lurie]] \textbf{$(\infty,1)$-topos} $\mathbf{H}$ is an \href{reflective%20sub-%28infinity,1%29-category#AccessibleReflectiveSubcategory}{accessible} \href{reflective+sub-%28infinity%2C1%29-category#ExactLocalizations}{left exact} [[reflective sub-(∞,1)-category]] of an [[(∞,1)-category of (∞,1)-presheaves]]

\begin{displaymath}
\mathbf{H}
  \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow}
  PSh_{(\infty,1)}(C)
  \,.
\end{displaymath}
If the localization in is a [[topological localization]] then $\mathbf{H}$ is an \textbf{[[(∞,1)-category of (∞,1)-sheaves]]}.

\end{defn}
\hypertarget{GiraudAxioms}{}\subsubsection*{{By Giraud-Rezk-Lurie axioms}}\label{GiraudAxioms}

Equivalently:

\begin{prop}
\label{}\hypertarget{}{}
An $(\infty,1)$-topos $\mathbf{H}$ is

an [[(∞,1)-category]] that satisfies the $(\infty,1)$-categorical analogs of [[Giraud's axioms]]:

\begin{itemize}%
\item $\mathbf{H}$ is [[presentable (infinity,1)-category|presentable]];
\item [[limit in quasi-categories|(∞,1)-colimits]] in $\mathbf{H}$ [[universal colimits|are universal]];
\item [[coproduct]]s in $\mathbf{H}$ are [[disjoint coproduct|disjoint]];
\item every [[groupoid object in an (infinity,1)-category|groupoid object]] in $\mathbf{H}$ is [[quotient object|effective]] (i.e. has a [[delooping]]).

\end{itemize}
\end{prop}
This is part of the statement of [[Higher Topos Theory|HTT, theorem 6.1.0.6]].

This is derived from the following equivalent one:

\begin{prop}
\label{CharacterizationByObjectClassifier}\hypertarget{CharacterizationByObjectClassifier}{}
An [[(∞,1)-topos]] is

\begin{itemize}%
\item a [[presentable (∞,1)-category]] with [[universal colimits]]


\item that has [[object classifiers]].



\end{itemize}
\end{prop}
\begin{remark}
\label{ReflectonOnCharacterizationByObjectClassifier}\hypertarget{ReflectonOnCharacterizationByObjectClassifier}{}
An [[object classifier]] is a (small) \emph{self-reflections} of the $\infty$-topos inside itself ([[type of types]], internal [[universe]]). See also (\href{Science+of+Logic#WesenAlsReflexionInIhmSelbst}{WdL, book 2, section 1}).

\end{remark}
A further equivalent one (essentially by an invocation of the adjoint functor theorem) is:

\begin{prop}
\label{}\hypertarget{}{}
An [[(∞,1)-topos]] is

\begin{itemize}%
\item a [[presentable (∞,1)-category]]


\item in which all colimits are [[van Kampen colimits]].



\end{itemize}
\end{prop}
\hypertarget{morphisms}{}\subsubsection*{{Morphisms}}\label{morphisms}

A [[morphism]] between $(\infty,1)$-toposes is an [[(∞,1)-geometric morphism]].

The [[(∞,1)-category]] of all $(\infty,1)$-topos is [[(∞,1)Toposes]].

\hypertarget{types_of_toposes}{}\subsection*{{Types of $(\infty,1)$-toposes}}\label{types_of_toposes}

\hypertarget{topological_localizations__sheaf_toposes}{}\subsubsection*{{Topological localizations / $(\infty,1)$-sheaf toposes}}\label{topological_localizations__sheaf_toposes}

for the moment see

\begin{itemize}%
\item [[topological localization]]

\end{itemize}
\hypertarget{hypercomplete_toposes}{}\subsubsection*{{Hypercomplete $(\infty,1)$-toposes}}\label{hypercomplete_toposes}

for the moment see

\begin{itemize}%
\item [[hypercomplete (∞,1)-topos]]

\end{itemize}
\hypertarget{cubical_type_theory}{}\subsubsection*{{Cubical type theory}}\label{cubical_type_theory}

The Cartesian cubical model of [[cubical type theory]] and [[homotopy type theory]] is \href{https://groups.google.com/d/msg/homotopytypetheory/RQkLWZ_83kQ/s6iazlFdBgAJ}{conjectured} to be an (∞,1)-topos not equivalent to (∞,1)-groupoids.

\hypertarget{models}{}\subsection*{{Models}}\label{models}

Another main theorem about $(\infty,1)$-toposes is that [[models for ∞-stack (∞,1)-toposes]] are given by the [[model structure on simplicial presheaves]]. See there for details

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{global_sections_geometric_morphism}{}\subsubsection*{{Global sections geometric morphism}}\label{global_sections_geometric_morphism}

Every [[∞-stack]] $(\infty,1)$-topos $\mathbf{H}$ has a canonical [[(∞,1)-geometric morphism]] to the terminal $\infty$-stack $(\infty,1)$-topos [[∞Grpd]]: the [[direct image]] is the [[global section]]s [[(∞,1)-functor]] $\Gamma$, the [[inverse image]] is the [[constant ∞-stack]] functor

\begin{displaymath}
(LConst \dashv \Gamma) : 
  \mathbf{H}
  \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}
  \infty Grpd
  \,.
\end{displaymath}
\hypertarget{Powering}{}\subsubsection*{{Powering and copowering over $\infty Grpd$ -- Hochschild homology}}\label{Powering}

Being a [[locally presentable (∞,1)-category]], an $(\infty,1)$-topos $\mathbf{H}$ is [[power]]ed and [[copower]]ed over [[∞Grpd]], as described at .

For any $K \in \infty Grpd$ and $X \in \mathbf{H}$ the powering is the [[(∞,1)-limit]] over the [[diagram]] constant on $X$

\begin{displaymath}
X^K = {\lim_\leftarrow}_K X
\end{displaymath}
and the $(\infty,1)$-copowering is is the [[(∞,1)-colimit]] over the diagram constant on $X$

\begin{displaymath}
K \cdot X = {\lim_{\to}}_K X
  \,.
\end{displaymath}
Under [[Isbell duality]] the powering operation corresponds to higher order [[Hochschild cohomology]] in $X$, as discussed there.

Below we discuss that the powering is equivalently given by the [[internal hom]] ([[mapping stack]]) out of the [[constant ∞-stack]] $LConst K$ on $K$:

\begin{displaymath}
X^K \simeq [LConst K, X]
  \,.
\end{displaymath}
\hypertarget{ClosedMonoidalStructure}{}\subsubsection*{{Closed monoidal structure}}\label{ClosedMonoidalStructure}

\begin{prop}
\label{}\hypertarget{}{}
Every $(\infty,1)$-topos is a [[cartesian closed (∞,1)-category]].

\end{prop}
\begin{proof}
By the fact that every $(\infty,1)$-topos $\mathbf{H}$ has [[universal colimits]] it follows that for every object $X$ the [[(∞,1)-functor]]

\begin{displaymath}
X \times (-) : \mathbf{H} \to \mathbf{H}
\end{displaymath}
preserves all [[(∞,1)-colimit]]s. Since every $(\infty,1)$-topos is a [[locally presentable (∞,1)-category]] it follows with the [[adjoint (∞,1)-functor theorem]] that there is a [[right adjoint|right]] [[adjoint (∞,1)-functor]]

\begin{displaymath}
(X \times (-) \dashv [X,-]) : \mathbf{H} \stackrel{\overset{X \times (-)}{\leftarrow}}{\underset{[X,-]}{\to}} \mathbf{H}
  \,.
\end{displaymath}
\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
For $C$ an [[(∞,1)-site]] for $\mathbf{H}$ we have that the [[internal hom]] ([[mapping stack]]) $[X,-]$ is given on $A \in \mathbf{H}$ by the [[(∞,1)-sheaf]]

\begin{displaymath}
[X,A] : U \mapsto \mathbf{H}(X \times L y(U), A)
  \,,
\end{displaymath}
where $y : C \to \mathbf{H}$ is the [[(∞,1)-Yoneda embedding]] and $L : PSh_C \to \mathbf{H}$ denotes [[∞-stackification]].

\end{prop}
\begin{proof}
The argument is entirely analogous to that of the [[closed monoidal structure on sheaves]].

We use the [[full and faithful (∞,1)-functor|full and faithful]] [[geometric embedding]] $(L \dashv i) : \mathbf{H} \hookrightarrow PSh_C$ and the [[(∞,1)-Yoneda lemma]] to find for all $U \in C$ the value

\begin{displaymath}
[X,A](U) \simeq PSh_C(y U, [X,A])
\end{displaymath}
and then the fact that [[∞-stackification]] $L$ is [[left adjoint]] to inclusion to get

\begin{displaymath}
\cdots \simeq \mathbf{H}(L y(U), [X,A])
  \,.
\end{displaymath}
Then the defining adjunction $(X \times (-) \dashv [X,-])$ gives

\begin{displaymath}
\cdots \simeq \mathbf{H}(X \times L y(U) , A)
  \,.
\end{displaymath}
\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Finite colimits may be taken out of the internal hom: For $I$ a finite $(\infty,1)$-category and $X : I \to \mathbf{H}$ a [[diagram]], we have for all $A \in \mathbf{H}$

\begin{displaymath}
[{\lim_\to}_i X_i, A]
   \simeq
  {\lim_\leftarrow}_i [X_i,A]
\end{displaymath}
\end{prop}
\begin{proof}
By the above proposition we have

\begin{displaymath}
[{\lim_\to}_i X_i, A](U) 
  \simeq
  \mathbf{H}(({\lim_\to}_i X_i) \times L y(U), A)
  \,.
\end{displaymath}
By [[universal colimits]] in $\mathbf{H}$ this is

\begin{displaymath}
\cdots \simeq
  \mathbf{H}({\lim_\to}_i X_i \times L y(U), A)
  \,.
\end{displaymath}
Using the fact that the [[hom-functor]] sends colimits in the first argument to limits this is

\begin{displaymath}
\cdots \simeq {\lim_\leftarrow}_i \mathbf{H}(X_i \times L y U, A)
  \,.
\end{displaymath}
By the internal hom adjunction and Yoneda this is

\begin{displaymath}
\cdots \simeq {\lim_\leftarrow}_i [X_i, A](U)
  \,.
\end{displaymath}
Since [[(∞,1)-limit]]s in the [[(∞,1)-category of (∞,1)-presheaves]] are computed objectwise, this is

\begin{displaymath}
\cdots \simeq ({\lim_\leftarrow}_i [X_i,A])(U) 
  \,.
\end{displaymath}
Finally, because $L$ is a [[left exact (∞,1)-functor]] this is also the [[(∞,1)-limit]] in $\mathbf{H}$.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
For $S \in$ [[∞Grpd]] write $LConst S$ for its [[inverse image]] under the [[global section]] [[(∞,1)-geometric morphism]] $(LConst \dashv \Gamma) : \mathbf{H} \to \infty Grpd$: the [[constant ∞-stack]] on $S$.

Then the internal hom $[LConst S,A]$ coincides with the \hyperlink{Powering}{(∞,1)-powering} of $A$ by $S$:

\begin{displaymath}
[LConst S, A] \simeq A^S
\end{displaymath}
\end{prop}
\begin{proof}
By the above we have

\begin{displaymath}
[LConst S, A](U) \simeq \mathbf{H}(LConst S \times L y(U), A)
  \,.
\end{displaymath}
As the notation indicates, $LConst S$ is precisely $L Const S$: the [[∞-stackification]] of the [[(∞,1)-presheaf]] that is literally constant on $S$. Morover $L$ is a [[left exact (∞,1)-functor]] and hence commutes with [[(∞,1)-product]]s, so that

\begin{displaymath}
\cdots \simeq \mathbf{H}(L(Const S \times y(U)), A)
  \,.
\end{displaymath}
By the defining geometric embedding $(L \dashv i)$ this is

\begin{displaymath}
\cdots \simeq PSh_C(Const S \times y(U), A)
  \,.
\end{displaymath}
Since limits of [[(∞,1)-presheaves]] are taken objectwise, we have in the first argument the [[tensoring]] of $y(U)$ over $S$

\begin{displaymath}
\cdots \simeq PSh_C(S \cdot y(U), A)
  \,.
\end{displaymath}
By the defining property of tensoring and cotensoring (or explicitly writing out $S \cdot y(U) = {\lim_\to}_{S} const y(U)$, taking the colimit out of the hom, thus turning it into a limit and then inserting that back in the second argument) this is

\begin{displaymath}
\cdots \simeq PSh_C(y(U), A^S)
  \,.
\end{displaymath}
So finally with the [[(∞,1)-Yoneda lemma]] we have

\begin{displaymath}
\cdots \simeq A^S(U)
  \,.
\end{displaymath}
\end{proof}
\hypertarget{overtoposes}{}\subsubsection*{{Over-$(\infty,1)$-toposes}}\label{overtoposes}

\begin{prop}
\label{}\hypertarget{}{}
For $\mathbf{H}$ an $(\infty,1)$-topos and $X \in \mathbf{H}$ an object, the [[over-(∞,1)-category]] $\mathbf{H}_{/X}$ is itself an $(\infty,1)$-topos -- an \textbf{[[over-(∞,1)-topos]]}. The projection $\pi_! : \mathbf{H}_{/X} \to \mathbf{H}$ part of an [[essential geometric morphism]]

\begin{displaymath}
\pi : \mathbf{H}_{/X} 
  \stackrel{\overset{\pi_!}{\to}}{\stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}}}
  \mathbf{H}
  \,.
\end{displaymath}
\end{prop}
This is [[Higher Topos Theory|HTT, prop. 6.3.5.1]].

The $(\infty,1)$-topos $\mathbf{H}_{/X}$ could be called the [[gros topos]] of $X$. A [[geometric morphism]] $\mathbf{K} \to \mathbf{H}$ that factors as $\mathbf{K} \to \mathbf{H}_{/X} \stackrel{\pi}{\to} \mathbf{H}$ is called an [[etale geometric morphism]].

\hypertarget{syntax_in_univalent_homotopy_type_theory}{}\subsubsection*{{Syntax in univalent homotopy type theory}}\label{syntax_in_univalent_homotopy_type_theory}

$(\infty,1)$-Toposes provide [[categorical semantics]] for [[homotopy type theory]] with a [[univalence|univalent]] Tarskian [[type of types]] (which inteprets as the [[object classifier]]).

For more on this see at

\begin{itemize}%
\item \emph{[[homotopytypetheory:model of type theory in an (infinity,1)-topos]]}


\item \emph{\href{relation+between+type+theory+and+category+theory#HomotopyWithUnivalence}{relation between type theory and category theory -- Univalent homotopy type theory and infinity-toposes}}



\end{itemize}
\hypertarget{ToposTheory}{}\subsection*{{$(\infty,1)$-Topos theory}}\label{ToposTheory}

Most of the standard constructions in [[topos theory]] have or should have immediate generalizations to the context of $(\infty,1)$-toposes, since all notions of [[category theory]] exist for [[(∞,1)-categories]].

For instance there are evident notions of

\begin{itemize}%
\item [[geometric morphism]]s between $(\infty,1)$-toposes, such as the [[global section]] geometric morphism to the [[terminal object|terminal]] [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaf]] $(\infty,1)$-topos [[? Grpd]].

\end{itemize}
Moreover, it turns out that $(\infty,1)$-toposes come with plenty of internal structures, more than canonically present in an ordinary topos. Every $(\infty,1)$-topos comes with its intrinsic notion of

\begin{itemize}%
\item [[cohomology|cohomology in an (∞,1)-topos]]

\end{itemize}
and with an intrinsic notion of

\begin{itemize}%
\item [[homotopy groups in an (∞,1)-topos|homotopy in an (∞,1)-topos]].

\end{itemize}
In classical topos theory, cohomology and homotopy of a topos $E$ are defined in terms of [[simplicial object]]s in $C$. If $E$ is a [[sheaf topos]] with [[site]] $C$ and [[point of a topos|enough point]]s, then this classical construction is secretly really a model for the intrinsic cohomology and homotopy in the above sense of the [[hypercomplete (∞,1)-topos]] of [[∞-stack]]s on $C$.

The beginning of a list of all the structures that exist intrinsically in a big $(\infty,1)$-topos is at

\begin{itemize}%
\item [[cohesive (∞,1)-topos]].

\end{itemize}
But \textbf{$(\infty,1)$-topos theory} in the style of an $\infty$-analog of the [[Elephant]] is only barely beginning to be conceived.

There are some indications as to what the

\begin{itemize}%
\item [[internal logic of an (∞,1)-topos]]

\end{itemize}
should be.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[(0,1)-topos]]


\item [[topos]]


\item [[2-topos]]


\item \textbf{$(\infty,1)$-topos}, [[elementary (∞,1)-topos]], [[(∞,1)-pretopos]]

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


\item [[(n,1)-topos]]


\item [[structured (∞,1)-topos]]


\item [[compact topos]], [[coherent (∞,1)-topos]]



\end{itemize}

\item [[category object in an (∞,1)-topos]]


\item [[(∞,2)-topos]]


\item [[(∞,n)-topos]]



\end{itemize}
[[!include locally presentable categories - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

In retrospect it turns out that the [[homotopy category of an (∞,1)-category|homotopy categories]] of [[(∞,1)-topos]]es have been known since

\begin{itemize}%
\item [[Kenneth Brown]], \emph{[[BrownAHT|Abstract homotopy theory and generalized sheaf cohomology]]} .

\end{itemize}
And the [[model category]] theory models have been known since [[Andre Joyal]] proposed the [[model structure on simplicial sheaves]] in his letter to [[Alexander Grothendieck]].

This work used 1-categorical [[site]]s. The generalization to [[(∞,1)-category|(∞,1)categorical sites]] -- modeled by [[model sites]] -- was discussed in

\begin{itemize}%
\item [[Bertrand Toën]], [[Gabriele Vezzosi]], \emph{Homotopical Algebraic Geometry I: Topos theory}, Advances in Mathematics 193.2 (2005): 257-372. (\href{http://arxiv.org/abs/math.AG/0207028}{arXiv:math.AG/0207028})

\end{itemize}
and ``model topos''-theory was also developed in

\begin{itemize}%
\item [[Charles Rezk]], \emph{Toposes and homotopy toposes}, 2010 (\href{http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf}{pdf})

\end{itemize}
The intrinsic [[higher category theory|category-theoretic]] definition of an [[(∞,1)-topos]] was given in section 6.1 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
building on ideas by [[Charles Rezk]]. There is is also proven that the Brown-Joyal-Jardine-To\"e{}n-Vezzosi models indeed precisely model $\infty$-stack $(\infty,1)$-toposes. Details on this relation are at [[models for ∞-stack (∞,1)-toposes]].

An overview is in

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{Cat\'e{}gories sup\'e{}rieures et th\'e{}orie des topos}, S\'e{}minaire Bourbaki, 21.3.2015, \href{http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf}{pdf}.

\end{itemize}
A useful collection of facts of [[simplicial homotopy theory]] and [[(infinity,1)-topos theory]] is in

\begin{itemize}%
\item [[Zhen Lin Low]], \emph{[[Notes on homotopical algebra]]}

\end{itemize}
A quick introduction to the topic is in

\begin{itemize}%
\item [[André Joyal]], \emph{[[A crash course in topos theory -- The big picture]]}, lecture series at \href{https://indico.math.cnrs.fr/event/747/}{Topos \`a{} l'IHES}, November 2015, Paris

\end{itemize}
\hypertarget{giraudrezklurie_axioms}{}\subsubsection*{{Giraud-Rezk-Lurie axioms}}\label{giraudrezklurie_axioms}

A discussion of the $(\infty,1)$-[[universal colimits]] in terms of [[model category]] presentations is due to

\begin{itemize}%
\item [[Charles Rezk]], \emph{Fibrations and homotopy colimits of simplicial sheaves} (\href{http://www.math.uiuc.edu/~rezk/rezk-sharp-maps.pdf}{pdf})

\end{itemize}
More on this with an eye on [[associated ∞-bundles]] is in

\begin{itemize}%
\item [[Matthias Wendt]], \emph{Classifying spaces and fibrations of simplicial sheaves} (\href{http://arxiv.org/abs/1009.2930}{arXiv})

\end{itemize}
\hypertarget{homotopy_type_theory}{}\subsubsection*{{Homotopy type theory}}\label{homotopy_type_theory}

Proof that all [[∞-stack]] [[(∞,1)-topos]] have [[presentable (∞,1)-category|presentations]] by [[model categories]] which interpret (provide [[categorical semantics]]) for [[homotopy type theory]] with [[univalence|univalent]] [[type universes]]:

\begin{itemize}%
\item [[Michael Shulman]], \emph{All $(\infty,1)$-toposes have strict univalent universes} (\href{https://arxiv.org/abs/1904.07004}{arXiv:1904.07004}).

\end{itemize}
[[!redirects (infinity,1)-topos]] [[!redirects (infinity,1)-topoi]] [[!redirects (infinity,1)-toposes]] [[!redirects (∞,1)-topos]] [[!redirects (∞,1)-topoi]] [[!redirects (∞,1)-toposes]]

[[!redirects Grothendieck (infinity,1)-topos]] [[!redirects Grothendieck (infinity,1)-topoi]] [[!redirects Grothendieck (infinity,1)-toposes]] [[!redirects Grothendieck (∞,1)-topos]] [[!redirects Grothendieck (∞,1)-topoi]] [[!redirects Grothendieck (∞,1)-toposes]]

[[!redirects Grothendieck-Rezk-Lurie (infinity,1)-topos]] [[!redirects Grothendieck-Rezk-Lurie (infinity,1)-topoi]] [[!redirects Grothendieck-Rezk-Lurie (infinity,1)-toposes]] [[!redirects Grothendieck-Rezk-Lurie (∞,1)-topos]] [[!redirects Grothendieck-Rezk-Lurie (∞,1)-topoi]] [[!redirects Grothendieck-Rezk-Lurie (∞,1)-toposes]] [[!redirects Grothendieck?Rezk?Lurie (infinity,1)-topos]] [[!redirects Grothendieck?Rezk?Lurie (infinity,1)-topoi]] [[!redirects Grothendieck?Rezk?Lurie (infinity,1)-toposes]] [[!redirects Grothendieck?Rezk?Lurie (∞,1)-topos]] [[!redirects Grothendieck?Rezk?Lurie (∞,1)-topoi]] [[!redirects Grothendieck?Rezk?Lurie (∞,1)-toposes]] [[!redirects Grothendieck--Rezk--Lurie (infinity,1)-topos]] [[!redirects Grothendieck--Rezk--Lurie (infinity,1)-topoi]] [[!redirects Grothendieck--Rezk--Lurie (infinity,1)-toposes]] [[!redirects Grothendieck--Rezk--Lurie (∞,1)-topos]] [[!redirects Grothendieck--Rezk--Lurie (∞,1)-topoi]] [[!redirects Grothendieck--Rezk--Lurie (∞,1)-toposes]]

[[!redirects (infinity,1)-Giraud theorem]] [[!redirects (∞,1)-Giraud theorem]]

[[!redirects infinity,1-topos]]



\end{document}