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

\begin{quote}%
This entry is about a refinement of the concept of [[cohesive topos]]. The definition here expresses an intuition not unrelated to that at [[cohesive (∞,1)-presheaf on E-∞ rings]] but the definitions are unrelated and apply in somewhat disjoint contexts.


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

\hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive $\infty$-Toposes}}\label{cohesive_toposes}

[[!include cohesive infinity-toposes - contents]]

\hypertarget{discrete_and_concrete_objects}{}\paragraph*{{Discrete and concrete objects}}\label{discrete_and_concrete_objects}

[[!include discrete and concrete objects - 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{ExternalDefinition}{Externally}\dotfill \pageref*{ExternalDefinition} \linebreak
\noindent\hyperlink{InternalDefinition}{Internally}\dotfill \pageref*{InternalDefinition} \linebreak
\noindent\hyperlink{DefinitionInHomotopyTypeTheory}{In homotopy type theory}\dotfill \pageref*{DefinitionInHomotopyTypeTheory} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{AsAPointLikeSpace}{As a point-like space}\dotfill \pageref*{AsAPointLikeSpace} \linebreak
\noindent\hyperlink{OverAnInfinityCohesiveSite}{Over an $\infty$-cohesive site}\dotfill \pageref*{OverAnInfinityCohesiveSite} \linebreak
\noindent\hyperlink{General}{General}\dotfill \pageref*{General} \linebreak
\noindent\hyperlink{OverArbitraryBases}{Base of discrete and codiscrete objects}\dotfill \pageref*{OverArbitraryBases} \linebreak
\noindent\hyperlink{FactorizationSystemsForPi}{Factorization systems associated to $\mathbf{\Pi}$}\dotfill \pageref*{FactorizationSystemsForPi} \linebreak
\noindent\hyperlink{Structures}{Structures in a cohesive $(\infty,1)$-topos}\dotfill \pageref*{Structures} \linebreak
\noindent\hyperlink{types_of_cohesion}{Types of cohesion}\dotfill \pageref*{types_of_cohesion} \linebreak
\noindent\hyperlink{infinitesimal_cohesion}{Infinitesimal cohesion}\dotfill \pageref*{infinitesimal_cohesion} \linebreak
\noindent\hyperlink{DifferentialCohesion}{Differential cohesion}\dotfill \pageref*{DifferentialCohesion} \linebreak
\noindent\hyperlink{locally_ringed_cohesion}{Locally ringed cohesion}\dotfill \pageref*{locally_ringed_cohesion} \linebreak
\noindent\hyperlink{Examples}{Examples of cohesive $\infty$-toposes}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{cohesive_diagram_toposes}{Cohesive diagram $(\infty,1)$-toposes}\dotfill \pageref*{cohesive_diagram_toposes} \linebreak
\noindent\hyperlink{CohesiveDiagramToposes}{Cohesive diagrams in a cohesive $(\infty,1)$-topos}\dotfill \pageref*{CohesiveDiagramToposes} \linebreak
\noindent\hyperlink{SimplicialObjctsInACohesiveTopos}{Simplicial objects in a cohesive $(\infty,1)$-topos}\dotfill \pageref*{SimplicialObjctsInACohesiveTopos} \linebreak
\noindent\hyperlink{bundles_of_cohesive_spectra}{Bundles of cohesive spectra}\dotfill \pageref*{bundles_of_cohesive_spectra} \linebreak
\noindent\hyperlink{global_equivariant_homotopy_theory}{Global equivariant homotopy theory}\dotfill \pageref*{global_equivariant_homotopy_theory} \linebreak
\noindent\hyperlink{ExamplesFromCohesiveSitesOnfDefinition}{From $\infty$-Cohesive sites of definition}\dotfill \pageref*{ExamplesFromCohesiveSitesOnfDefinition} \linebreak
\noindent\hyperlink{DiscreteInfinityGroupoids}{Discrete $\infty$-groupoids}\dotfill \pageref*{DiscreteInfinityGroupoids} \linebreak
\noindent\hyperlink{topological_groupoids}{Topological $\infty$-groupoids}\dotfill \pageref*{topological_groupoids} \linebreak
\noindent\hyperlink{smooth_groupoids}{Smooth $\infty$-groupoids}\dotfill \pageref*{smooth_groupoids} \linebreak
\noindent\hyperlink{complex_analytic_groupoids}{Complex analytic $\infty$-groupoids}\dotfill \pageref*{complex_analytic_groupoids} \linebreak
\noindent\hyperlink{formal_smooth_groupoids}{Formal smooth $\infty$-groupoids}\dotfill \pageref*{formal_smooth_groupoids} \linebreak
\noindent\hyperlink{smooth_super_groupoids}{Smooth Super $\infty$-groupoids}\dotfill \pageref*{smooth_super_groupoids} \linebreak
\noindent\hyperlink{smooth_groupoids_over_algebraic_stacks}{Smooth $\infty$-groupoids over algebraic $\infty$-stacks}\dotfill \pageref*{smooth_groupoids_over_algebraic_stacks} \linebreak
\noindent\hyperlink{arithmetic_groupoids}{$E_\infty$-Arithmetic $\infty$-groupoids}\dotfill \pageref*{arithmetic_groupoids} \linebreak
\noindent\hyperlink{smooth_cohesion_over_arbitrary_base_toposes}{Smooth cohesion over arbitrary base $\infty$-toposes}\dotfill \pageref*{smooth_cohesion_over_arbitrary_base_toposes} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{precursors}{Precursors}\dotfill \pageref*{precursors} \linebreak
\noindent\hyperlink{on_cohesive_toposes_proper}{On cohesive $(\infty,1)$-toposes proper}\dotfill \pageref*{on_cohesive_toposes_proper} \linebreak
\noindent\hyperlink{ReferencesInHoTT}{In homotopy type theory}\dotfill \pageref*{ReferencesInHoTT} \linebreak


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

A \emph{cohesive $(\infty,1)$-topos} is a [[big topos|gros]] [[(∞,1)-topos]] $\mathbf{H}$ that provides a context of generalized [[spaces]] in which [[higher geometry]] makes sense, in particular [[higher differential geometry]]. See also at \emph{[[motivation for cohesive toposes]]} for a non-technical discussion.

Technically, it is an $(\infty,1)$-topos whose [[global section]] [[(∞,1)-geometric morphism]] $(Disc \dashv \Gamma): \mathbf{H} \stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}}$ [[∞Grpd]] admits a further [[left adjoint|left]] [[adjoint (∞,1)-functor]] $\Pi$ and a further right adjoint $coDisc$:

\begin{displaymath}
(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \mathbf{H} \to \infty Grpd
\end{displaymath}
with $Disc$ and $coDisc$ both [[full and faithful (∞,1)-functor]]s and such that $\Pi$ moreover preserves finite [[(∞,1)-limit|(∞,1)-product]]s. Here

\begin{enumerate}%
\item the existence of $coDisc$ induces a sub-[[(∞,1)-quasitopos]] $Conc(\mathbf{H}) \hookrightarrow \mathbf{H}$ of \emph{[[concrete (∞,1)-sheaf|concrete objects]]} that behave like [[∞-groupoid]]s \emph{equipped with extra cohesive structure} , such as with [[topology|continuous structure]], [[smooth structure]], etc.


\item the existence of $\Pi$ induces a notion of [[geometric homotopy groups in an (∞,1)-topos|geometric]] [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos|fundamental ∞-groupoid]], hence under $|-| : \infty Grpd \simeq$ [[Top]] of [[geometric realization]] $|\Pi(-)|$ of objects in $\mathbf{H}$.



\end{enumerate}
The functor $\Gamma$ itself may be thought of as sending a cohesive [[∞-groupoid]] $X$ to its underlying bare $\infty$-groupoid $\Gamma(X)$. This is $X$ with all \emph{cohesion forgotten} (for instance with the continuous or the smooth structure forgotten).

Conversely, $Disc$ and $CoDisc$ send an $\infty$-groupoid $A$ either to the [[discrete space|discrete ∞-groupoid]] $Disc(A)$ with \emph{discrete} cohesive structure (for instance with [[discrete topology]]) or to the [[codiscrete space|codiscrete ∞-groupoid]] $Codisc(A)$ with the \emph{codiscrete} cohesive structure (for instance with [[codiscrete topology]]).

This kind of [[adjoint quadruple]], directly analogous to the structure introduced by [[William Lawvere]] for [[cohesive toposes]], induces three [[adjoint modalities]] which in turn are adjoint to each other, to yield the adjoint string [[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]] $\int \dashv \flat \dashv \sharp$, which may be thought of as expressing ``continuum'' and ``quantity'' in the sense of [[Georg Hegel]]`s \emph{[[Science of Logic]]} (as explained in detail there.)

The existence of such an [[adjoint quadruple]] of adjoint $(\infty,1)$-functors alone implies a rich [[internalization|internal]] [[higher geometry]] in $\mathbf{H}$ that comes with its internal notion of [[Galois theory]], [[Lie theory]], [[differential cohomology]], [[Chern-Weil theory]].

Examples of cohesive $(\infty,1)$-toposes include

\begin{itemize}%
\item the $(\infty,1)$-topos $\mathbf{H} =$[[Disc∞Grpd]] of [[discrete ∞-groupoid]]s;


\item the $(\infty,1)$-topos $\mathbf{H} =$ [[ETop∞Grpd]] of [[Euclidean-topological ∞-groupoids]];


\item the $(\infty,1)$-topos $\mathbf{H} =$ [[Smooth∞Grpd]] of [[smooth ∞-groupoids]];


\item the $(\infty,1)$-topos $\mathbf{H} =$ [[SynthDiff∞Grpd]] of [[formal smooth ∞-groupoids]];


\item the $(\infty,1)$-topos $\mathbf{H} =$ [[Super∞Grpd]] of [[super ∞-groupoids]];


\item the $(\infty,1)$-topos $\mathbf{H} =$ [[SmoothSuper∞Grpd]] of [[smooth super ∞-groupoids]].



\end{itemize}
In [[ETop∞Grpd]] and those contexts containing it, the internal notions of [[geometric realization]], [[geometric homotopy groups in an (infinity,1)-topos|geometric homotopy]] and [[Galois theory]] subsume the usual ones (over well-behaved topological spaces). In [[Smooth∞Grpd]] also the notions of [[Lie theory]], [[differential cohomology]] and [[Chern-Weil theory]] subsume the usual ones. In [[SynthDiff∞Grpd]] the internal notion of [[Lie algebra]] and [[Lie algebroid]] subsumes the traditional one --- and generalizes them to higher smooth geometry.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

We state the definition in several equivalent ways.

\begin{enumerate}%
\item \hyperlink{ExternalDefinition}{externally} in the ambient context;


\item \hyperlink{InternalDefinition}{internally} to the cohesive $(\infty,1)$-topos itselfs;


\item internally and \hyperlink{DefinitionInHomotopyTypeTheory}{formulated in homotopy type theory}



\end{enumerate}
\hypertarget{ExternalDefinition}{}\subsubsection*{{Externally}}\label{ExternalDefinition}

The definition is the immediate analog of the definition of a [[cohesive topos]].

\begin{defn}
\label{CohesiveInfinTopos}\hypertarget{CohesiveInfinTopos}{}
An [[(∞,1)-topos]] $\mathbf{H}$ is \textbf{cohesive} if

\begin{enumerate}%
\item it is a [[strongly ∞-connected (∞,1)-topos]];


\item it is a [[local (∞,1)-topos]].



\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
As always in [[topos theory]] and [[higher topos theory]], such definitions can be made sense of over any \emph{base} . Here: over any [[base (∞,1)-topos]].

Over the canonical base [[∞Grpd]] of [[∞-groupoid]]s, the definition of a cohesive $(\infty,1)$-topos is equivalently the following:

the [[global section]] [[(∞,1)-geometric morphism]] $\Gamma : \mathbf{H} \to \infty Grpd$ lifts to an [[adjoint quadruple]] of [[adjoint (∞,1)-functor]]s

\begin{displaymath}
(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : 
  \mathbf{H}
   \stackrel{\stackrel{\overset{\Pi}{\longrightarrow}}{\overset{Disc}{\leftarrow}}}{\stackrel{\underset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}
  \infty Grpd
  \;
\end{displaymath}
where $\Pi$ preserves [[finite limit|finite]] [[(∞,1)-product]]s.

Often we will tacitly assume to work over [[∞Grpd]]. But most statements and constructions have straightforward generalizations to arbitrary bases. In particular, below in the \hyperlink{InternalDefinition}{internal definition} of cohesion, it takes more to fix the base topos than to leave it arbitrary.

Every [[adjoint quadruple]] induces an [[adjoint triple]] of endofunctors.

\begin{displaymath}
(
   \mathbf{\Pi}
   \dashv 
   \mathbf{\flat}
   \dashv 
   \mathbf{#}
  )
  :
  \mathbf{H}
    \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{\Disc}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}}}
  \infty Grpd
   \stackrel{\overset{Disc}{\longrightarrow}}{\stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\longrightarrow}}}
  \mathbf{H}
  \,.
\end{displaymath}
Here ``$\mathbf{\flat}$'' is meant to be pronounced ``flat''. The interpretation of these three functors is discussed in detail at \emph{[[cohesive (∞,1)-topos -- structures]]}.

\end{remark}
Sometimes it is desireable to add further axioms, such as the following.

\begin{defn}
\label{PiecesHavePoints}\hypertarget{PiecesHavePoints}{}
We say that \textbf{pieces have points} for an object $X$ in a cohesive $(\infty,1)$-topos $\mathbf{H}$ if the [[points-to-pieces transform]]

\begin{displaymath}
\Gamma X \to \Gamma Disc \Pi X \simeq \Pi X
\end{displaymath}
is an [[effective epimorphism in an (∞,1)-category]], equivalently (as discussed there) such that this is an [[epimorphism]] on [[connected]] components.

\end{defn}
Here the first morphismism is the image under $\Gamma$ of the $(Disc \dashv \Gamma)$-[[unit of an adjunction|unit]] and the second is an inverse of the $(\Pi \dashv Disc)$-counit (which is invertible because $Disc$ is [[full and faithful (∞,1)-functor|full and faithful]] in a [[local (∞,1)-topos]].)

\begin{defn}
\label{DiscreteObjectsAreConcrete}\hypertarget{DiscreteObjectsAreConcrete}{}
We say \textbf{discrete objects are concrete} in $\mathbf{H}$ if for all $S \in$[[∞Grpd]] the morphism

\begin{displaymath}
Disc S \to coDisc \Gamma Disc S \stackrel{\simeq}{\longrightarrow} coDisc S
\end{displaymath}
induces [[monomorphism]]s on all [[categorical homotopy groups in an (infinity,1)-topos|homotopy sheaves]].

\end{defn}
The following extra condition ensures that the [[shape modality]] is explicitly given by a geometric [[path ∞-groupoid]] construction.

\begin{defn}
\label{A1ExhibitingCohesion}\hypertarget{A1ExhibitingCohesion}{}
Given a cohesive $\infty$-topos $\mathbf{H}$ and an object $\mathbb{A}^1 \in \mathbf{H}$, we say that $\mathbb{A}^1$ \emph{exhibits the cohesion} if the [[shape modality]] $\Pi$ is equivalent to ``[[A1-homotopy theory|A1-localization]]'' $L_{\mathbb{A}^1}$, hence to [[localization of an (∞,1)-category|localization]] of $\mathbf{H}$ at the class of [[projection]] morphisms of the form $(-)\times \mathbb{A}^1 \longrightarrow (-)$, i.e. if

\begin{displaymath}
\Pi \simeq L_{\mathbb{A}^1}
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
In the example of [[smooth ∞-groupoid|smooth cohesion]] and its variants such as [[Euclidean topological infinity-groupoid|Euclidean topological cohesion]], the standard [[real line]] $\mathbb{A}^1 = \mathbb{R}^1$ exhibits cohesion in the sense of def. \ref{A1ExhibitingCohesion} (by \href{cohesive+%28infinity%2C1%29-topos+--+structures#RealLineIsTheContinuum}{this discussion}). Therefore one might also say that if an object $\mathbb{A}^1$ in a cohesive $\infty$-topos exhibits the cohesion, then it plays the role of \emph{the [[continuum]]} in analogy of the traditional use of this term in geometry. See also at \emph{\href{continuum#InCohesiveHomotopyTypeTheory}{continuum -- In cohesive homotopy theory}}.

\end{remark}
Another extra axiom is (see at \emph{[[Aufhebung]]} for more):

\begin{defn}
\label{AufhebungOfBecoming}\hypertarget{AufhebungOfBecoming}{}
Say that a cohesive $\infty$-topos, def. \ref{CohesiveInfinTopos}, has \emph{[[Aufhebung]] of [[becoming]]} if the [[sharp modality]] preserves the [[initial object]]

\begin{displaymath}
\sharp \emptyset \simeq \emptyset
  \,.
\end{displaymath}
\end{defn}
\hypertarget{InternalDefinition}{}\subsubsection*{{Internally}}\label{InternalDefinition}

We reformulate the \hyperlink{ExternalDefinition}{above axioms} for a cohesive $(\infty,1)$-topos without references to functors \emph{on} it, and instead entirely in terms of structures \emph{in} it.

\begin{lemma}
\label{ReflectiveEmbeddingInternally}\hypertarget{ReflectiveEmbeddingInternally}{}
A [[full sub-(∞,1)-category]] $\mathbf{B} \hookrightarrow \mathbf{H}$ is

\begin{itemize}%
\item [[reflective sub-(∞,1)-category|reflectively embedded]] precisely if for every [[object]] $X \in \mathbf{H}$ there is a [[morphism]]

\begin{displaymath}
loc_X : X \to L X
\end{displaymath}
(the [[unit of an adjunction|unit]]) with $L X \in \mathbf{B} \hookrightarrow \mathbf{H}$, such that for all $Y \in \mathbf{B} \hookrightarrow \mathbf{H}$ the value of the [[(∞,1)-categorical hom-space]]-functor

\begin{displaymath}
\mathbf{H}(loc_X, Y) : 
  \mathbf{H}(L X, Y)
  \stackrel{\simeq}{\longrightarrow}
  \mathbf{H}(X, Y)
\end{displaymath}
is an [[equivalence in an (∞,1)-category|equivalence]] (of [[∞-groupoid]]s).


\item coreflectively embedded precisely if for every [[object]] $Y \in \mathbf{H}$ there is a [[morphism]]

\begin{displaymath}
coloc_Y : R Y \to Y
\end{displaymath}
(the [[unit of an adjunction|counit]]) with $R Y \in \mathbf{B} \hookrightarrow \mathbf{H}$ such that for all $X \in \mathbf{B} \hookrightarrow \mathbf{H}$ the value of the [[(∞,1)-categorical hom-space]]-functor

\begin{displaymath}
\mathbf{H}(X, coloc_Y) : 
  \mathbf{H}(X, R Y)
  \stackrel{\simeq}{\to}
  \mathbf{H}(X, Y)
\end{displaymath}
is an [[equivalence in an (∞,1)-category|equivalence]] (of [[∞-groupoid]]s).



\end{itemize}
\end{lemma}
This is proven \href{http://ncatlab.org/nlab/show/reflective+sub-%28infinity,1%29-category#CharacterizationOfReflectors}{here}.

\begin{lemma}
\label{}\hypertarget{}{}
A reflective embedding

\begin{displaymath}
coDisc : \mathbf{B}_{cod} \stackrel{\overset{\tilde \Gamma}{\leftarrow}}{\underset{coDisc}{\hookrightarrow}} \mathbf{H}
\end{displaymath}
and a coreflective embedding

\begin{displaymath}
Disc : \mathbf{B}_{disc} \stackrel{\overset{Disc}{\hookrightarrow}}{\underset{\Gamma}{\leftarrow}} \mathbf{H}
\end{displaymath}
fit into a single [[adjoint triple]]

\begin{displaymath}
\mathbf{H}
    \stackrel{
      \overset{Disc}{\hookleftarrow}
    }{
     \stackrel{  
       \overset{\Gamma}{\to}
     }{
       \underset{coDisc}{\hookleftarrow}
     }
   }
  \mathbf{B}
\end{displaymath}
(hence there is an equivalence $\mathbf{B}_{disc} \simeq \mathbf{B}_{cod}$ that moreover makes the coreflector $\tilde\Gamma$ of $Disc$ coincide with the reflector $\Gamma$ of $coDisc$) precisely if for the [[unit of an adjunction|unit]] and counit given by lemma \ref{ReflectiveEmbeddingInternally} we have that the morphisms on the left of

\begin{equation}
coDisc \tilde \Gamma (Disc \Gamma X \to X)
  \;
  =:
  \;
  (coDisc \tilde \Gamma Disc \Gamma X

   \stackrel{\simeq}{\to}
  coDisc \tilde \Gamma X)
\label{FirstNaturalEquivalence}\end{equation}
\begin{equation}
Disc \Gamma (X \to coDisc \tilde \Gamma X)
  \;\;
   =:
  \;\;
  (
    Disc \Gamma X 
     \stackrel{\simeq}{\to}
    Disc \Gamma coDisc \tilde \Gamma
  )
\label{SecondNaturalEquivalence}\end{equation}
are (natural) [[equivalence in an (∞,1)-category|equivalences]] for all objects $X \in \mathbf{H}$, as indicated on the right.

\end{lemma}
\begin{proof}
It is clear that if we have an [[adjoint triple]], then \eqref{FirstNaturalEquivalence} and \eqref{SecondNaturalEquivalence} are implied. We discuss now the converse.

First notice that the two embeddings always combine into an adjunction of the form

\begin{displaymath}
\mathbf{B}_{disc}
   \stackrel{\overset{Disc}{\hookrightarrow}}{\underset{\Gamma}{\leftarrow}}
  \mathbf{H}
   \stackrel{\overset{\tilde \Gamma}{\to}}{\underset{coDisc}{\hookleftarrow}}
  \mathbf{B}_{cod}    
  \,.
\end{displaymath}
The natural equivalence \eqref{FirstNaturalEquivalence} applied to a codiscrete object $X := coDisc A$ gives that $coDisc$ of the counit of this composite adjunction is an equivalence

\begin{displaymath}
coDisc \tilde \Gamma Disc \Gamma coDisc A
    \stackrel{\simeq}{\to}
  coDisc \tilde \Gamma coDisc A
    \stackrel{\simeq}{\to}
  coDisc A
\end{displaymath}
for all $A$, and since $coDisc$ is [[full and faithful (infinity,1)-functor|full and faithful]], so is the composite counit

\begin{displaymath}
\tilde \Gamma Disc \Gamma coDisc A
    \stackrel{\simeq}{\to}
  \tilde \Gamma coDisc A
    \stackrel{\simeq}{\to}
  A
\end{displaymath}
itself. Analogously, \eqref{SecondNaturalEquivalence} implies that the unit of the composite adjunction is an equivalence. Therefore \eqref{FirstNaturalEquivalence} and \eqref{SecondNaturalEquivalence} together imply that the adjunction itself exhibits an [[equivalence of (∞,1)-categories|equivalence]] $\mathbf{B}_{disc} \simeq \mathbf{B}_{cod}$.

Using this we then find for each $X \in \mathbf{H}$ a composite natural equivalence

\begin{displaymath}
Disc \tilde \Gamma X
   \stackrel{\simeq}{\to}
  Disc \Gamma coDisc \tilde \Gamma X
   \stackrel{\simeq}{\to}
  Disc \Gamma X
\end{displaymath}
where the first morphism uses the above equivalence on the codiscrete object $\tilde \Gamma X$ and the second is a choice of natural inverse of \eqref{SecondNaturalEquivalence}.

Since $Disc$ is full and faithful, this mean that we have equivalences

\begin{displaymath}
\tilde \Gamma X 
   \simeq 
  \Gamma X
\end{displaymath}
natural in $X$, hence that $\tilde \Gamma \simeq \Gamma$.

\end{proof}
\begin{remark}
\label{AdjunctionIsoFromInternalReflectionCoreflection}\hypertarget{AdjunctionIsoFromInternalReflectionCoreflection}{}
In the above situation, the defining [[adjoint (∞,1)-functor|adjunction equivalence]]

\begin{displaymath}
\mathbf{H}(\mathbf{\Pi} X , A)
    \simeq 
  \mathbf{H}(X, \flat A)
\end{displaymath}
exhibiting $(\mathbf{\Pi} \dashv \flat)$ is given by the composite of natural equivalences

\begin{displaymath}
\mathbf{H}(\mathbf{\Pi}X, A)
     \underoverset{\simeq}{\mathbf{H}(\mathbf{\Pi}X, coloc_A)^{-1}}{\to}
     \mathbf{H}(\mathbf{\Pi} X, \flat A)
   \underoverset{\simeq}{\mathbf{H}(loc_X, \flat A)}{\to}
    \mathbf{H}(X, \flat A)
\end{displaymath}
from lemma \ref{ReflectiveEmbeddingInternally}, using that both $\mathbf{\Pi} X$ as well as $\flat A$ are discrete.

\end{remark}
Using these lemmas we can now restate cohesiveness internally.

\begin{corollary}
\label{}\hypertarget{}{}
For $Disc : \mathbf{B} \hookrightarrow \mathbf{H}$ a [[sub-(∞,1)-category]], the inclusion extends to an [[adjoint quadruple]] of the form

\begin{displaymath}
\mathbf{H}
    \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\hookleftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\hookleftarrow}}}}
  \mathbf{B}
\end{displaymath}
precisely if there exists for each object $X \in \mathbf{H}$

\begin{enumerate}%
\item a morphism $X \to \mathbf{\Pi}(X)$ with $\mathbf{\Pi}(X) \in \mathbf{B} \stackrel{Disc}{\hookrightarrow} \mathbf{H}$;


\item a morphism $\mathbf{\flat} X \to X$ with $\mathbf{\flat} X \in \mathbf{B} \stackrel{Disc}{\hookrightarrow} \mathbf{H}$;


\item a morphism $X \to #X$ with $# X \in \mathbf{B} \stackrel{coDisc}{\hookrightarrow} \mathbf{H}$



\end{enumerate}
such that for all $Y \in \mathbf{B} \stackrel{Disc}{\hookrightarrow} \mathbf{H}$ and $\tilde Y \in \mathbf{B} \stackrel{coDisc}{\hookrightarrow} \mathbf{H}$ the induced morphisms

\begin{enumerate}%
\item $\mathbf{H}(\mathbf{\Pi}X , Y) \stackrel{\simeq}{\to} \mathbf{H}(X,Y)$;


\item $\mathbf{H}(Y, \mathbf{\flat}X) \stackrel{\simeq}{\to} \mathbf{H}(Y,X)$;


\item $\mathbf{H}(# X , \tilde Y) \stackrel{\simeq}{\to} \mathbf{H}(X,\tilde Y)$;


\item $# (\flat X \to X)$;


\item $\flat (X \to # X)$



\end{enumerate}
are [[equivalence in an (infinity,1)-category|equivalences]] (the first three of [[∞-groupoids]] the last two in $\mathbf{H}$).

Moreover, if $\mathbf{H}$ is a [[cartesian closed category]], then $\Pi$ preserves finite products precisely if the $Disc$-inclusion is an [[exponential ideal]].

\end{corollary}
The last statement follows from the $(\infty,1)$-category analog of the discussion \href{http://ncatlab.org/nlab/show/reflective+subcategory#ReflectiveSubcategoriesOfCartesianClosedCategotries}{here}.

\hypertarget{DefinitionInHomotopyTypeTheory}{}\subsubsection*{{In homotopy type theory}}\label{DefinitionInHomotopyTypeTheory}

The axioms for cohesion, in the \hyperlink{InternalDefinition}{internal version}, can be formulated in [[homotopy type theory]], the [[internal language of an (∞,1)-topos]].

The corresponging [[Coq]]-[[HoTT]] code is in (\hyperlink{Shulman}{Shulman}).

For more see \emph{[[cohesive homotopy type theory]]}.

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

We discuss basic properties implied by the axioms for cohesive $(\infty,1)$-toposes in

\begin{itemize}%
\item \hyperlink{AsAPointLikeSpace}{As a point-like space};


\item \hyperlink{General}{General}.


\item \hyperlink{OverArbitraryBases}{Over arbitrary bases}



\end{itemize}
Then we discuss presentations over special [[site]]s in

\begin{itemize}%
\item \hyperlink{OverAnInfinityCohesiveSite}{Over an ∞-cohesive site}.

\end{itemize}
\hypertarget{AsAPointLikeSpace}{}\subsubsection*{{As a point-like space}}\label{AsAPointLikeSpace}

\begin{prop}
\label{PointLike}\hypertarget{PointLike}{}
A nontrivial cohesive $(\infty,1)$-topos

\begin{enumerate}%
\item has the [[shape of an (∞,1)-topos|shape]] of the point;


\item has [[homotopy dimension]] 0;


\item has [[cohomology dimension]] 0.



\end{enumerate}
\end{prop}
\begin{proof}
The first holds for every [[∞-connected (∞,1)-topos]], see there.

The second holds for every [[local (∞,1)-topos]], see there.

The third follows from the second, see [[homotopy dimension]].

\end{proof}
\begin{remark}
\label{ThickenedPoint}\hypertarget{ThickenedPoint}{}
This says that a cohesive $(\infty,1)$-topos $\mathbf{H}$ is, when itself regarded as a [[little topos]], a generalized [[space]], a \emph{thickened point} . We may think of it as the standard [[point]] equipped with a \emph{cohesive neighbourhood} .

In this sense every space $X$ \emph{modeled on} the cohesive structure defined by $\mathbf{H}$ is an [[étale space]] over $X$: its [[petit topos|petit]] $(\infty,1)$-topos $\mathbf{H}/X$ sits by a [[locally homeomorphic geometric morphism]] over $\mathbf{H}$

\begin{displaymath}
\mathbf{H}/X \stackrel{local\;homeo}{\to} \mathbf{H}
  \,.
\end{displaymath}
\end{remark}
\hypertarget{OverAnInfinityCohesiveSite}{}\subsubsection*{{Over an $\infty$-cohesive site}}\label{OverAnInfinityCohesiveSite}

We discuss a [[presentable (∞,1)-category|presentation]] of classes of cohesive [[(∞,1)-topos]]es by a [[model structure on simplicial presheaves]] over a suitable [[site]].

\begin{prop}
\label{SimplicialPresheavesOverInfinityCohesviveSite}\hypertarget{SimplicialPresheavesOverInfinityCohesviveSite}{}
For $C$ an [[∞-cohesive site]] the [[(∞,1)-category of (∞,1)-sheaves]] $(\infty,1)Sh(C)$ over $C$ is a cohesive $(\infty,1)$-topos satisfying the two axioms \emph{\hyperlink{PiecesHavePoints}{pieces have points}} and \emph{\hyperlink{DiscreteObjectsAreConcrete}{discrete objects are concrete}} .

\end{prop}
The detailed discussion is at [[∞-cohesive site]].

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

\begin{prop}
\label{HypercompleteProperty}\hypertarget{HypercompleteProperty}{}
Every cohesive $(\infty,1)$-topos over [[∞Grpd]] is a [[hypercomplete (∞,1)-topos]].

\end{prop}
\begin{proof}
By the \hyperlink{PointLike}{above proposition} it has finite [[homotopy dimension]]. This implies hypercompeteness. See there.

\end{proof}
\begin{prop}
\label{Cohesive1Topos}\hypertarget{Cohesive1Topos}{}
For $\mathbf{H}$ a cohesive $(\infty,1)$-topos, the [[(n,1)-topos|(1,1)-topos]] $\tau_{\leq 1-1} \mathbf{H}$ of 0-[[truncated]] objects is a [[cohesive topos]].

\end{prop}
\begin{prop}
\label{PiPreservesPullbacksOverDiscretes}\hypertarget{PiPreservesPullbacksOverDiscretes}{}
For a cohesive $(\infty,1)$-topos $(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \mathbf{H} \to \infty Grpd$ over an [[∞-cohesive site]], the functor $\Pi$ preserves [[(∞,1)-pullbacks]] over [[discrete objects]].

\end{prop}
We first consider a lemma. Notice that for $A \in \infty Grpd$ the [[(∞,1)-Grothendieck construction]] gives an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
\infty Grpd_{/A} \simeq Func(A, \infty Grpd)
\end{displaymath}
from the [[over (∞,1)-category]] of $\infty Grpd$ over $A$ to the [[(∞,1)-functor (∞,1)-category]] from $A$ to $\infty Grpd$.

\begin{lemma}
\label{GrothendieckConstrInCohesivefinityTopos}\hypertarget{GrothendieckConstrInCohesivefinityTopos}{}
For $\mathbf{H}$ a cohesive $(\infty,1)$-topos over an [[∞-cohesive site]] and for $A \in \infty Grpd$, we have an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
\mathbf{H}_{/ Disc A} \simeq Func(A, \mathbf{H})
  \,.
\end{displaymath}
\end{lemma}
\begin{proof}
We establish this via a [[presentable (infinity,1)-category|presentation]] of $\mathbf{H}$ by a [[model structure on simplicial presheaves]].

Let $C$ be an [[∞-cohesive site]] of definition for $\mathbf{H}$. Then by the discussion there we have

\begin{displaymath}
\mathbf{H} \simeq ([C^{op}, sSet]_{proj,loc})^\circ
  \,.
\end{displaymath}
Moreover, picking a [[Kan complex]] presentation for $A$, which we shall denote by the same symbol, we have that the constant simplicial presheaf $const A \in [C^{op}, sSet]_{proj, loc}$ is fibrant. Therefore by \href{http://ncatlab.org/nlab/show/model+structure+on+an+over+category#PresentationOfSliceInfinityCat}{this proposition} the induced [[model structure on an overcategory]] on $[C^{op}, sSet]/const A$ presents the given [[over (∞,1)-category]]

\begin{displaymath}
\mathbf{H}_{/ Disc A}
  \simeq
  \left(
   \left([C^{op}, sSet]/const A\right)_{(proj,loc)/const A}
  \right)^\circ
  \,.
\end{displaymath}
Now observe that we have an ordinary [[equivalence of categories]]

\begin{displaymath}
[C^{op}, sSet]/const A 
   \simeq
  [C^op, sSet/A]
\end{displaymath}
under which the model structure becomes that of the local projective [[model structure on functors]] with values in the model structure $(sSet/A)_{Quillen/A}$ that presents $\infty Grpd_{/ A}$.

Let then $sSet^+/A$ denote the [[model structure for left fibrations]]. By the discussion there, this also presents $\infty Grpd_{/ A}$. Hence by \href{http://ncatlab.org/nlab/show/model+structure+on+functors#PresentationOfInfinityFunctors}{this proposition} we have an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
\begin{aligned}
   \mathbf{H}_{Disc A}
   & \simeq
  \left(
    [C^{op}, sSet/A]_{proj,loc}
  \right)^\circ
  \\
  & \simeq
  \left(
   [C^{op}, sSet^+/A]_{proj,loc}
  \right)^\circ
  \end{aligned}
  \,.
\end{displaymath}
This allows now to apply \href{http://ncatlab.org/nlab/show/model%20structure%20for%20left%20fibrations#GrothendieckConstruction}{this presentation} of the [[(∞,1)-Grothendieck construction]] to find

\begin{displaymath}
\cdots \simeq
  \left(
   [C^{op}, [w(A), sSet_{Quillen}]_{proj}]_{proj,loc}
  \right)^\circ
  \,,
\end{displaymath}
where $w(A)$ is the [[simplicially enriched category]] corresponding to $A$ (as discussed at \emph{[[relation between quasi-categories and simplicial categories]]} ) and $[w(A), sSet_{Quillen}]_{proj}$ is the global [[model structure on sSet-enriched presheaves]].

Then using the [[cartesian closed category|cartesian closure]] of the category of simplicial presheaves (which is a [[topos]]) inside the $(-)^\circ$ we have

\begin{displaymath}
\cdots
  \simeq
  \left(
   [w(A), [C^{op}, sSet_{Quillen}]_{proj,loc}]_{proj}
  \right)^\circ
  \,.
\end{displaymath}
Finally this implies the claim using \href{http://ncatlab.org/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-functors#PresentationByModelStructuresOnFunctors}{this proposition}.

\end{proof}
With this lemma we can now give the proof of prop. \ref{PiPreservesPullbacksOverDiscretes}.

\begin{proof}
By the discussion at \href{http://ncatlab.org/nlab/show/adjoint+%28infinity,1%29-functor#OnSlices}{adjoint (∞,1)-functors on slices} we have that $(\Pi \dashv Disc)$ induces an adjoint pair

\begin{displaymath}
(\Pi/Disc A \dashv Disc / Disc A)
  : 
  \mathbf{H}_{Disc A}
  \to 
  \infty Grpd_{/A}

  \,.
\end{displaymath}
Under the equivalence from lemma \ref{GrothendieckConstrInCohesivefinityTopos} the functor $\Pi/Disc A$ maps to

\begin{displaymath}
Func(A, \Pi) : 
  Func(A, \mathbf{H}) \to 
  Func(A, \infty Grpd)
  \,.
\end{displaymath}
Since [[products]] of $(\infty,1)$-functor $(\infty,1)$-categories are computed objectwise, and since $\Pi$ preserves finite products by the axioms of cohesion, also $Func(A, \Pi)$ preserves finite products, and hence so does $\Pi/Disc A$. But products in the slice over $Disc A$ are [[(∞,1)-pullbacks]] over $Disc A$. So this proves the claim.

\end{proof}
\hypertarget{OverArbitraryBases}{}\subsubsection*{{Base of discrete and codiscrete objects}}\label{OverArbitraryBases}

In the \hyperlink{InternalDefinition}{internal definition} the base of discrete/codiscrete objects is not explicitly axiomatized to be an [[(∞,1)-topos]] itself (the [[base (∞,1)-topos]]), but this is implied by the axioms. We deduce that and related properties in stages.

In the following, let $\mathbf{H}$ be an [[(∞,1)-topos]] equipped with an [[adjoint quadruple]] of functors to an [[(∞,1)-category]] $\mathbf{B}$ -- the \emph{base of cohesion}, where $Disc$ and $coDisc$ are full and faithful.

\begin{prop}
\label{LimitsInBase}\hypertarget{LimitsInBase}{}
The base $\mathbf{B}$ of cohesion has all [[(∞,1)-limits]] and [[(∞,1)-colimits]].

\end{prop}
\begin{proof}
This is a general property of a reflectively and coreflectively embedded subcategory. The limits are computed by computing them in $\mathbf{H}$ and then applying $\Gamma$ and the colimits are computed by computing them in $\mathbf{H}$ and then applying $\Pi$. For $X : I \to \mathbf{B}$ any [[diagram]] we have

\begin{displaymath}
\begin{aligned}
    \Gamma \lim_{\leftarrow_i} Disc X_i
    & \simeq
    \lim_{\leftarrow_i} \Gamma Disc X_i 
    \\
    & \simeq 
    \lim_{\leftarrow_i}  X_i 
  \end{aligned}
\end{displaymath}
\begin{displaymath}
\begin{aligned}
    \Pi \lim_{\to_i} Disc X_i
    & \simeq
    \lim_{\leftarrow_i} \Pi Disc X_i 
    \\
    & \simeq 
    \lim_{\leftarrow_i}  X_i 
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Since $Disc$, being both a [[left adjoint]] as well as a [[right adjoint]] preserves limits and colimits, it follows that a (co)limit of discrete objects computed in $\mathbf{H}$ is itself again discrete and is the image under $Disc$ of the coresponding (co)limit computed in $\mathbf{B}$.

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
Since [[loop space objects]] are [[(∞,1)-limits]] it follows that the loop space object of any discrete object is itself again a discrete object.

\end{example}
We have also the following stronger statement.

\begin{prop}
\label{}\hypertarget{}{}
The base of cohesion $\mathbf{B}$ is a [[presentable (∞,1)-category]] and in fact an [[(∞,1)-topos]] itself.

\end{prop}
\begin{proof}
By one of the equivalent characterizations of [[presentable (∞,1)-categories]] these are [[reflective sub-(∞,1)-categories]] of [[(∞,1)-categories of (∞,1)-presheaves]] where the embedding is by an [[accessible (∞,1)-functor]].

Since $\mathbf{H}$ is itself accessibly and reflectively embedded into the presheaves $PSh(C)$ on a [[(∞,1)-site]] of definition, we have a composite reflective inclusions

\begin{displaymath}
\mathbf{B}
  \stackrel{Disc}{\hookrightarrow}
  \mathbf{H}
  \hookrightarrow
  PSh(C)
  \,.
\end{displaymath}
Since $Disc$ even preserves all [[(∞,1)-colimits]], it is in particular an [[accessible (∞,1)-functor]], hence so is the above composite.

Finally, since $\Gamma$ preserves all [[(∞,1)-limits]], hence in particular the [[finite limits]], $(\Gamma, coDisc)$ is a [[geometric embedding]] that exhibits an sub-[[(∞,1)-topos]].

\end{proof}
Notice that the reflection $(\Pi \dashv Disc)$ does not in general constitute a [[geometric embedding]], since $\Pi$ is only required to preserve finite products (and in interesting examples rarely preserves more limits than that).

The following statement and its proof about [[cohesive topos|cohesive 1-toposes]] should hold verbatim also for cohesive $(\infty,1)$-toposes.

\begin{prop}
\label{}\hypertarget{}{}
The [[reflective subcategories]] of discrete objects and of codiscrete objects are both [[exponential ideals]].

\end{prop}
\begin{proof}
By the discussion at [[exponential ideal]] a reflective subcategories of a [[cartesian closed category]] is an exponential ideal precisely if the [[reflector]] preserves [[products]]. For the codiscrete objects the reflector $\Gamma$ preserves even all [[limits]] and for the discrete objects the reflector $\Pi$ does so by assumotion of strong connectedness.

\end{proof}
\hypertarget{FactorizationSystemsForPi}{}\subsubsection*{{Factorization systems associated to $\mathbf{\Pi}$}}\label{FactorizationSystemsForPi}

We discuss [[orthogonal factorization system in an (infinity,1)-category|orthogonal factorization systems]] in a cohesive $(\infty,1)$-topos that characterize or are characterized by the [[reflective sub-(infinity,1)-category|reflective subcategory]] of dicrete objects, with reflector $\mathbf{\Pi} : \mathbf{H} \stackrel{\Pi}{\to} \infty Grpd \stackrel{Disc}{\hookrightarrow} \mathbf{H}$.

\begin{defn}
\label{PiClosure}\hypertarget{PiClosure}{}
For $f : X \to Y$ a morphism in $\mathbf{H}$, write $c_{\mathbf{\Pi}} f \to Y$ for the [[(∞,1)-pullback]] in

\begin{displaymath}
\itexarray{
    c_{\mathbf{\Pi}} f &\to & \mathbf{\Pi} X
    \\
    \downarrow && \downarrow^{\mathrlap{\mathbf{\Pi} f}}
    \\
    Y &\to& \mathbf{\Pi} Y
  }
  \,,
\end{displaymath}
where the bottom morphism is the $(\Pi \dashv Disc)$-[[unit of an adjunction|unit]]. We say that $c_{\mathbf{\Pi}} f$ is the \textbf{$\mathbf{\Pi}$-closure} of $f$, and that $f$ is \textbf{$\mathbf{\Pi}$[[Pi-closed morphism|-closed]]} if $X \simeq c_{\mathbf{\Pi}} f$.

\end{defn}
\begin{prop}
\label{FactorizationPiEquivalencePiClosed}\hypertarget{FactorizationPiEquivalencePiClosed}{}
If $\mathbf{H}$ has an [[∞-cohesive site]] of definition, then every morphism $f : X \to Y$ in $\mathbf{H}$ factors as

\begin{displaymath}
\itexarray{
     X &&\stackrel{f}{\to}&& Y
     \\
     & \searrow && \nearrow
     \\
     && c_{\mathbf{\Pi}}f 
  }
  \,,
\end{displaymath}
such that $X \to c_{\mathbf{\Pi}} f$ is a \emph{$\mathbf{\Pi}$-equivalence} in that it is inverted by $\mathbf{\Pi}$.

\end{prop}
\begin{proof}
The factorization is given by the naturality of $\mathbf{\Pi}$ and the universal property of the $(\infty,1)$-pullback in def. \ref{PiClosure}.

\begin{displaymath}
\itexarray{
    X &\to & c_{\mathbf{\Pi}} f &\to & \mathbf{\Pi} X
    \\
    &{}_{\mathllap{f}}\searrow & \downarrow && \downarrow^{\mathrlap{\mathbf{\Pi} f}}
    \\
    && Y &\to& \mathbf{\Pi} Y
  }
  \,.
\end{displaymath}
Then by prop. \ref{PiPreservesPullbacksOverDiscretes} the functor $\mathbf{\Pi}$ preserves the $(\infty,1)$-pullback over the discrete object $\mathbf{\Pi}Y$ and since $\mathbf{\Pi}(X \to \mathbf{\Pi}X)$ is an equivalence, it follows that $\mathbf{\Pi}(X \to c_{\mathbf{\Pi}f})$ is an equivalence.

\end{proof}
\begin{prop}
\label{PiEquivalencePiClosedFactorizationSystem}\hypertarget{PiEquivalencePiClosedFactorizationSystem}{}
The pair of classes

\begin{displaymath}
(\mathbf{\Pi}-equivalences, \mathbf{\Pi}-closed morphisms)
\end{displaymath}
is an [[orthogonal factorization system in an (infinity,1)-category|orthogonal factorization system]] in $\mathbf{H}$.

\end{prop}
\begin{proof}
This follows by the general reasoning discussed at [[reflective factorization system]]:

By prop. \ref{FactorizationPiEquivalencePiClosed} we have the required factorization. It remains to check the orthogonality.

So let

\begin{displaymath}
\itexarray{
     A &\to& X 
     \\
     \downarrow && \downarrow 
     \\
     B &\to& Y
  }
\end{displaymath}
be a square diagram in $\mathbf{H}$ where the left morphism is a $\mathbf{\Pi}$-equivalence and the right morphism is $\mathbf{\Pi}$-closed. Then by assumption there is a pullback square on the right in

\begin{displaymath}
\itexarray{
     A &\to& X &\to& \mathbf{\Pi}X 
     \\
     \downarrow && \downarrow  && \downarrow
     \\
     B &\to& Y &\to& \mathbf{\Pi}Y
     \,.
  }
  \,.
\end{displaymath}
By naturality of the [[unit of an adjunction|adjunction unit]], the total rectangle is equivalent to

\begin{displaymath}
\itexarray{
    A &\to& \mathbf{\Pi} A &\to & \mathbf{\Pi} Y
    \\
    \downarrow && \downarrow^{\mathrlap{\simeq}} && \downarrow
    \\
    B &\to& \mathbf{\Pi} B &\to& \mathbf{\Pi}X
  }
  \,.
\end{displaymath}
Here by assumption the middle morphism is an equivalence. Therefore there is an essentially unique lift in the square on the right and hence a lift in the total square. Again by the universality of the adjunction, any such lift factors through $\mathbf{\Pi} B$ and hence also this lift is essentially unique.

Finally by universality of the pullback, this induces an essentially unique lift $\sigma$ in

\begin{displaymath}
\itexarray{
     A &\to& X &\to& \mathbf{\Pi}X 
     \\
     \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow  && \downarrow
     \\
     B &\to& Y &\to& \mathbf{\Pi}Y
     \,.
  }
  \,.
\end{displaymath}
\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
For $f : X \to Y$ a $\mathbf{\Pi}$-closed morphism and $y : * \to Y$ a [[global element]], the [[homotopy fiber]] $X_y := y^* X$ is a discrete object.

\end{prop}
\begin{proof}
By the def. \ref{PiClosure} and the [[pasting law]] we have that $y^* X$ is equivalently the $\infty$-pullback in

\begin{displaymath}
\itexarray{
    y^* X &\to& &\to& \mathbf{\Pi} X
    \\
    \downarrow && && \downarrow
    \\
    * &\stackrel{y}{\to}& Y &\stackrel{}{\to}& \mathbf{\Pi}Y
  }
  \,.
\end{displaymath}
Since the [[terminal object in an (infinity,1)-category|terminal object]] is discrete, and since the [[right adjoint]] $Disc$ preserves $\infty$-pullbacks, this exhibits $y^* X$ as the image under $Disc$ of an $\infty$-pullback of $\infty$-groupoids.

\end{proof}
\hypertarget{Structures}{}\subsection*{{Structures in a cohesive $(\infty,1)$-topos}}\label{Structures}

A cohesive $(\infty,1)$-topos is a general context for [[higher geometry]] with good [[cohomology]] and [[homotopy]] properties. We list fundamental structures and constructions that exist in every cohesvive $(\infty,1)$-topos.

This section is at

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

\end{itemize}
\hypertarget{types_of_cohesion}{}\subsection*{{Types of cohesion}}\label{types_of_cohesion}

\hypertarget{infinitesimal_cohesion}{}\subsubsection*{{Infinitesimal cohesion}}\label{infinitesimal_cohesion}

\begin{itemize}%
\item [[infinitesimal cohesion]]

\end{itemize}
\hypertarget{DifferentialCohesion}{}\subsubsection*{{Differential cohesion}}\label{DifferentialCohesion}

We discuss [[extra structure]] on a cohesive $(\infty,1)$-topos that encodes a refinement of the corresponding notion of cohesion to \emph{infinitesimal cohesion} . More precisely, we consider inclusions $\mathbf{H} \hookrightarrow \mathbf{H}_{th}$ of cohesive $(\infty,1)$-toposes that exhibit the objects of $\mathbf{H}_{th}$ as infinitesimal cohesive neighbourhoods of objects in $\mathbf{H}$.

This section is at

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

\end{itemize}
\hypertarget{locally_ringed_cohesion}{}\subsubsection*{{Locally ringed cohesion}}\label{locally_ringed_cohesion}

Every cohesive $(\infty,1)$-topos $\mathbf{H}$ equipped with \hyperlink{DifferentialCohesion}{differential cohesion} comes canonically equipped with a notion of [[formally étale morphism]]s (as discussed there). Combined with the canonical interpretation of $\mathbf{H}$ as the [[classifying topos]] of a [[theory]] of [[local algebra|local T-algebra]]s, this canonically induces a notion of [[locally algebra-ed topos|locally algebra-ed (∞,1)-toposes]] with cohesive structure, generalizing the notion of [[locally ringed space]]s and [[locally ringed topos]]es.

This section is at

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

\end{itemize}
\hypertarget{Examples}{}\subsection*{{Examples of cohesive $\infty$-toposes}}\label{Examples}

We list examples of cohesive $(\infty,1)$-toposes, both specific ones as well as classes of examples constructed in a certain way.

\hypertarget{cohesive_diagram_toposes}{}\subsubsection*{{Cohesive diagram $(\infty,1)$-toposes}}\label{cohesive_diagram_toposes}

\hypertarget{CohesiveDiagramToposes}{}\paragraph*{{Cohesive diagrams in a cohesive $(\infty,1)$-topos}}\label{CohesiveDiagramToposes}

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathbf{H}$ be an cohesive $(\infty,1)$-topos.

Let $D$ be a [[small category]] ([[diagram]]) with [[initial object]] $\bottom$ and [[terminal object]] $\top$, or else a [[presentable (∞,1)-category]]. Write

\begin{displaymath}
(\bottom \dashv p \dashv \top)
  :
  D \stackrel{\overset{\bottom}{\hookleftarrow}}{\stackrel{\overset{p}{\to}}{\underset{\top}{\hookleftarrow}}}
  *
\end{displaymath}
for the [[adjoint triple|triple]] of [[adjoint (∞,1)-functors]] given by including $\bottom$ and $\top$.

Then the [[(∞,1)-category of (∞,1)-functors|(∞,1)-functor category]] $\mathbf{H}^D$ is again a cohesive $(\infty,1)$-topos, exhibited by the [[adjoint quadruple]] which is the composite

\begin{displaymath}
\mathbf{H}^D
   \stackrel{\overset{\top^*}{\to}}{\stackrel{\overset{p^*}{\hookleftarrow}}{\stackrel{\overset{\bottom^*}{\to}}{\underset{\bottom_*}{\hookleftarrow}}}}
  \mathbf{H}
   \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\hookleftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\hookleftarrow}}}}
  \infty Grpd
  \,,
\end{displaymath}
where the [[adjoint quadruple]] on the left is that induced under [[(∞,1)-Kan extension]] from $(\bottom \dashv p \dashv \top)$.

\end{prop}
\begin{proof}
By the discussion at [[(∞,1)-Kan extension]] each of the original three functors induces adjoint triples $(\bottom_! \dashv \bottom^* \dashv \bottom_*)$ etc, as indicated. In particular $\top^*$ is a [[right adjoint] and therefore preserves finite products (and all small [[(∞,1)-limits]], even).

By the original adjunctions one finds that $\bottom_! \simeq p^*$ and $p_! \simeq \top^*$, which implies the adjoint quadruple as indicated above by essential uniqueness of adjoints.

Finally it is clear that $\top^* p^* \simeq Id$, which implies that $p^*$ is a [[full and faithful (∞,1)-functor]] (and hence so is $\bottom_*$).

\end{proof}
In particular we have

\begin{cor}
\label{}\hypertarget{}{}
For $\mathbf{H}$ a cohesive $(\infty,1)$-topos, also its [[arrow category|arrow (∞,1)-category]] $\mathbf{H}^{\Delta[1]}$ is cohesive.

\end{cor}
\begin{example}
\label{SierpinskiTopos}\hypertarget{SierpinskiTopos}{}
For $\mathbf{H} =$ [[∞Grpd]] (``discrete cohesion'', see \hyperlink{DiscreteInfinityGroupoids}{below}) the corresponding cohesive $(\infty,1)$-topos $\infty Grpd^{\Delta[1]}$ is known as the \emph{[[Sierpinski (∞,1)-topos]]}.

\end{example}
\hypertarget{SimplicialObjctsInACohesiveTopos}{}\paragraph*{{Simplicial objects in a cohesive $(\infty,1)$-topos}}\label{SimplicialObjctsInACohesiveTopos}

For $\mathbf{H}$ a cohesive $(\infty,1)$-topos its [[category of simplicial objects|(∞,1)-category of simplicial objects]] $\mathbf{H}^{\Delta^{op}}$ is cohesive over $\mathbf{H}$

\begin{displaymath}
\mathbf{H}^{\Delta^{op}}
  \stackrel{\Pi_I}{\stackrel{\longrightarrow}{\stackrel{\overset{Disc_I}{\longleftarrow}}{\stackrel{\overset{\Gamma_I}{\longrightarrow}}{\underset{coDisc_I}{\longleftarrow}}}}}
  \mathbf{H}
  \,.
\end{displaymath}
Here

\begin{itemize}%
\item $\Pi_I$ sends a [[simplicial object]] to the [[homotopy colimit]] over its components, hence to its ``[[geometric realization]]'' as seen in $\mathbf{H}$.


\item $\Gamma_I$ evaluates on the 0-simplex;


\item $Disc_I$ sends an object in $\mathbf{H}$ to the simplicial object which is simplicially constant on $A$.



\end{itemize}
Hence cohesion of $\mathbf{H}^{\Delta^{op}}$ relative to $\mathbf{H}$ expresses the existence of a discrete and directed notion of path.

The simplicial interval $\Delta^1 \in \mathbf{H}^{\Delta^{op}}$ (regarded under [[(∞,1)-Yoneda embedding]]) \emph{exhibits the cohesion} of $\mathbf{H}^{\Delta^{op}}$ over $\mathbf{H}$ in the sense of def. \ref{A1ExhibitingCohesion}, in that the relative [[shape modality]] $\Pi_I$ is equivalent to the ``[[A1-homotopy theory|A1-localization]]'' at $\mathbb{A}^1 = \Delta^1$

\begin{displaymath}
\Pi_I \simeq L_{\Delta^1}
  \,.
\end{displaymath}
Notice that there is an inclusion

\begin{displaymath}
Grpd(\mathbf{H})
  \hookrightarrow
  Cat(\mathbf{H})
  \hookrightarrow
  \mathbf{H}^{\Delta^{op}}
\end{displaymath}
of the [[groupoid object in an (∞,1)-category|groupoid objects]] internal to $\mathbf{H}$ and of the [[category object in an (∞,1)-category|category objects]] internal to $\mathbf{H}$ inside $\mathbf{H}^{\Delta^{op}}$.

Here $\mathbf{H}^{\Delta^{op}}$ is also the [[classifying topos]] for [[linear intervals]]. Its [[homotopy type theory]] [[internal language]] is equipped with an interval [[type]].

For more see at \emph{[[simplicial object in an (∞,1)-category]]}.

\hypertarget{bundles_of_cohesive_spectra}{}\paragraph*{{Bundles of cohesive spectra}}\label{bundles_of_cohesive_spectra}

The [[tangent (∞,1)-category]] $T\mathbf{H}$ to a cohesive $\infty$-topos is itself cohesive, (by the discussion at \emph{\href{tangent+%28infinity%2C1%29-category#TangentTopos}{tangent ∞-category -- Examples -- Of an ∞-topos}}), the \emph{[[tangent cohesive (∞,1)-topos]]}.

This $T \mathbf{H}$ the $\infty$-topos of [[parameterized spectra]] in $\mathbf{H}$, hence is context for cohesive [[stable homotopy theory]].

\begin{displaymath}
\itexarray{
    Stab(\mathbf{H})
    &
    \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
    &
    Stab(\infty Grpd)
    \\
    \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}}
    \\
    T \mathbf{H}
    &
    \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
  &
  T \infty Grpd
  \\
  {}^{\mathllap{base}}\downarrow {}^{\mathllap{0}}\uparrow \downarrow^{\mathrlap{base}} \uparrow^{\mathrlap{0}}
  && 
  {}^{\mathllap{base}}\downarrow {}^{\mathllap{0}}\uparrow \downarrow^{\mathrlap{base}} \uparrow^{\mathrlap{0}}
  \\
    \mathbf{H}
    &
    \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}
  &
  \infty Grpd
  }
  \,.
\end{displaymath}
\hypertarget{global_equivariant_homotopy_theory}{}\subsubsection*{{Global equivariant homotopy theory}}\label{global_equivariant_homotopy_theory}

[[!include equivariant homotopy theory -- table]]

\hypertarget{ExamplesFromCohesiveSitesOnfDefinition}{}\subsubsection*{{From $\infty$-Cohesive sites of definition}}\label{ExamplesFromCohesiveSitesOnfDefinition}

\begin{prop}
\label{}\hypertarget{}{}
Examples of [[∞-cohesive site]]s are

\begin{itemize}%
\item any category with finite [[product]]s and equipped with the trivial [[coverage]].


\item the [[full subcategory]] $CartSp_{top} \subset$ [[Top]] on [[open ball]]s with the [[good open cover]] [[coverage]];


\item the site [[CartSp]] of [[Cartesian space]]s with [[smooth function]]s between them and [[good open cover]] [[coverage]].


\item the [[Cahiers topos]]-site [[ThCartSp]] of infinitesimally thickened Cartesian spaces.



\end{itemize}
\end{prop}
From this one obtains the following list of examples of cohesive $(\infty,1)$-toposes.

\hypertarget{DiscreteInfinityGroupoids}{}\paragraph*{{Discrete $\infty$-groupoids}}\label{DiscreteInfinityGroupoids}

\begin{itemize}%
\item [[discrete ∞-groupoid]]

\end{itemize}
\hypertarget{topological_groupoids}{}\paragraph*{{Topological $\infty$-groupoids}}\label{topological_groupoids}

\begin{itemize}%
\item [[Euclidean-topological ∞-groupoid]]


\item [[locally contractible topological ∞-groupoid]]



\end{itemize}
\hypertarget{smooth_groupoids}{}\paragraph*{{Smooth $\infty$-groupoids}}\label{smooth_groupoids}

\begin{itemize}%
\item [[smooth ∞-groupoid]]

\end{itemize}
\hypertarget{complex_analytic_groupoids}{}\paragraph*{{Complex analytic $\infty$-groupoids}}\label{complex_analytic_groupoids}

\begin{itemize}%
\item [[complex analytic ∞-groupoid]]

\end{itemize}
\hypertarget{formal_smooth_groupoids}{}\paragraph*{{Formal smooth $\infty$-groupoids}}\label{formal_smooth_groupoids}

\begin{itemize}%
\item [[formal smooth ∞-groupoid]]

\end{itemize}
\hypertarget{smooth_super_groupoids}{}\paragraph*{{Smooth Super $\infty$-groupoids}}\label{smooth_super_groupoids}

\begin{itemize}%
\item [[super ∞-groupoid]]


\item [[smooth super ∞-groupoid]]


\item [[synthetic differential super ∞-groupoid]]



\end{itemize}
\hypertarget{smooth_groupoids_over_algebraic_stacks}{}\paragraph*{{Smooth $\infty$-groupoids over algebraic $\infty$-stacks}}\label{smooth_groupoids_over_algebraic_stacks}

One can consider the [[tangent (∞,1)-topos]] of the [[cohesive (∞,1)-topos]]

\begin{displaymath}
Sh_\infty\left(SmthMfd, Sh_\infty\left(Sch_{\mathbb{Z}}\right)\right)
  \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}
  Sh_\infty(Sch_{\mathbb{Z}})
\end{displaymath}
of [[∞-stacks]] on the [[site]] of [[smooth manifolds]] with values in turn in [[∞-stack]] over a [[site]] of [[arithmetic schemes]], hence by [[smooth ∞-groupoids]] but over a [[base (∞,1)-topos]] of algebraic [[∞-stacks]].

This leads to [[differential algebraic K-theory]]. See there for details.

\hypertarget{arithmetic_groupoids}{}\subsubsection*{{$E_\infty$-Arithmetic $\infty$-groupoids}}\label{arithmetic_groupoids}

see \emph{[[differential cohesion and idelic structure]]}

[[!include arithmetic cohesion -- table]]

\hypertarget{smooth_cohesion_over_arbitrary_base_toposes}{}\subsubsection*{{Smooth cohesion over arbitrary base $\infty$-toposes}}\label{smooth_cohesion_over_arbitrary_base_toposes}

The above examples relativize to \hyperlink{OverArbitraryBases}{arbitrary bases}.

For instance [[smooth infinity-groupoid|smooth cohesion]] over an [[(∞,1)-topos]] over a suitable [[site]] of [[schemes]] is the natural context for [[differential algebraic K-theory]]. See there for more.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

As a context for geometric spaces and paths in geometric spaces, cohesive $(\infty,1)$-toposes are a natural context in which to formulate fundamental fundamental [[physics]]. See [[higher category theory and physics]] for more on this.

See also

\begin{itemize}%
\item [[∞-Chern-Weil theory]]


\item [[schreiber:∞-Chern-Simons theory]]



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

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

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


\item [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]


\item [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]



\end{itemize}

\item [[local topos]] / [[local (∞,1)-topos]]

\begin{itemize}%
\item [[Pi modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]]

\end{itemize}

\item [[cohesive topos]] / \textbf{cohesive (∞,1)-topos}



\end{itemize}
and

\begin{itemize}%
\item [[locally connected site]], [[locally ∞-connected site]]


\item [[connected site]]


\item [[local site]]


\item [[cohesive site]], [[(∞,1)-cohesive site]]



\end{itemize}
\emph{unrelated} is the notion of \emph{[[cohesive ∞-prestack]]}

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

\hypertarget{precursors}{}\subsubsection*{{Precursors}}\label{precursors}

The [[category theory|category-theoretic]] definition of [[cohesive topos]] was proposed by [[Bill Lawvere]]. See the references at \emph{[[cohesive topos]]}.

The observation that the further left adjoint $\Pi$ in a [[locally ∞-connected (∞,1)-topos]] defines an intrinsic notion of paths and [[geometric homotopy groups in an (∞,1)-topos]] was suggested by [[Richard Williamson]].

The observation that the further right adjoint $coDisc$ in a [[local (∞,1)-topos]] serves to characterize [[concrete sheaf|concrete (∞,1)-sheaves]] was amplified by [[David Carchedi]].

Some aspects of the discussion here are, more or less explicitly, in

\begin{itemize}%
\item [[Carlos Simpson]], [[Constantin Teleman]], \emph{deRham theorem for $\infty$-stacks} (\href{http://math.berkeley.edu/~teleman/math/simpson.pdf}{pdf})

\end{itemize}
For instance something similar to the notion of [[infinity-connected (infinity,1)-site|∞-connected site]] and the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] is the content of section 2.16. The \hyperlink{LieTheory}{infinitesimal path ∞-groupoid adjunction} $(\mathbf{Red} \dashv \mathbf{\Pi}_{inf} \dashv \mathbf{\flat}_{inf})$ is essentially discussed in section 3. The notion of geometric realization (see ), is touched on around remark 2.22, referring to

\begin{itemize}%
\item [[Carlos Simpson]], \emph{The topological realization of a simplicial presheaf} , (\href{http://arxiv.org/abs/q-alg/9609004}{arXiv:q-alg/9609004}).

\end{itemize}
But, more or less explicitly, the presentation of geometric realization of simplicial presheaves is much older, going back to Artin-Mazur. See [[geometric homotopy groups in an (∞,1)-topos]] for a detailed commented list of literature.

A characterization of infinitesimal extensions and formal smoothness by adjoint functors (discussed at [[infinitesimal cohesion]]) is considered in

\begin{itemize}%
\item [[Maxim Kontsevich]], [[Alexander Rosenberg]], \emph{Noncommutative spaces}, preprint MPI-2004-35 (\href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2331}{ps}, \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2303}{dvi})

\end{itemize}
in the context of \emph{[[Q-categories]]} .

\hypertarget{on_cohesive_toposes_proper}{}\subsubsection*{{On cohesive $(\infty,1)$-toposes proper}}\label{on_cohesive_toposes_proper}

The material presented here is also in section 3 of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} (\href{http://arxiv.org/abs/1310.7930}{arXiv:1310.7930})

\end{itemize}
\hypertarget{ReferencesInHoTT}{}\subsubsection*{{In homotopy type theory}}\label{ReferencesInHoTT}

Expositions and discussion of the formalization of cohesion in [[homotopy type theory]] is in

\begin{itemize}%
\item [[Mike Shulman]], \emph{Axiomatic cohesion in HoTT} (\href{http://homotopytypetheory.org/2011/11/02/axiomatic-cohesion-in-hott/}{blog post})

\end{itemize}
The corresponding [[Coq]]-code is in

\begin{itemize}%
\item [[Mike Shulman]], \emph{\href{https://github.com/mikeshulman/HoTT/tree/master/Coq/Subcategories}{HoTT/Coq/Subcategories}}

\end{itemize}
Exposition is at

\begin{itemize}%
\item [[Mike Shulman]], \emph{Internalizing the External, or The Joys of Codiscreteness} (\href{http://golem.ph.utexas.edu/category/2011/11/internalizing_the_external_or.html}{blog post})

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

[[!redirects cohesive (∞,1)-toposes]] [[!redirects cohesive (infinity,1)-toposes]]

[[!redirects cohesive (∞,1)-topoi]] [[!redirects cohesive (infinity,1)-topoi]]

[[!redirects cohesive ∞-topos]] [[!redirects cohesive ∞-toposes]]

[[!redirects cohesive infinity-topos]] [[!redirects cohesive infinity-toposes]]

[[!redirects cohesive ∞-groupoid]] [[!redirects cohesive ∞-groupoids]]

[[!redirects cohesive infinity-groupoid]] [[!redirects cohesive infinity-groupoids]]

[[!redirects cohesive homotopy theory]] [[!redirects cohesive homotopy theories]]

[[!redirects cohesive model topos]] [[!redirects cohesive model toposes]]



\end{document}