shape of an (infinity,1)-topos

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#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
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#Contents#
* table of contents
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## Idea
 {#Idea}

To the extent that an [[(∞,1)-topos]] may be thought of as representing a *[[space]]*, the *shape* of the $\infty$-topos is the underlying [[pro-homotopy theory|pro-]][[homotopy type]] of this space (Def. \ref{ShapeOfAnInfinityTopos} below). Equivalently, the shape of an $\infty$-topos is the generalized [[étale homotopy]]-type of its [[terminal object]] (Def. \ref{ShapeOfToposAsImageOfItsPointUnderProLeftAdjointToLConst}, Prop. \ref{ShapeOfToposIsShapeOfItsTerminalObject} below).

In the special case that the [[(∞,1)-topos]] is that of [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaves]] on (the [[site]] of [[category of open subsets|open subsets]] of) a [[paracompact space|paracompact]] [[topological space]], its shape coincides with the [[strong shape]] of $X$ in the classical sense of [[shape theory]].

In the special case that the $\infty$-topos is a [[slice (infinity,1)-topos|slice]] of a [[cohesive (infinity,1)-topos|cohesive $(\infty,1)$-topos]] $\mathbf{H}$ over some object, its shape is the [[shape modality|cohesive shape]] of that object (Prop. \ref{ShapeOfSliceOfCohesiveInfinityTopos} below).

## Definition
 {#Definition}

We state three somewhat different-looking definitions, and show that they are all equivalent to each other.

\begin{definition}\label{ShapeOfAnInfinityTopos}
**(shape of an $\infty$-topos -- [Toën-Vezzosi 2002, Def. 5.3.2](#ToenVezzosi02))**
\linebreak
The composite [[(∞,1)-functor]]

$$
  Shp
  \;\colon\;
  (\infty,1)Topos
     \stackrel{Y}{\to}
  Func((\infty,1)Topos, \infty Grpd)^{op}
      \stackrel{Lex(PSh(-), \infty Grpd)}{\to}
      AccLex(\infty Grpd, \infty Grpd)^{op}
     \simeq
       Pro \infty Grpd
$$

is the **shape functor** . Its value

$$
  Shp(\mathbf{H}) =
  (\infty,1)Topos
  \big(
    \mathbf{H}
    ,\,
    PSh(-)
  \big)
$$

on an $(\infty,1)$-topos $\mathbf{H}$ is the **shape** of $\mathbf{H}$.
\end{definition}

Here:

* [[(∞,1)Topos]] is the [[(∞,1)-category]] of [[(∞,1)-topos]]es;

* [[∞Grpd]] is the $(\infty,1)$-category of [[∞-groupoid]]s;

* $Y$ is the [[(∞,1)-Yoneda embedding]];

* $Func(-,-)$ is the [[(∞,1)-category of (∞,1)-functors]];

* $AccLex(-,-) \subset (\infty,1)Func(-,-)$ is the full [[sub-(∞,1)-category]]
  of the [[(∞,1)-category of (∞,1)-functors]] on those which are left [[exact functor]]s (preserve finite [[(∞,1)-limit]]s) and also [[accessible (∞,1)-functor|accessible]].

* $PSh(-) : \infty Grpd \to (\infty,1)Topos$ is the functor that
  produces the [[(∞,1)-category of (∞,1)-presheaves]] $Func(X^{op}, \infty Grpd)$ on $X$ (equivalently on the equivalent [[opposite (∞,1)-category|opposite ∞-groupoid]] $X^{op}$);

* $Pro \infty Grpd$ is the [[pro-object in an (∞,1)-category|(∞,1)-category of pro-objects]] in $\infty Grpd$.

That this does indeed land in accessible left exact functors follows from the equivalence to the following definition, see Rem. \ref{ShapeIsLexAndAccessible} below.

Notice (see [here](https://ncatlab.org/nlab/show/infinity1-topos#GlobalSectionsGeometricMorphism)) that for every [[(∞,1)-topos]] $\mathbf{H}$ there is a unique [[geometric morphism]]

$$
  (LConst \dashv \Gamma)
  \;\colon\;
  \mathbf{H}
  \underoverset
    { \underset{\Gamma}{\longrightarrow} }
    { \overset{LConst}{\longleftarrow} }
    { \bot }
  \infty Grpd
  \,,
$$

where

* [[∞Grpd]] is the $(\infty,1)$-topos of [[∞-groupoids]]

* $\Gamma$ is the [[global sections]] [[(∞,1)-functor]]

* $LConst$ is the [[constant ∞-stack]] functor.

\begin{definition}\label{LurieShapeOfAnInfinityTopos}
**(shape of an $\infty$-topos -- [Lurie 2009, Def. 7.1.6.3](#Lurie09))**
\linebreak
The *shape* of an [[(infinity,1)-topos|$(\infty,1)$-topos]] $\mathbf{H}$ is the composite [[(infinity,1)-functor|$(\infty,1)$-functor]]

$$
  Shp(\mathbf{H})
  \;\coloneqq\;
  \Gamma \circ LConst
  \;\;:\;\;
  \infty Grpd \stackrel{LConst}{\to}
  \mathbf{H}
  \stackrel{\;\;\Gamma\;\;}{\to}
  \infty Grpd
$$

regarded as a [[pro-object in an (infinity,1)-category|pro-]][[infinity-groupoid|$\infty$-groupoid]]:

$$
  Shp(\mathbf{H})
  \;\in\;
  Pro(\infty Grpd)
  \;=\;
  Lex(\infty Grpd, \infty Grpd)^{op}
  \,.
$$
\end{definition}

\begin{proposition}\label{ShapeIsGlobalSectionsOfLConst}
Def. \ref{ShapeOfAnInfinityTopos}
and Def. \ref{LurieShapeOfAnInfinityTopos} are equivalent.
\end{proposition}
\begin{proof}
For $X \in $ [[∞Grpd]] we have by the [[(∞,1)-Grothendieck construction]]-theorem and using that up to equivalence every morphism of $\infty$-groupoids is a [[Cartesian fibration]] (see [there](infinity1-Grothendieck+construction#GrpdFibsOverGrpds)) that

$$
  Func(X,\infty Grpd)
  \;\simeq\;
  \infty Grpd_{/X}
$$

is the [[slice (∞,1)-category]].  Moreover, by [this theorem](limit+in+a+quasi-category#WithValInooGrpd) about limits in ∞Grpd we have that the terminal geometric morphism $Hom(*,-): [X, \infty Grpd] \to \infty Grpd$ is the canonical projection $\infty Grpd/ X \to \infty Grpd$. This means that it is an [[etale geometric morphism]]. So for any [[geometric morphism]] $f : \mathbf{H} \to [X, \infty Grpd]$  we have a system of [[adjoint (∞,1)-functor]]s

$$
  (LConst \dashv \Gamma)
  \;\colon\;
  \mathbf{H}
    \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}
   \infty Grpd/X
    \stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}}
   \infty Grpd
  \,.
$$

whose composite is the [[global section]] geometric morphism as indicated, because that is terminal.

Notice that in $\infty Grpd/X$ there is a canonical morphism

$$
  (* \to  \pi^* X)
  \;\coloneqq\;
  (X \stackrel{(Id,Id)}{\to} X \times X)
  \,.
$$

The image of this under $f^*$ is (using that this preserves the terminal object) a morphism  in $\mathbf{H}$ of the form

$$
  * \to f^* \pi^* X = LConst X
  \,.
$$

Conversely, given a morphism of the form $* \to LConst X$ in $\mathbf{H}$ we obtain the [[base change geometric morphism]] of [[slice (infinity,1)-topos|slice $(\infty,1)$-toposes]]

$$
  \mathbf{H}
  \,\simeq\,
   \mathbf{H}_{/*}
   \xrightarrow{\;\;}
   \mathbf{H}_{/LConst X}
   \xrightarrow{\;\; \Gamma\;\;}
  \infty Grpd_{/X}
  \,.
$$

One checks that these constructions establish an equivalence

$$
  (\infty,1)Topos
  \big(
    \mathbf{H}
    ,\,
    \infty Grpd_{/X}
  \big)
  \;\simeq\;
  \mathbf{H}
  (*, LConst X)
  \,.
$$

{#UsingThis} Using this, we find the following sequence of [[equivalence in an (infinity,1)-category|equiavlences]]:

$$
  \begin{aligned}
    Shp (\mathbf{H})
    \;\colon\;
    X
    \;\;
   \mapsto
   \;\;
   &
   (\infty,1)Topos
   \big(
     \mathbf{H}
     ,\,
     \infty Grpd_{/X}
   \big)
     \\
      & \simeq \mathbf{H}(*,LConst X)
     \\
      & \simeq \mathbf{H}(LConst *, LConst X)
     \\
      & \simeq \infty Grpd(*, \Gamma LConst X)
    \\
      & \simeq \Gamma \circ LConst(X)
  \end{aligned}
  \,.
$$

The [[composition|composite]] of these is the equivalence to be shown.
\end{proof}

\begin{remark}\label{ShapeIsLexAndAccessible}
Prop. \ref{ShapeIsGlobalSectionsOfLConst} immediately implies that $Shp(\mathbf{H}) \;\colon\; \infty Grpd \to \infty Grpd$ according to Def. \ref{ShapeOfAnInfinityTopos} does preserve finite $(\infty,1)$-limits, since in the equivalent Def. \ref{LurieShapeOfAnInfinityTopos} this is manifest:
There $\Gamma$ clearly preserves all limits, [[right adjoints preserve limits|since]] it is a [[right adjoint]], and $LConst$ preserves finite limits, since it is a [[left exact functor]] by definition.
Similarly, this makes manifest that $Shp(\mathbf{H})$ is [[accessible (infinity,1)-functor|accessible]], since $\Gamma$ and $LConst$ are both accessible.
\end{remark}

\begin{definition}\label{ShapeOfToposAsImageOfItsPointUnderProLeftAdjointToLConst}
**(shape of an $\infty$-topos -- [Hoyois 2013, p. 3](#Hoyois13))**
\linebreak
  For $(\Gamma \dashv LConst) \;\colon\; \mathbf{H} \to \infty Grpd$ an [[(infinity,1)-topos|$(\infty,1)$-topos]],
  write (with convenient overloading of notation)
  \[
    \label{ProLeftAdjointToLConst}
    \array{
      Shp
      \;\colon\;
      &
      \mathbf{H}
      &\longrightarrow&
      Pro(\infty Grpd)
      \\
      &
      X
      &\mapsto&
      \mathbf{H}
      \big(
        X
        ,\,
        LConst(-)
      \big)
    }
  \]
  for the [[pro-left adjoint]] to $LConst \,\colon\, \infty Grpd \to \mathbf{H},$ and say that the *shape of $\mathbf{H}$* is the image under this pro-left adjoint of its [[terminal object]] $\ast_{\mathbf{H}}$:
$$
  Shp(\mathbf{H})
  \;\coloneqq\;
  Shp(\ast_{\mathbf{H}})
  \,.
$$
\end{definition}

\begin{prop}\label{ShapeOfToposIsShapeOfItsTerminalObject}
  Def. \ref{ShapeOfToposAsImageOfItsPointUnderProLeftAdjointToLConst}
  is equivalent to Def. \ref{LurieShapeOfAnInfinityTopos}
\end{prop}
\begin{proof}
Consider the following sequence of [[natural equivalence|natural]] [[equivalence in an (infinity,1)-category|equivalences]]:
$$
\begin{aligned}
  Shp(\mathbf{H})
  &
  \;\simeq\;
  \Gamma \circ LConst(-)
  \\
  &
  \;\simeq\;
  \infty Grpd
  \big(
    \ast
    ,\,
    \Gamma \circ LConst(-)
  \big)
  \\
  &
  \;\simeq\;
  \mathbf{H}
  \big(
    LConst(\ast)
    ,\,
    LConst(-)
  \big)
  \\
  & \;\simeq\;
  \mathbf{H}
  \big(
    \ast_{\mathbf{H}}
    ,\,
    LConst(-)
  \big)
  \\
  &
  \;\simeq\;
  Shp(\ast_{\mathbf{H}})
\end{aligned}
$$
Here the line is just Def. \ref{LurieShapeOfAnInfinityTopos} (alternatively: is Prop. \ref{ShapeIsGlobalSectionsOfLConst}), and the second line follows by the [[cartesian monoidal (infinity,1)-category|cartesian]] [[closed monoidal (infinity,1)-category|closure]] of [[∞Grpd]].
The third line is the characteristic [hom-equivalences](adjoint+infinity1-functor#CharacterizationInTermsOfHomEquivalences) of the [[adjoint (infinity,1)-functor|adjunction]] $\Gamma \dashv LConst$ . In the second but last step we use that $LConst$, being a [[left exact functor]], [[preserved limit|preserves]] [[terminal object in an (infinity,1)-category|terminal objects]]. The last step is the definition (eq:ProLeftAdjointToLConst) of the [[pro-left adjoint]].
\end{proof}

\linebreak

## Examples

### Shape of locally $\infty$-connected toposes

\begin{prop}
If $\mathbf{H}$ is [[locally ∞-connected (∞,1)-topos|locally ∞-connected]], in that $\LConst$ has an actual [[left adjoint|left]] [[adjoint (infinity,1)-functor|$(\infty,1)$-functor]] $Shp$ (see at *[[shape modality]]*, in constrast to just a [[pro-left adjoint]] (eq:ProLeftAdjointToLConst))
then the shape of $\mathbf{H}$ is $Shp(*) \in \infty Grpd \xhookrightarrow Prop \infty Grpd$ (under the embedding [here](pro-object+in+an+infinity1-category#PlainObjectsAmongProObjects)):

$$
  Shp(\mathbf{H})
  \;\;
    \simeq
  \;\;
  Shp(\ast)
$$
\end{prop}
\begin{proof}
This follows immediately from Prop. \ref{ShapeOfToposIsShapeOfItsTerminalObject} and the observation that an actual left adjoint, when it exists, of coincides with the [[pro-left adjoint]] under the embedding $\infty Grpd \hookrightarrow Pro(\infty Grpd)$ ([here](pro-object+in+an+infinity1-category#PlainObjectsAmongProObjects))

But we may also see explicitly that we have the following sequence of [[natural equivalence|natural]] [[equivalences of ∞-groupoids]], starting with Def. \ref{LurieShapeOfAnInfinityTopos} (alternatively: Prop. \ref{ShapeIsGlobalSectionsOfLConst}(.

$$
\begin{aligned}
  Shape(\mathbf{H})(A)
    &\simeq \Gamma(LConst(A))
    \\
  &\simeq Hom_{\infty Grpd}(*, \Gamma(LConst(A)))\\
  &\simeq Hom_{\mathbf{H}}(LConst(*), LConst(A)) \\
  &\simeq Hom_{\mathbf{H}}(*, LConst(A)) \\
  &\simeq Hom_{\infty Grpd}(Shp(*),A).
\end{aligned}
$$

\end{proof}

Thus, if we regard $Shp(*)$ as "the [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos|fundamental ∞-groupoid]] of $\mathbf{H}$" --- which is reasonable since when $\mathbf{H}=Sh(X)$ consists of sheaves on a locally contractible [[topological space]] $X$, $Shp_{\mathbf{H}}(*)$ is equivalent to the usual [[fundamental ∞-groupoid]] of $X$ --- then we can regard the shape of an $(\infty,1)$-topos as a generalized version of the "homotopy $\infty$-groupoid" which nevertheless makes sense even for non-locally-contractible toposes, by taking values in the larger category of "pro-$\infty$-groupoids."

It follows also that $\mathbf{H}$ is not only locally ∞-connected but also [[∞-connected (∞,1)-topos|∞-connected]], then it has the shape of a point.

More generally:

\begin{remark}
Let $(f_! \dashv f^* \dashv f_*) : \mathbf{H} \xrightarrow{\;\;} \mathbf{B}$ be an [[essential geometric morphism]] of $(\infty,1)$-toposes that exhibits $\mathbf{B}$ as an essential [[retract]] of $\mathbf{H}$ in that

$$
  (Id \dashv Id)
  \;\;
  \simeq
  \;\;
  \mathbf{B}
    \underoverset
      {\underset{f^*}{\longrightarrow}}
      {\overset{f_!}{\longleftarrow}}
      {\;\;\;\bot\;\;\;}
  \mathbf{H}
    \underoverset
      {\underset{f_*}{\longrightarrow}}
      {\overset{f^*}{\longleftarrow}}
      {\;\;\;\bot\;\;\;}
  \mathbf{B}
  \,.
$$

Then the shape of $\mathbf{B}$ is equivalent to that of $\mathbf{H}$.
\end{remark}
\begin{proof}
Since $\infty Grpd$ is the [[terminal object in an (infinity,1)-category|terminal object]] in the [[(infinity,1)-category|$(\infty,1)$-category]] of [[infinity-stack (infinity,1)-topos|Grothendieck $(\infty,1)$-toposes]] and [[geometric morphisms]] (see [here](infinity1-topos#GlobalSectionsGeometricMorphism)), we have

$$
  \begin{aligned}
    &
    (\infty Grpd
      \xrightarrow{\;LConst_{\mathbf{B}}\;}
     \mathbf{B}
       \xrightarrow{\;\Gamma_\mathbf{B}\;}
     \infty Grpd)
    \\
    &
    \simeq
    (\infty Grpd
      \xrightarrow{\;LConst_{\mathbf{B}}\;}
      \mathbf{B}
      \xrightarrow{\;f^*\;}
      \mathbf{H}
      \xrightarrow{f_*}
      \mathbf{B}
      \xrightarrow{\Gamma_\mathbf{B}}
     \infty Grpd)
    \\
    &
    \simeq
    (\infty Grpd
     \xrightarrow{LConst_\mathbf{H}}
     \mathbf{H}
     \xrightarrow{\Gamma_\mathbf{H}}
     \infty Grpd)
  \end{aligned}
  \,.
$$
\end{proof}

\begin{example}\label{CohesiveInfinityToposHasTrivialShape}
**([[cohesive (∞,1)-topos]] has trivial shape)**
\linebreak
Every

* $\infty$-topos which is both [[locally ∞-connected (∞,1)-topos|locally ∞-connected]] and [[∞-connected (∞,1)-topos|$\infty$-connected]],

* in particular every [[cohesive (∞,1)-topos]]

over $\infty Grpd$ has the shape of the point.
\end{example}
\begin{proof}
By definition $\mathbf{H}$ is $\infty$-connected if the [[constant ∞-stack]] [[inverse image]] $f^* = L Const$ is

1. not only a left but also a [[right adjoint]];

1. is a [[full and faithful (∞,1)-functor]].

By standard properties of [[adjoint (∞,1)-functor]]s we have that a [[right adjoint]] $f^*$ is a [[full and faithful (∞,1)-functor]] precisely if the counit $f_! f^* \to Id$ is an [[equivalence in a quasi-category|equivalence]].

Equivalently, we can observe that a locally ∞-connected (∞,1)-topos is ∞-connected precisely when $Shp$ preserves the terminal object, and apply the above observation that the shape of a locally ∞-connected (∞,1)-topos is represented by $Shp(*)$.
\end{proof}

\begin{remark}\label{TrivialShapeAndGrosToposes}
**(trivial shape of gros $\infty$-toposes)**
\linebreak
  That [[cohesive (infinity,1)-toposes|cohesive $(\infty,1)$-toposes]] have trivial shape (Exp. \ref{CohesiveInfinityToposHasTrivialShape}) is a reflection of their characteristic nature as [[gros toposes]]: Rather than representing a single specific non-trivial space, cohesive $\infty$-toposes are gros $\infty$-categories *of spaces* (of geometric/cohesive spaces, specifically).

This is further brought out by Prop. \ref{ShapeOfSliceOfCohesiveInfinityTopos} below, in view of which Exp. \ref{CohesiveInfinityToposHasTrivialShape}, may be read as saying that the shape of a cohesive $\infty$-topos does not interfere with the cohesive shapes of its objects.
\end{remark}

### Shape of slice of a cohesive $\infty$-topos
 {#ShapeOfSlicesOfCohesiveInfinityToposes}

\begin{proposition}\label{ShapeOfSliceOfCohesiveInfinityTopos}
  If $\mathbf{H}$ is a [[cohesive (infinity,1)-topos|cohesive $(\infty,1)$-topos]] with [[shape modality]]
$$
  ʃ
  \;\;
   \colon
  \;\;
  \mathbf{H}
  \xrightarrow{\;Shp\;}
  \Grp_\infty
  \xrightarrow{\;Disc\;}
  \mathbf{H}
$$
then for every [[object]] $X \,\in\, \mathbf{H}$ the shape of the [[slice (infinity,1)-topos|slice $(\infty,1)$-topos]] $\mathbf{H}_{/X}$, according to Def. \ref{ShapeOfAnInfinityTopos}, is [[equivalence in an (infinity,1)-category|equivalently]] the [[shape modality|cohesive shape]] of $X$:
$$
  Shp
  \big(
    \mathbf{H}_{/X}
  \big)
  \;\;
    \simeq
  \;\;
  Shp(X)
  \;\;\;\;\;
  \in
  \;
  Grp_\infty
  \xhookrightarrow{\;}
  Pro(Grp_\infty)
  \,.
$$
\end{proposition}
\begin{proof}
For $\mathbf{H}$ any [[(infinity,1)-topos|$(\infty,1)$-topos]]
and $X \,\in\, \mathbf{H}$ an [[object]], the [[slice (infinity,1)-topos|slice $(\infty,1)$-topos]] $\mathbf{H}_{/X}$ is related to $\mathbf{H}$ by the [[base change]] [[adjoint triple]] shown on the left here, together with, on the right, part of the [[adjoint quadruple]] that exhibits the [[cohesion]] of $\mathbf{H}$:

\begin{tikzcd}[column sep=large]
    \Gamma_X
    \;\;\;
    \colon
    &[-40pt]
    \mathbf{H}_{/X}
    \ar[
      rr,
      phantom,
      shift left=10+10pt,
      "{\scalebox{.7}{$\bot$}}"
    ]
    \ar[
      rr,
      phantom,
      shift left=-10+10pt,
      "{\scalebox{.7}{$\bot$}}"
    ]
    \ar[
      rr,
      shift left=20+10pt,
      "{ \sum_X }"{description}
    ]
    \ar[
      rr,
      shift left=-20+10pt,
      "{ \prod_X }"{description}
    ]
    &&
    \mathbf{H}
    \ar[
      ll,
      shift right=+10pt,
      "{ X \times (-) }"{description}
    ]
    \ar[
      rr,
      phantom,
      shift left=10+10pt,
      "{\scalebox{.7}{$\bot$}}"
    ]
    \ar[
      rr,
      phantom,
      shift left=-10+10pt,
      "{\scalebox{.7}{$\bot$}}"
    ]
    \ar[
      rr,
      shift left=20+10pt,
      "{ \mathrm{Shp} }"{description},
      "{\mathclap{\times}}"{description, pos=0}
    ]
    \ar[
      rr,
      shift left=-20+10pt,
      "{ \mathrm{Pnts} }"{description}
    ]
    &&
    \mathrm{Grp}_\infty
    \ar[
      ll,
      hook',
      shift right=+10pt,
      "{ \mathrm{Disc} }"{description}
    ]
    &[-40pt]
    \colon
    \;\;\;
    \mathrm{LConst}_X
\end{tikzcd}

By essential uniqueness of [[adjoint (infinity,1)-functors|adjoint $(\infty,1)$-functors]] ([here](adjoint+infinity1-functor#UniquenessOfAdjoints)) and of the terminal [[(infinity,1)-geometric morphism|$(\infty,1)$-geometric morphism]] ([here](infinity1-topos#GlobalSectionsGeometricMorphism)), the composite adjunction is the [[global section]] geometric morphism of the slice topos:

$$
  \big(
    \mathrm{LConst}_X
    \;\;
      \dashv
    \;\;
    \Gamma_X
  \big)
  \;\;\;\;\simeq\;\;\;\;
  \big(
    X \times \mathrm{Disc} (-)
    \;\;
    \dashv
    \;\;
    \mathrm{Pnts} \circ \prod_X
  \big)
  \,.
$$

Hence, by Prop. \ref{ShapeIsGlobalSectionsOfLConst}, we need to exhibit a [[natural equivalence]] of this form:

\[
  \label{CohesiveShapeCorepresentsShapeOfSlice}
  \Gamma_X \circ LConst_X(-)
  \;\;
  \simeq
  \;\;
  Grp_\infty
  \big(
    Shp(X)
    ,\,
    -
  \big)
\]

Therefore, consider, for $S_1, S_2 \,\in\, Grpd_\infty$,
the following sequence of [[natural equivalences|natural]] [[equivalence in an (infinity,1)-category|equivalences]]:

$$
  \begin{aligned}
    &
    \mathrm{Grp}_\infty
    \big(
      S_1
      ,\,
      \Gamma_X
      \circ
      \mathrm{LConst}_X(S_2)
    \big)
    \\
    &
    \;\simeq\;
    \mathbf{H}_{/X}
    \big(
      \mathrm{LConst}_X( S_1 )
      ,\,
      \mathrm{LConst}_X(S_2)
    \big)
    \\
    & \;\simeq\;
    \mathbf{H}_{/X}
    \big(
      X \times \mathrm{Disc}(S_1)
      ,\,
      X \times \mathrm{Disc}(S_2)
    \big)
    \\
    &
    \;\simeq\;
    \mathbf{H}
    \big(
      X \times \mathrm{Disc}(S_1)
      ,\,
      \mathrm{Disc}(S_2)
    \big)
    \\
    & \;\simeq\;
    \mathrm{Grp}_\infty
    \Big(
      \mathrm{Shp}
      \big(
        X \times \mathrm{Disc}(S_1)
      \big)
      ,\,
      S_2
    \Big)
    \\
    & \;\simeq\;
    \mathrm{Grp}_\infty
    \big(
      \mathrm{Shp}(X)
      \times
      S_1
      ,\,
      S_2
    \big)
    \\
    & \;\simeq\;
    \mathrm{Grp}_\infty
    \Big(
      S_1
      ,\,
      \mathrm{Grp}_\infty
      \big(
        \mathrm{Shp}(X)
        ,\,
        S_2
      \big)
    \Big)
  \end{aligned}
$$

Here the first four steps use the [hom-equivalences](adjoint+infinity1-functor#CharacterizationInTermsOfHomEquivalences) of the above adjunctions. The second but last step uses [[cohesion]], namely that the [[shape modality]] preserves [[finite product|finite]] [[homotopy products]] and is [[idempotent (infinity,1)-monad|idempotent]]. The last step is the [hom-equivalence](adjoint+infinity1-functor#CharacterizationInTermsOfHomEquivalences) for the [[cartesian monoidal (infinity,1)-category|cartesian monoidal]]-structure of [[∞Grpd]].

By the [[(infinity,1)-Yoneda lemma|$(\infty,1)$-Yoneda lemma]], this implies the equivalence (eq:CohesiveShapeCorepresentsShapeOfSlice) and hence the claim to be proven.
\end{proof}

### Shape of a topological space
 {#ShapeOfATopologicalSpace}

To any topological space $X$ is associated the
its [[(infinity,1)-category of (infinity,1)-sheaves|$\infty$-category of $\infty$-sheaves]] (with respect its [[site]] of [[open subsets]]), which is an [[(infinity,1)-topos|$\infty$-topos]].

At least when $X$ is a [[compact Hausdorff space]], then its strong shape in the classical sense of [Mardešić & Segal 1971](shape+theory#MardesicSegal71) does agree with the [[shape of an (infinity,1)-topos|$\infty$-topos theoretic shape]] of its [[(infinity,1)-category of (infinity,1)-sheaves|$\infty$-category of $\infty$-sheaves]].

This fact must have motivated the terminology in [Toën-Vezzosi 2002](#ToenVezzosi02) and in [Lurie 2009, Sec. 7.1.6](#Lurie09); it is made explicit in [Hoyois 2013, Rem. 2.13](#Hoyois13).

## Related concepts

* [[étale homotopy]]

* [[shape modality]]

* [[coshape of an (∞,1)-topos]]

## References

The notion of shape of an $\infty$-topos first appears, in essentially the form of the above Def. \ref{ShapeOfAnInfinityTopos}, in:

* {#ToenVezzosi02} [[Bertrand Toën]], [[Gabriele Vezzosi]],  Def. 5.3.2 in: _Segal topoi and stacks over Segal categories_, in: Proceedings of the Program _Stacks, Intersection theory and Non-abelian Hodge Theory_, MSRI, Berkeley, January-May 2002 ([arXiv:math/0212330](http://arxiv.org/abs/math/0212330)).

The concise formulation as $\Gamma \circ LConst(-)$, as in the above Def. \ref{LurieShapeOfAnInfinityTopos},
and discussion of relation to classical ([[strong shape|strong]]) [[shape theory]], of [[topological spaces]], is due to

* {#Lurie09} [[Jacob Lurie]], Section 7.1.6 of: _[[Higher Topos Theory]]_, 2009

The further re-formulation as the image of the [[terminal object]] under the [[pro-left adjoint]] to $LConst$ is highlighted in

* {#Hoyois13} [[Marc Hoyois]], Def. 2.3 in: _Higher Galois theory_, Algebraic & Geometric Topology __17__ 1 (2017) 567-643 ([arxiv/1506.07155](https://arxiv.org/abs/1506.07155), [doi:10.2140/agt.2017.17.567](https://doi.org/10.2140/agt.2017.17.567))

See also

* [[Ilan Barnea]], [[Yonatan Harpaz]], [[Geoffroy Horel]], Section 6 of: _Pro-categories in homotopy theory_, Algebraic & Geometric Topology 17 (2017) 567–643 ([arxiv:1507.01564](https://arxiv.org/abs/1507.01564), [doi:10.2140/agt.2017.17.567](https://doi.org/10.2140/agt.2017.17.567))

[[!redirects shape of an (∞,1)-topos]]

[[!redirects shape of an infinity1-topos]]


shape theory

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#Contents#
* table of contents
{:toc}

## Motivation

While the idea of a [[homotopy type]]  is very suitable for the study of (locally) good [[topological space|topological spaces]], the [[weak homotopy type]] fails to give useful information for 'bad' spaces, of which classical examples include  the [[Warsaw Circle]], [Sierpinski gasket](http://en.wikipedia.org/wiki/Sierpinski_triangle), [[p-adic solenoid]] and so on. Even if our initial and principal interest is more often than not in good spaces, bad spaces arise naturally in their study. For example, in the study of dynamical systems on manifolds, an important issue is the study of  the attractors of such systems, which are typically fractal sets, and thus not 'locally nice' at all! The intuitive idea of shape theory is to define invariants of quite general topological spaces by approximating them with 'good' spaces, either by  embedding them into good spaces, and looking at open or polyhedral neighborhoods of them, or by considering abstract inverse systems of good spaces. The two approaches are closely related.

## Idea

If there are few maps from polyhedra (e.g. from spheres) into the space, then the [[weak homotopy type]] may tell too little about the space.
Therefore one "expands" the space into a successive system of spaces which are good recipients of maps from polyhedra (e.g. absolute neighborhood retracts (ANRs), polyhedra) and one adapts the homotopy theory to such expansions. The analogue of (strong) homotopy type in this setting is the shape of a space; the shape is an invariant of the strong homotopy type and agrees with it on the ANRs for metric spaces and on the polyhedra. It is more crude for other spaces, but more suitable than the weak homotopy type, or more exactly gives complementary information. Instead of embedding a space, one may abstractly expand or resolve the space or its homotopy class into a pro-object in a category of nice spaces. [[Strong shape theory]] is a variant which is closer to the usual kind of  homotopy, is more geometric and has more homotopy theoretic constructions available in its 'toolkit'. It differs by passing to the homotopy category at a later stage in the theory, so one gets homotopy coherent approximating systems rather than homotopy commutative ones.

## History

Shape theory was first explicitly introduced by Polish mathematician [Karol Borsuk](http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Borsuk.html) in the 1960s, although Christie, a student of Lefshetz, had done some initial development work on the same basic idea much earlier. One of the modern versions of shape theory is developed in terms of inverse systems of absolute neighbourhood retracts (ANRs) (which are [[pro-object|pro-objects]] in the homotopy category of polyhedra).  These were introduced in this setting by [[Sibe Mardesic|S. Mardešić]], and [[Jack Segal|J. Segal]] (1971) and independently, in a slightly different form, by [[Tim Porter]] (thesis, 1971), using the more combinatorial framework of [[pro-objects]] in the category of simplicial sets. This latter approach also indicated the possible link with the étale homotopy theory of Artin and Mazur, (Springer Lecture Notes 100).

Shape theory is a '[[Cech methods|Čech homotopy theory]]', having a similar relationship to Čech homology as homotopy theory, based on the singular complex construction, has to singular homology. In fact, as mentioned above, the origins of both shape theory and strong shape theory go back than further Borsuk's initial papers  to work by [Lefshetz](http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lefschetz.html) and his student, D. Christie (thesis plus article, D.E. Christie, _Net homotopy for compacta_,  [Trans. Amer. Math. Soc., 56 (1944) 275--308](www.ams.org/journals/tran/1944-056-00/home.html)). Christie considered a 2-truncated form of strong shape theory, categorically this corresponds to a lax or op-lax 2-categorical version of shape theory.
Although many of the initial ideas were developed by Christie, the paper went unnoticed until Borsuk developed his slightly different approach in the late 1960s.

For many applications one needs more refined invariants which build up [[strong shape theory]], while sometimes more crude versions may be useful, for example the recent theory of [[coarse shape]].

Strong Shape Theory developed in the 1970s through the work of Edwards and Hastings (lecture notes, see below), Porter, Quigley, and others.  It has, especially in the approach pioneered by Edwards and Hastings, strong links to [[proper homotopy theory]].  The links are a form of duality related to some of the more geometric duality theorems of classical cohomology.

M. Batanin further elucidated strong shape theory from a categorical and 2-categorical point of view, but his approach is as yet not much used. His 1997 paper, shows the connections between this theory and a homotopy theory of simplicial [[distributor|distributors]] linked to $A_{\infty}$-categories.

The structure of the [[strong shape theory]] of compact spaces is related to certain structure and constructions on the corresponding (commutative) $C^*$-algebras of functions.  These are related to the algebraic K-theory of such commutative $C^*$-algebras.  Extensions to non-commutative $C^*$-algebras have been made; see ([Blackadar](#Blackadar)) and ([Dadarkat](#Dadarlat)) below, for a start.

As shape theory is a [[Čech methods|Čech homotopy theory]], its corresponding homology is Čech homology, but what is the corresponding construction for strong shape? The answer is Steenrod--Sitnikov homology. This is discussed in Mardešić's book, _Strong Shape and Homology_, (see below).  Many of the themes of homotopy coherence and related ideas occur in this theory and this suggests an infinity categorical approach (closely related to Batanin's) may be important.  This seems to be emerging with interpretations of work by Toen and Vezzosi, and by Lurie, and perhaps suggests a review of Batanin's work from that new viewpoint.

## Borsuk's shape theory

This was the original form and applies to compact metric spaces.  It uses the fact that any compact metric space can be embedded in the [[Hilbert Cube]]. For any such embedded compact metric spaces, $X$ and $Y$, one considers **shape maps** from the collection of open neighbourhoods of $X$ to those of $Y$.  These shape maps are families of continuous maps satisfying a compatibility relationship ' up to homotopy'.  These compose nicely and form the Borsuk shape category.  Two spaces have the same shape if they are isomorphic in this category. Full details of the definition of such shape morphisms are given in the separate entry, [[Borsuk shape theory]].

A remarkable and beautiful theorem of Chapman (the Chapman complement theorem) shows that the shape of two compact metric spaces, $X$ and $Y$ embedded in the [[Hilbert cube|pseudo-interior]] of the Hilbert cube, $Q$, have the same shape if and only if their complements $Q\setminus X$ and $Q\setminus Y$ are homeomorphic.

## Strong shape via $\infty$-topos theory
 {#StrongShapeViaInfinityToposTheory}

To any topological space $X$ there is associated
its [[(infinity,1)-category of (infinity,1)-sheaves|$\infty$-category of $\infty$-sheaves]] (with respect to its [[site]] of [[open subsets]]), which is an [[(infinity,1)-topos|$\infty$-topos]]. Many properties of "[[spaces]]" are shared or may be modeled by [[(infinity,1)-topos|$\infty$-toposes]], in particular there is a general and canonical notion of [[shape of an (infinity,1)-topos|shape of an $\infty$-topos]] (see there for more).

At least when $X$ is a [[compact Hausdorff space]], then its strong shape in the classical sense of [Mardešić & Segal 1971](#MardesicSegal71) does agree with the [[shape of an (infinity,1)-topos|$\infty$-topos theoretic shape]] of its [[(infinity,1)-category of (infinity,1)-sheaves|$\infty$-category of $\infty$-sheaves]].

This fact must have motivated the terminology in [Toën-Vezzosi 2002](shape+of+an+infinity1-topos#ToenVezzosi02) and in [Lurie 2009, Sec. 7.1.6](#Lurie09); it is made explicit in [Hoyois 2013, Rem. 2.13](#Hoyois13).

## Abstract shape category

### Idea

The idea of abstract shape theory is very simple. You have a category, $C$, of objects that you want to study.  (In Borsuk's classical topological case this was the (homotopy) category of compact metric spaces.) You  have a well behaved set of methods that work well for some subcategory, $D$, of those objects (polyhedra in Borsuk's case, where the methods were those of homotopy theory). The categorical idea that can be glimpsed behind the topological constructions of topological shape theory is that of replacing an object $X$ of $C$ with approximations to $X$ by objects of $D$, (so 'approximating' a compact metric space by polyhedra, for instance). Categorically this replaces the object $X$ by the [[comma category]], $(X/D)$, which comes with a projection functor to $D$, which 'records' the approximating $D$-object for each approximation.
You then use your invariants for objects in $D$ to define (and study) the more general objects in $C$. This does not come without consequences as you obtain new types of maps, (shape maps) between the objects of $C$, namely functors between the comma categories that respect the projections.  The objects of $C$ together with your new shape maps form the shape category of your situation.

### Definition

The **shape category** $Shape(C,D)$ is associated to a pair $(C,D)$ of a category $C$ and a [[dense subcategory]] $D$.

Here _dense subcategory_ is used in the second sense of that term: for every object $X$ in $C$ there is its $D$-expansion, which is the object $\bar{X}$ in the category $pro D$ of [[pro-object|pro-objects]] in $D$ that is universal ([[initial object|initial]]) with the property that it is equipped with a morphism $X\to\bar{X}$ in $pro D$.

The shape category $Shape(C,D)$ has

* the same objects as $C$

* its morphisms are equivalence classes of maps between
  the $D$-expansions.

A more [[categorical shape theory|categorical form of shape theory]] was studied by Deleanu and Hilton in a series of papers in the 1970s. They consider a more general setting of a functor $K : D \to C$, which in the classical Borsuk case would be the inclusion of the homotopy category of compact [[simplicial complex|polyhedra]] into that of all compact [[metric space|metric spaces]].

This was developed further by Bourn and Cordier, and a strong shape version was then found by Batanin.

### Examples

#### Pro-spaces in a shape context

The classical application of shape theoretic idea is to the study of [[topological space|topological spaces]] that do not have the [[homotopy type]] of a [[CW-complex]]. This is the case obtained from the above general setup by choosing

* $C =$ [[HoTop]]${}_{he}$ the [[homotopy category]] of the category [[Top]] of all [[topological space|topological spaces]] localised at the **[[homotopy equivalence|homotopy equivalences]]**;

* $D = Ho(CW Cplx)$ the full [[subcategory]] given by [[CW-complex|CW-complexes]].

More on this is in the section [Shape theory for topological spaces](ForTopSpaces) below and in [[Cech homotopy]].

#### Profinite groups

Consider the category $C =$[[Grp]] of groups and its subcategory $D$ of finite group. A shape map between two groups is a map between their [[profinite completion of a group|profinite completions]].  This sort of behaviour is quite general as this form of abstract shape theory is related to equational completions; see

* Gildenhuys and Kennison, _Equational completions, model induced triples and pro-objects_, J. Pure Applied Algebra, 4 (1971) 317-346.

This aspect is explored reasonably fully in the book by Cordier and Porter (see below).

A different terminology and slightly different emphasis is often used within the shape theoretic literature as it corresponds more to the geometric intuition needed there, deriving originally from the important classical motivation of Borsuk, [[Sibe Mardesic|Mardešić]], and [[J. Segal|Segal]].

## Applications of Shape Theory

### Geometric Topology

### Dynamical Systems

In a [[dynamical system]], the [[attractor|attractors]] are rarely polyhedra and their homotopy properties correspond more nearly to shape theoretic ones than to standard homotopy theoretic ones.  This seems first to have been studied by [[Hastings]] in 1988, (see references) and more recently has been explored in papers by [[José Sanjurjo]] and his coworkers, see below.  Adapting this idea [[Luis Javier Hernandez|Hernandez]], [[Teresa Rivas]] and [[José García Calcines]] have been using ideas developed for [[proper homotopy theory]] and shape involving [[pro-spaces]],  to describe limiting properties of dynamical systems.

## Related concepts

* [[shape fibration]], [[approximate fibration]]

* [[shape]], [[shape modality]], [[shape of an (infinity,1)-topos]]

* [[strong shape theory]], [[Borsuk's shape theory]]

* [[geometric realization]]

* [[topology]]

## References

The original references:

* [[Karol Borsuk]], _Concerning homotopy properties of compacta_, Fund Math. 62 (1968) 223-254

* [[Karol Borsuk]], _Theory of Shape_, Monografie Matematyczne Tom 59, Warszawa 1975 ([mathscinet:0418088](https://mathscinet.ams.org/mathscinet-getitem?mr=0418088))

* [[Karol Borsuk]], [[Jerzy Dydak]], _What is the theory of shape?_, Bulletin of the Australian Mathematical Society, 1980, 22(2), 161-198, ([doi:10.1017/S000497270000647X](https://doi.org/10.1017/S000497270000647X))

* {#MardesicSegal71} [[Sibe Mardesic|S. Mardešić]], [[Jack Segal]], _Shapes of compacta and ANR-systems, Fund. Math. 72 (1971) 41-59 ([dml:214361](https://eudml.org/doc/214361), [pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm72/fm7214.pdf))

* [[Sibe Mardesic|S. Mardešić]], [[Jack Segal]], _Shape theory. The inverse system approach_, North-Holland Mathematical Library, 26. North-Holland, Amsterdam - New York, 1982.

Textbook accounts:

* {#DydakSegal78} [[Jerzy Dydak]], [[Jack Segal]], _Shape Theory_, Lecture Notes in Mathematics, Springer 1978 ([doi:10.1007/BFb0067572](https://link.springer.com/book/10.1007/BFb0067572))

Relation of this classical shape theory to ordinary [[homotopy type]] and to the notion of [[shape of an (infinity,1)-topos|shape of an$\infty$-topos]]:

* {#Lurie09} [[Jacob Lurie]], Section 7.1.6 in: _[[Higher Topos Theory]]_

* {#Hoyois13} [[Marc Hoyois]], Def. 2.3 in: _Higher Galois theory_, Algebraic & Geometric Topology __17__ 1 (2017) 567-643 ([arxiv/1506.07155](https://arxiv.org/abs/1506.07155), [doi:10.2140/agt.2017.17.567](https://doi.org/10.2140/agt.2017.17.567))

See also:

* {#Wang17} Jintao Wang, Theorem 4.6 of: _Theory of Compact Hausdorff Shape_ ([arXiv:1708.07346](https://arxiv.org/abs/1708.07346))

\linebreak

A general [[category theory|category theoretic]]-approach is given in

* W. Holsztyński, _An extension and axiomatic characterization of Borsuk's theory of shape_, Fund. Math. 70 (1971) no. 2, 157–168 [pdf](https://www.impan.pl/shop/en/publication/transaction/download/product/79189)

The more or less equivalent pro-object approach was independently developed by Porter in

* [[Tim Porter|T. Porter]], _Čech homotopy I_, Jour. London Math. Soc., 1, 6, 1973, pp. 429-436 [doi](https://dx.doi.org/10.1112/jlms/s2-6.3.429)

* [[Tim Porter|T. Porter]], _Čech homotopy II_, Jour. London Math. Soc., 2, 6, 1973, pp. 667-675 [doi](https://dx.doi.org/10.1112/jlms/s2-6.4.667)

References relating more to strong shape  theory include:

* D.A. Edwards and H. M. Hastings, (1976), Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Maths. 542, Springer-Verlag, [pdf](http://www.math.uga.edu/~davide/Cech_and_Steenrod_Homotopy_Theories_with_Applications_to_Geometric_Topology.pdf)

* J.T. Lisica and [[Sibe Mardesic|S. Mardešić]], Coherent prohomotopy and strong shape theory, Glasnik Mat. 19(39) (1984) 335--399.

* [[Sibe Mardesic|S. Mardešić]], _Strong shape and homology_, Springer monographs in mathematics, Springer-Verlag.

* [[Michael Batanin]], [Categorical strong shape theory](http://www.numdam.org/item/CTGDC_1997__38_1_3_0/), Cahiers Topologie Géom. Différentielle Catég. 38 (1997), no. 1, 3--66.

* [[T. Porter]], _Stability Results for Topological Spaces_, Math. Zeit. 150, 1974, pp. 1-21.

* [[T. Porter]], _Abstract homotopy theory in procategories_, Cahiers Top. Géom. Diff., 17, 1976, pp. 113-124, [numdam](http://archive.numdam.org/article/CTGDC_1976__17_2_113_0.pdf)

* [[T. Porter]],  _Coherent prohomotopical algebra_, [numdam](http://archive.numdam.org/article/CTGDC_1977__18_2_139_0.pdf), Cahiers Top. Géom. Diff. __18__, (1978) pp. 139-179;

* [[T. Porter]], _Coherent prohomotopy theory_, Cahiers Top. Géom. Diff. __19__, (1978) pp. 3-46, [numdam](http://archive.numdam.org/article/CTGDC_1978__19_1_3_0.pdf).

These last three papers developed a version of the [[BrownAHT]] to pro-categories of simplicial sets and of chain complexes, so as to give [[strong shape theory]] a better foundation and toolbox of homotopical methods. These methods were complementary to those of Edwards and Hastings, (listed above), who used a Quillen model category structure on the pro-category.

References to the categorical forms of shape theory include

* A. Deleanu, P.J. Hilton, _On the categorical shape of a functor_, Fund. Math. 97 (1977) 157 - 176.

* A. Deleanu, P.J. Hilton, _Borsuk's shape and Grothendieck categories of pro-objects_, Math. Proc. Camb. Phil. Soc. 79 (1976) 473-482.

* [[Dominique Bourn]], [[Jean-Marc Cordier]], _Distributeurs et théorie de la forme_, Cah. Top. Géom. Diff. Cat. **21** no.2 (1980) pp.161-189. ([numdam:CTGDC_1980__21_2_161_0](http://www.numdam.org/item/CTGDC_1980__21_2_161_0), [pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1980__21_2/CTGDC_1980__21_2_161_0/CTGDC_1980__21_2_161_0.pdf))

  > (via [[profunctors]])

and

* [[Jean-Marc Cordier|J.-M. Cordier]] and [[Tim Porter|T. Porter]], (1989), Shape Theory: Categorical Methods of Approximation, Mathematics and its Applications, Ellis Horwood. Reprinted Dover (2008),

which explores categorical methods in the area.

The relationship between invariants of $C^*$-algebras and the shape of their spectra was explored in

* {#Blackadar} [[Bruce Blackadar]], _Shape theory for $C^*$-algebras_, Math. Scand. 56 (1985) 249 -  275, [journal link](http://www.mscand.dk/article.php?id=2767).

The links are with K-theory and Kasparov's theory. This connection and a related one to 'asymptotic morphisms' is explored in some neat notes by Anderson and Grodal:

* [Noncommutative topology - homotopy functors and E-theory](http://www.math.ku.dk/~jg/papers/etheory.html)

That connection with [[asymptotic morphisms]] is fully explored in the work of [[Marius Dadarlat|Dadarlat]]; see his papers,

* [[Marius Dadarlat|Marius Dādārlat]], _Shape theory and asymptotic morphisms for C$^\ast$-algebras_,  Duke Math. J. 73(3):687-711, 1994, [MR95c:46117](http://www.ams.org/mathscinet-getitem?mr=1262931), [doi](http://dx.doi.org/10.1215/S0012-7094-94-07327-4)
 {#Dadarlat}

* [[Marius Dadarlat|Marius Dādārlat]], Terry A. Loring, _Deformations of topological spaces predicted by E-theory_, in Algebraic methods in operator theory, pages 316-327. Birkhäuser 1994.

For links with dynamical systems see the early paper

*  [[H. M. Hastings]], _Shape theory and dynamical systems_ in M.G.Markely and
W.Perizzo: The structure of attractors in dynamical systems, Lect. Notes in
Math. 688 (1978) 150-160. Springer-Verlag.

and more recently

*  [[Joel W. Robbin]], [[Dietmar A. Salamon]], _Dynamical systems, shape theory and the Conley  index_, Ergodic Theory Dynam. Systems 8 (1988) 375 -  393,

* A. Giraldo, M. A. Morón, F. R. Ruiz del Portal, [[J. M. R. Sanjurjo]], _Shape of global attractors in topological spaces_, Nonlinear Analysis 60 (2005) 837 - 847 [doi](https://dx.doi.org/10.1016/j.na.2004.03.036)

* J. J. Sánchez-Gabites, [[J. M. R. Sanjurjo|José M. R. Sanjurjo]], _Shape properties of the boundary of attractors_, Glas. Mat. Ser. III 42(62) (2007), no. 1, 117–130

*  J. García Calcines, [[Luis Javier Hernández|L. J. Hernández]] and [[Teresa Rivas|M.T. Rivas]],  _Limit and end functors of dynamical systems via exterior spaces, _ [Preprint (2011)](http://www.unirioja.es/cu/luhernan/puben.html).

[[!redirects shape of a topological space]]


shape via cohesive path ∞-groupoid

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## Idea
 {#Idea}

In any [[cohesive ∞-topos]], the [[shape modality]] ʃ may be interpreted as sending any cohesive [[space]] to its [[path ∞-groupoid]]. This follows by the defining [[adjoint (∞,1)-functor|adjunction]] $ʃ \dashv \flat$ from the interpretation of the [[flat modality]] as sending any [[delooping]] $\mathbf{B} \mathcal{G}$ to the [[moduli stack]] for [[flat ∞-connections]] with [[structure group|structure]] [[∞-group]] $\mathcal{G}$:

$$
  \frac{
    ʃ X
      \xrightarrow{\;\nabla\;}
    \phantom{\flat}
    \mathbf{B}\mathcal{G}
  }{
    \phantom{ʃ}
    X
      \xrightarrow{ \;\;\; }
    \flat
    \mathbf{B}\mathcal{G}
  }
  \;\;\;\;
  {
    { \text{higher parallel transport} }
    \atop
    { \text{higher flat bundles} }\,,
  }
$$

because this identifies morphisms out of $ʃ X$ as [[higher parallel transport]]-functors for [[gauge group]] $\mathcal{G}$.

Concretely, in the [[cohesive ∞-topos]] of [[smooth ∞-groupoids]] it is readily checked ([[schreiber:differential cohomology in a cohesive topos|dcct]], see around [here](smooth+infinity-groupoid+--+structures#FlatCoefficientsForDeloopedLieGroup) at *[[smooth ∞-groupoid -- structures]]*) that $\flat \mathbf{B}\mathcal{G}$ indeed classifies [[flat ∞-connections]] in the traditional sense.

It requires more work to show, even in this concrete smooth case, that $ʃ X$ is indeed modeled by a [[path ∞-groupoid]] in the traditional sense (say based on the idea of the smooth [[singular simplicial complex]]). That this is indeed the case is Prop. \ref{SmoothShapeModelityGivenBySmoothPathInfinityGroupoid} below, due to [[Dmitri Pavlov]] et al.

This fact has some remarkable consequences, which we develop [further below](#Consequences).

## Details

Throughout,

\[
  \label{TheAmbientInfinityTopos}
  \begin{aligned}
    \mathbf{H}
    &\,\coloneqq\,
    SmoothGroupoids_\infty
    \\
    &
    \;\simeq\;
    Sh_\infty(CartesianSpaces)
    \;\simeq\;
    Sh_\infty(SmoothManifolds)
    \;\hookrightarrow\;
    PSh_\infty(SmoothManifolds)
  \end{aligned}
\]

denotes the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]], which is the [[hypercomplete (∞,1)-topos]] over the [[site]] of [[smooth manifolds]] or equivalently over the [[dense subsite]] of [[Cartesian spaces]].

### Smooth path $\infty$-groupoid

\begin{definition}\label{SmoothExtendedSimplices}
**(smooth extended [[simplices]])** \linebreak
  Write
  $$
    \Delta^\bullet_{smooth}
    \;\colon\;
    \Delta
      \longrightarrow
    CartesianSpaces
      \xhookrightarrow{\;\;\y\;\;}
    \mathbf{H}
  $$
  for the [[cosimplicial object]] of smooth extended [[simplices]], hence with

$$
  \Delta^n_{smth}
  \;\coloneqq\;
  \Big\{
    (x_0, \cdots, x_n)
    \,\in\,
    \mathbb{R}^{n+1}
    \,\left\vert\,
    \sum_i x_i \,=\, 1
    \right.
  \Big\}
  \;\subset\;
  \mathbb{R}^{n+1}
$$

and with co-degeneracy and co-face maps given by addition of consecutive variables, and by insertion of zeros, respectively.
\end{definition}

\begin{definition}\label{PathInfinityGroupoid}
**([[path ∞-groupoid]])** \linebreak
  For $A \,\in\, SmoothGroupoids_\infty = Sh_\infty(CartesianSpaces)$,
  write
  \[
    \label{SimplicialFormulaForPathInfinityGroupoid}
    \begin{aligned}
    Sing(A)
    &
    \;\coloneqq\;
    \underset{\underset{[n] \in \Delta^{op}}{\longrightarrow}}{\lim}
    X
    \big(
       \Delta^n_{smth}
    \big)
    \\
    & \;\simeq\;
    \underset{\underset{[n] \in \Delta^{op}}{\longrightarrow}}{\lim}
    \mathbf{H}
    \big(
      \Delta^n_{smth},
      X
    \big)
    \;
    \;\;\;
    \in
    Groupoids_\infty
    \end{aligned}
  \]
  for the [[homotopy colimit]] (in [[∞-groupoids]]) of the evaluation of $A$ (as an [[(∞,1)-presheaf]]) on the smooth extended simplices (Def. \ref{SmoothExtendedSimplices}).

Here in the second line we recall, just for emphasis, how, under the [[(∞,1)-Yoneda lemma]], these values are the [[∞-groupoids]] of the $\Delta^n$-shaped *[[plots]]* of the [[generalized smooth space]] $X$.
\end{definition}

\begin{example}
  For $X \,\in\, DiffeologicalSpaces \xhookrightarrow{\;\;y\;\;} SmoothGroupoids_\infty$ a [[diffeological space]] (in particular: a [[smooth manifold]]) its [[path ∞-groupoid]] in the sense of Def. \ref{PathInfinityGroupoid}) is its ordinary smooth [[singular simplicial complex]] (e.g. [Christensen & Wu 13, Def. 4.3](#ChristensenWu13)).

This follows by

1. the [[sub-(infinity,1)-category|fully faithful embedding]] of [[diffeological spaces]] as the [[concrete object|concrete]] [[n-truncated object of an (infinity,1)-category|0-truncated objects]] (see [here](diffeological+space#EmbeddingOfDiffeologicalSpacesIntoTheSheafTopos));

1. the fact that [every simplicial set is the homotopy colimt over its cells](homotopy+limit#SimplicialSetIsHomotopyColimitOverItself).

\end{example}

### Equivalence to smooth shape modality

\begin{proposition}\label{SmoothShapeModelityGivenBySmoothPathInfinityGroupoid}
**(smooth [[shape modality]] given by smooth [[path ∞-groupoid]])**

  For $A \,\in\, SmoothGroupoids_\infty = Sh_\infty(SmthMfd)$,
  the **smooth path $\infty$-groupoid**, namely the [[(∞,1)-presheaf]]

  $$
    \mathbf{Sing}(A)
    \;\colon\;
    U
    \;\mapsto\;
    Sing
    \big(
      [U,A]
    \big)
    \;=\;
    \underset{\underset{[n] \in \Delta^n_{smoth}}{\longrightarrow}}{lim}
    \Big(
      A
      \big(
        \Delta^n_{smth} \times U
      \big)
    \Big)
  $$

of smoothly parameterized path $\infty$-groupoids in $A$ (Def. \ref{PathInfinityGroupoid}),

1. is in fact an [[(∞,1)-sheaf]]

   \[
     \label{SmoothSingIsInfinitySheaf}
     \mathbf{Sing}(A)
     \;\in\;
     Sh_\infty(SmthMfds)
     \xhookrightarrow{\;\;\;}
     PSh_\infty(SmthMfds)
   \]

1. as such is equivalent to the [[shape modality|shape]] of $A$:

   \[
     \label{ShapeEquivalentToSmoothlyParameterizedPathInfinityGroupoid}
     ʃ A
     \;\simeq\;
     \mathbf{Sing}(A)
     \,,
   \]

1. which means in particular (by $ʃ \simeq Disc \circ Shape$) that it is equivalent to the plain [[path ∞-groupoid]] from Def. \ref{PathInfinityGroupoid}, equipped with [[discrete object|discrete]] smooth structure:

   \[
     \label{SmoothlyParameterizePathInfinityGroupoidIsDiscrete}
     \mathbf{Sing}(A) \;\simeq\; Disc \circ Sing(A)
     \,,
   \]

   hence, with (eq:ShapeEquivalentToSmoothlyParameterizedPathInfinityGroupoid), that

   \[
     \label{ShapeEquivalentToDiscreteSmoothSingularComplex}
     ʃ A
     \;\simeq\;
     Disc \circ Sing(A)
     \,.
   \]

\end{proposition}

This is due to [Pavlov et al. 2019, Thm. 1.1](#BEBP19) (see discussion [here](https://nforum.ncatlab.org/discussion/11361/shape-via-cohesive-path-groupoid/?Focus=93004#Comment_93004)), announced in [Pavlov 2014, Thm. 0.2](#Pavlov). The analogous statement for [[sheaves of spectra]] (which follows essentially formally) was observed in [Bunke-Nikolaus-Völkl 2013, Lem. 7.5](#BunkeNikolausVoelkl13) (see at *[[differential cohomology hexagon]]* for more on this case). The particular conclusion (eq:ShapeEquivalentToDiscreteSmoothSingularComplex) is also claimed as  [Bunk 2020, Prop. 3.6 with Prop. 3.11](#Bunk20).

\linebreak

## Consequences
 {#Consequences}

### Smooth Oka principle
 {#ConsequenceSmoothOkaPrinciple}

{#TheFollowingMayBeCompared} The following may be compared ([SS21](#SS21), [p. 7](https://arxiv.org/pdf/2112.13654.pdf#page=7)) to the [[Oka principle]] in [[complex analytic geometry]], now seen in [[higher differential geometry|higher]] [[differential topology]]:

\begin{proposition}\label{SmoothOkaPrinciple}
**(smooth Oka principle)**
  For

  * $X \,\in\, SmoothManifolds \xhookrightarrow{\;\;y\;\;} SmoothGroupoids_\infty$ a [[smooth manifold]] [[(∞,1)-Yoneda embedding|regarded as]] a [[smooth ∞-groupoid]],

  * $A \,\in\, SmoothGroupoids_\infty$ any [[smooth ∞-groupoid]]

we have a [[natural equivalence]]

\[
  \label{SmoothOkaPrincipleEquivalence}
  \big[
    X,
    \,
    ʃ A
  \big]
  \;\;
  \simeq
  \;\;
  ʃ
  [X,A]
  \;\;\;\;\;
  \in
  \;
  SmoothGroupoids_\infty
\]

between the [[mapping stack]] from $X$ to the [[shape modality|shape]] of $A$ and the [[shape modality|shape]] of the [[mapping stack]] into $A$ itself.
\end{proposition}

For precursor discussion see
[nForum comment 55210 (Nov 2015)](https://nforum.ncatlab.org/discussion/6816/the-shape-of-function-objects/?Focus=55210#Comment_55210).

The following proof (from [June 2021](https://ncatlab.org/nlab/revision/shape+via+cohesive+path+∞-groupoid/6), [here](https://nforum.ncatlab.org/discussion/11361/shape-via-cohesive-path-groupoid/#Comment_93145)) as a direct consequence of Prop. \ref{SmoothShapeModelityGivenBySmoothPathInfinityGroupoid} is that appearing in [SS21, §3.3.1](#SS21). We later learned that the statement also appears as [Clough 2021, Theorem B/Cor. 6.4.8](#Clough21), where it is proven independently, now implying a proof of Prop. \ref{SmoothShapeModelityGivenBySmoothPathInfinityGroupoid}.

\begin{proof}
For any $U \in SmoothManifolds$ consider the following sequence of [[natural equivalences]]  of values ([[∞-groupoids]]) of [[(∞,1)-presheaves]] on [[SmoothManifolds]] (eq:TheAmbientInfinityTopos):
$$
  \begin{aligned}
    [X, ʃA](U)
    &
    \;\simeq\;
    Sh_\infty(SmthMfds)
    \big(
      X \times U, ʃA
    \big)
    \\
    & \;\simeq\;
    PSh_\infty(SmthMfds)
    \Big(
      X \times U,
      \,
      \big(
      U'
      \,\mapsto\,
      \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}}
      A(U' \times \Delta^n_{\mathrm{smth}})
      \big)
    \Big)
    \\
    & \;\simeq\;
    \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}}
    PSh_\infty(SmthMfds)
    \Big(
      X \times U,
      \,
      \big(
      U'
      \,\mapsto\,
      A(U' \times \Delta^n_{\mathrm{smth}})
      \big)
    \Big)
    \\
    & \;\simeq\;
    \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}}
    PSh_\infty(SmthMfds)
    \Big(
      X \times U,
      \,
      [\Delta^n_{\mathrm{smth}}, A]
    \Big)
    \\
    & \;\simeq\;
    \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}}
    PSh_\infty(SmthMfds)
    \Big(
      \Delta^n_{\mathrm{smth}} \times X \times U,
      \,
      A
    \Big)
    \\
    & \;\simeq\;
    \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}}
    \big(
      [X,A]( \Delta^n_{\mathrm{smth}} \times U  )
    \big)
    \\
    & \;\simeq\;
    \big(
      ʃ [X,A]
    \big)(U)
    \,.
  \end{aligned}
$$

Here

* the first step uses the [[closed monoidal structure on presheaves]];

* the second step is Prop. \ref{SmoothShapeModelityGivenBySmoothPathInfinityGroupoid};

* the third step uses that with $X \in SmoothManifolds$, by assumption, and with $\Delta^n_{smth} \in CartesianSpaces \hookrightarrow SmoothManifolds$ by Def. \ref{SmoothExtendedSimplices}, also the [[Cartesian product]] $X \times \Delta^n_{smth} \,\in\, SmoothManifolds \xhookrightarrow{y} PSh_\infty(SmoothManifolds)$ is still [[representable functor|representable]], so that by the [[(∞,1)-Yoneda lemma]], we may [[limits of presheaves are computed objectwise|compute the colimit of presheaves objectwise]];

* the fourth just makes explicit the [[internal hom]] in order to make manifest another appeal to the [[closed monoidal structure on presheaves]] in the fifth and sixth step;

* the last step applies again Prop. \ref{SmoothShapeModelityGivenBySmoothPathInfinityGroupoid}.

Since the composite of these equivalences is still [[natural equivalence|natural]] in $U$, the statement (eq:SmoothOkaPrincipleEquivalence) follows by the [[(∞,1)-Yoneda embedding]].
\end{proof}

### Classification of principal $\infty$-bundles
  {#ClassificationOfPrincipalInfinityBundles}

\begin{definition}
  \label{ClassifyingSpaceAsShapeOfDelooping}
  For $\mathcal{G} \,\in\, Groups(SmoothGroupoids_\infty)$
  any [[smooth ∞-group]], write
  $$
    B \mathcal{G}
    \;\;
    \coloneqq
    \;\;
    ʃ
    \mathbf{B}
    \mathcal{G}
    \;\;
    \simeq
    \mathbf{B}
    ʃ
    \mathcal{G}
  $$
  for the [[shape modality|shape]] of its [[delooping]], equivalently (since [[shape modality|shape]] preserves the simplicial [[(∞,1)-colimits]] and the [[finite products]] involved in defining the [[delooping]] as the realization of the [[Cech nerve]] of the [[effective epimorphism]] $\ast \to \mathbf{B}\mathcal{G}$) the delooping of its shape.
\end{definition}

\begin{remark}
Notice that the [[delooping]] $\mathbf{B}\mathcal{G} \in \mathbf{H}$ of an [[∞-group]] $\mathcal{G} \in Groups(\mathbf{H})$ is the [[moduli ∞-stack]] of $\mathcal{G}$-[[principal ∞-bundles]], in that ([[schreiber:Principal infinity-bundles|NSS 12]])
for $X \in \mathbf{H}$ there is a [[natural equivalence]] (of [[∞-groupoids]])

\[
  \label{ModulationOfPrincipalInfinityBundles}
  \mathcal{G}PrinBund_X
  \;\;
  \simeq
  \;\;
  \mathbf{H}(X,\mathbf{B}\mathcal{G})
  \;\;\;
  \in
  \;
  Groupoids_\infty
  \,.
\]

In particular, upon [[n-truncated object of an (infinity,1)-category|0-truncation]] $\tau_0$, this means that the [[moduli stack]] $\mathbf{B}\mathcal{G}$ classifies [[equivalence classes]] of [[principal ∞-bundles]]:

$$
  \big(\mathcal{G}PrinBund_X\big)_{/\sim_{equ}}
  \;\;
  \simeq
  \;\;
  \tau_0\, \mathbf{H}(X,\mathbf{B}\mathcal{G})
  \;\;\;
  \in
  \;
  Sets
  \,.
$$

But for this reduced information much less than the full [[moduli stack]] may be necessary: A *[[classifying space]]* for $\mathcal{G}$-[[principal ∞-bundles]] is a [[discrete object]] in $Groupoids_\infty \xhookrightarrow{Disc} \mathbf{H}$, such that [[homotopy classes]] of maps into it still correspond to [[equivalence classes]] of [[principal ∞-bundles]], at least over suitable domains (traditionally: [[paracompact topological spaces]], hence in particular [[smooth manifolds]]).
\end{remark}

Slightly coarser than plain equivalence classes of principal $\infty$-bundles are [[concordance classes]] of principal $\infty$-bundles (in fact, often the two notions coincide, see e.g. [Roberts-Stevenson 16, Cor. 15](#nonabelian+cohomology#RobertsStevenson16)):

\begin{definition}\label{ConcordanceClassesOfPrincipalInfinityBundles}
  **([[concordance classes]] of [[principal ∞-bundles]])** \linebreak
  For $\mathcal{G} \in Groups(SmoothGroupoids)_\infty$
  and any $X \in SMoothGroupoids_\infty$,
  we say that a *[[concordance]]* between two [[principal ∞-bundles]]
  $$
    P_0, P_1 \;\in\; \mathcal{G}PrinBund(SmoothGroupoids|infty)_X
  $$
  is a principal $\infty$-bundle on the [[cylinder]] over $X$
  $$
    \widehat P \;\in\; \mathcal{G}PrinBund_{X \times \mathbb{R}}
  $$
  whose restriction to the points $0,1 \,\in\, \mathbb{R}$ is
  [[equivalence in an (∞,1)-category|equivalent]] to the
  given pair of bundles:
  $$
    {\widehat P}_{\vert X \times \{0\}}
    \;\simeq\;
    P_0
    \;\;\;\;\;\;\;
    \text{and}
    \;\;\;\;\;\;\;
    {\widehat P}_{\vert X \times \{1\}}
    \;\simeq\;
    P_1
    \,.
  $$
  We write
  \[
    \label{SetOfConcordanceClassesOfPrincipalInfinityBundles}
    \big(
       \mathcal{G}PrinBund_X
    \big)_{/\sim_{conc}}
    \;\;
      \coloneqq
    \;\;
    \big(
       \mathcal{G}PrinBund_X
    \big)
    \big/
    \big(
       \mathcal{G}PrinBund_{X \times \mathbb{R}}
    \big)
  \]
  for the set of [[equivalence classes]] of [[principal ∞-bundles]] under this concordance [[relation]].
\end{definition}

\begin{proposition}\label{ClassificationOfSmoothPrincipalInfinityBundlesUpToConcordance}
**([[classifying spaces]] for [[smooth ∞-groupoid|smooth]] [[principal ∞-bundles]] up to [[concordance]])**
  For

  * any [[smooth ∞-group]] $\mathcal{G} \,\in\, Groups(SmoothGroupoids_\infty)$;

 * any [[smooth manifold]] $X \,\in\, SmoothManifolds \xhookrightarrow{\;\;y\;\;} SmoothGroupoids_\infty$

the space $B \mathcal{G} \,\in\, Groupoids_\infty \xhookrightarrow{Disc} SmoothGroupoids_\infty$ (Def. \ref{ClassifyingSpaceAsShapeOfDelooping}) is a [[classifying space]] for $\mathcal{G}$-[[principal ∞-bundles]] over $X$, up to [[concordance]] (Def. \ref{ConcordanceClassesOfPrincipalInfinityBundles}), in that we have a [[natural bijection]]:

$$
  \big(
    \mathcal{G}PrinBund_X
  \big)_{/\sim_{conc}}
  \;\;
  \simeq
  \;\;
  \tau_0
  \,
  \mathbf{H}
  \big(
    X,\,
    B \mathcal{G}
  \big)
  \,.
$$

\end{proposition}

\begin{proof}
This follows as the [[composition]] of the following sequence of [[natural bijections]]:

$$
\begin{aligned}
    \tau_0
    \,
    \mathbf{H}
    \big(
      X, \, B \mathcal{G}
    \big)
    & \;\simeq\;
    \tau_0
    \,
    Pnts
    \,
    \big[
      X, \, B \mathcal{G}
    \big]
    \\
    &
    \;\simeq\;
    \tau_0
    \,
    Pnts
    \,
    \big[
      X,
      \,
      ʃ
      \,
      \mathbf{B}\mathcal{G}
    \big]
    \\
    & \;\simeq\;
    \tau_0
    \,
    Pnts
    \,
    ʃ
    \,
    \big[
      X,
      \,
      \mathbf{B}\mathcal{G}
    \big]
    \\
    & \;\simeq\;
    \tau_0
    \,
    Pnts
    \,
    Disc
    \,
    \mathrm{Sing}
    \big[
      X,
      \,
      \mathbf{B}\mathcal{G}
    \big]
    \\
    & \;\simeq\;
    \tau_0
    \,
    \mathrm{Sing}
    \big[
      X,
      \,
      \mathbf{B}\mathcal{G}
    \big]
    \\
    & \;\simeq\;
    \tau_0
    \,
    \underset{\longrightarrow}{\mathrm{lim}}
    \Big(
    \big[
      X,
      \,
      \mathbf{B}\mathcal{G}
    \big]
    (\Delta^\bullet_{\mathrm{smth}})
    \Big)
    \\
    & \;\simeq\;
    \tau_0
    \,
    \underset{\longrightarrow}{\mathrm{lim}}
    \Big(
    \mathbf{H}
    \big(
      X
      \times
      \Delta^\bullet_{\mathrm{smth}}
      \,
      \mathbf{B}\mathcal{G}
    \big)
    \Big)
    \\
    & \;\simeq\;
    \tau_0
    \,
    \underset{\longrightarrow}{\mathrm{lim}}
    \big(
      {\mathcal{G}}PrinBund_{X \times \Delta^\bullet_{\mathrm{smth}}}
    \big)
    \\
    & \;\simeq\;
    \underset{\longrightarrow}{\mathrm{lim}}
    \,
    \tau_0
    \big(
    {\mathcal{G}}PrinBund_{X \times \Delta^\bullet_{\mathrm{smth}}}
    \big)
    \\
    & \;\simeq\;
    \Big(
      \tau_0
      \big(
        {\mathcal{G}}PrinBund_{ X }
      \big)
    \Big)
    \big/
    \Big(
      \tau_0
      \big(
      {\mathcal{G}}PrinBund_{X \times \Delta^1_{\mathrm{smth}}}
      \big)
    \Big)
    \\
    & \;\simeq\;
    \big(
      {\mathcal{G}}PrinBund_X
    \big)_{\sim_{\mathrm{conc}}}
  \end{aligned}
$$

Here:

* the first step is the relation between the [[(∞,1)-categorical hom-space]] and the [[internal hom]] via the Points-functor $Pts(X) \coloneqq \mathbf{H}(\ast,X)$;

* the second step is the definition of the classifying space (Def. \ref{ClassifyingSpaceAsShapeOfDelooping});

* the third step is the smooth Oka principle (Prop. \ref{SmoothOkaPrinciple});

* the fourth step is the relation $ʃ \;\simeq\; Disc \circ Sing$ (eq:SmoothlyParameterizePathInfinityGroupoidIsDiscrete) from Prop. \ref{SmoothShapeModelityGivenBySmoothPathInfinityGroupoid};

* the fifth step is the relation $Pnts \circ Disc \simeq id$, valied in any [[cohesive (∞,1)-topos]];

* the sixth step is (eq:SimplicialFormulaForPathInfinityGroupoid) from Def. \ref{PathInfinityGroupoid};

* the seventh step is the formula for the [[internal hom]] by the [[closed monoidal structure on presheaves]];

* the eighth step is the modulation of principal $\infty$-bundles by their [[moduli stack]] (eq:ModulationOfPrincipalInfinityBundles);

* the ninth step uses that [[n-truncated object in an (infinity,1)-category|n-truncation]] is a [[left adjoint|left]] [[adjoint (∞,1)-functor]] and hence commutes with [[(∞,1)-colimits]];

* the tenth step uses that the inclusion of the first two face maps into the [[opposite category|opposite]] of the [[simplex category]] is a [[final functor]] (by [this example](final+functor#FirstPairOfFaceMapsFinalInOppositeSimplexCategory), noting that this happens in the [[1-category]] of [[n-truncated object in an (infinity,1)-category|0-truncated]] [[∞-groupoids]], hence in [[Sets]]);

* the eleventh step invokes the definition (eq:SetOfConcordanceClassesOfPrincipalInfinityBundles) of [[concordance]] of principal $\infty$-bundles (Def. \ref{ConcordanceClassesOfPrincipalInfinityBundles}).

\end{proof}

## Related concepts

* [[concordance]]

* [[higher parallel transport]]

## References

The identification in Prop. \ref{SmoothShapeModelityGivenBySmoothPathInfinityGroupoid} is due to

* {#Pavlov14} [[Dmitri Pavlov]], _Structured Brown representability via concordance_, 2014 ([pdf](https://dmitripavlov.org/concordance.pdf), [[Pavlov_Concordance.pdf:file]])

* {#BEBP19} [[Daniel Berwick-Evans]], [[Pedro Boavida de Brito]], [[Dmitri Pavlov]], _Classifying spaces of infinity-sheaves_ ([arXiv:1912.10544](https://arxiv.org/abs/1912.10544))

with a precursor observation in:

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], Lem. 7.5 of: _Differential cohomology theories as sheaves of spectra_, Journal of Homotopy and Related Structures **11** 1 (2016) ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188), [doi:10.1007/s40062-014-0092-5](https://doi.org/10.1007/s40062-014-0092-5))

The particular conclusion (eq:ShapeEquivalentToDiscreteSmoothSingularComplex) is also claimed in:

* {#Bunk20} [[Severin Bunk]], Section 3 of: *The $\mathbb{R}$-Local Homotopy Theory of Smooth Spaces*,  	J. Homotopy Relat. Struct. (2022) [[arXiv:2007.06039](https://arxiv.org/abs/2007.06039), [doi:10.1007/s40062-022-00318-7](https://doi.org/10.1007/s40062-022-00318-7)]

The "smooth Oka principle" is also proven (Theorem B) in:

* {#Clough21} [[Adrian Clough]], *A Convenient Category for Geometric Topology*, PhD thesis, UT Austin (2021) [[pdf](https://repositories.lib.utexas.edu/bitstream/handle/2152/114981/CLOUGH-DISSERTATION-2021.pdf), [[Clough-ConvenientCategory.pdf:file]]]

   *The homotopy theory of differentiable sheaves* [[arXiv:2309.01757](https://arxiv.org/abs/2309.01757)]

Talk notes:

* [[Adrian Clough]], *The smooth Oka principle*, talk at [[CQTS]] (Oct 2022) [[pdf](https://github.com/adrianclough/adrianclough.github.io/raw/main/The%20smooth%20Oka%20principle%206.pdf)]

The special case of the smooth path $\infty$-groupoid (Def. \ref{PathInfinityGroupoid})
applied to [[diffeological spaces]] (among all [[smooth infinity-groupoids|smooth $\infty$-groupoids]]), was considered also in:

* {#ChristensenWu13} [[J. Daniel Christensen|J. Daniel Christensen]],  [[Enxin Wu]], Def. 4.3 of: _The homotopy theory of diffeological spaces, I. Fibrant and cofibrant objects_, New York J. Math. 20 (2014), 1269-1303 ([arXiv:1311.6394](http://arxiv.org/abs/1311.6394))

Application of the theorem to a (orbi-)smooth Oka principle and to the classification of [[equivariant bundles|equivariant]] [[principal bundles]]:

* {#SS21} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Equivariant principal infinity-bundles|Equivariant principal $\infty$-bundles]]* $[$[arXiv:2112.13654](https://arxiv.org/abs/2112.13654)$]$

[[!redirects shape via cohesive path ∞-groupoids]]

[[!redirects shape via cohesive path infinity-groupoid]]

[[!redirects shape via the cohesive path ∞-groupoid]]
[[!redirects shape via the cohesive path infinity-groupoid]]

[[!redirects smooth Oka principle]]


sharing graph

_Sharing graphs_ are another kind of [[string diagram]] which focuses on the syntax of a language.  Given a [BNF](http://en.wikipedia.org/wiki/Backus-Naur_form) description of a [[context-free grammar]], we get a category whose objects are (roughly) lists of subterms and whose morphisms are rules for composing the subterms into terms.

#Example: Untyped lambda calculus
Lambda calculus terms are recursively defined as follows:

: $t$ ::= $v | t(t) | \lambda v.t$

where $v$ is a countably infinite set of variables.  (That is, a term is either a given variable, a given term applied to a given term, or the $\lambda$-abstraction of a given term as a function of a given variable.)  There are three equivalences on terms, $\alpha$ (for handling dummy variables), $\beta$ (substitution of terms), and $\eta$ (extensionality).  The corresponding category has

* two generating types, $V$ and $T$
* six function symbols

  * $\lambda:V^* \times T \to T$.  Lambda takes an antivariable and a term that may use the corresponding variable.
  * $\cap:1 \to V^* \times T$.  This turns an antivariable $x$ introduced by lambda into the term $x$.
  * $A:T \times T \to T$. (Application) This takes $f$ and $g$ and produces $f(g)$.
  * $swap:T \times T \to T \times T$.
  * $!:T \to 1$.
  * $\Delta:T \to T \times T$.

* relations

  * The last three function symbols come with relations that make the category cartesian (and justify the use of $\times$ above).  Leaving off the last two would give linear lambda calculus.

  * $\beta\eta: A \circ (\lambda \times 1_T) \circ (1_{V^*} \times P \times 1_T) \circ (\cap \times 1_{T^n} \times 1_T) \Leftrightarrow P \circ swap_{1_{T^n}, 1_T}$

    This is the categorical encoding of

    $(\lambda x.P)(y) \Leftrightarrow P[y/x]$

    where the free variables of the term $P$ are $\{x, x_1, \ldots, x_n\}$.

The string diagrams corresponding to this category are called sharing graphs.

#Discussion

Peter Selinger [said](https://golem.ph.utexas.edu/category/2006/08/categorifying_cccs_seeing_comp.html#c021631) of this example:

"Sharing graphs" are extremely well-studied. See e.g.

(1) [Stefano Guerrini, A general theory of sharing graphs (1997)](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.530)

There is a reduction strategy on sharing graphs that is provably the most efficient possible reduction strategy for lambda calculus; this goes back to the work of Levy and Lamping (both cited in the above paper).

Guerrini also gives an algebraic semantics of these sharing graphs, see section 7. I don't know whether it is related to your categorical view.

I also like the book by Peyton-Jones:

(2) [Simon Peyton Jones, The Implementation of Functional Programming Languages (1987)](http://research.microsoft.com/en-us/um/people/simonpj/papers/slpj-book-1987/)

It shows how to use sharing graphs as the basis for a practical implementation of lazy programming languages. As far as I know, this is still state-of-the-art and is used in the implementation of Haskell. In keeping with the practical nature of the book, the sharing graphs
are represented in slightly different form (with syntactic variables rather than backpointers), but this is of course equivalent.

As for the categorical semantics, what you have in mind is a kind of abstract syntax with variable binding. To put this into perspective, the semantics of ordinary abstract syntax (i.e., without variable binding), is given by an object $A$ in a cartesian category, together with interpretations for each $n$-ary function symbol $f : A^n \to A$. One can then inductively define the interpretation of terms, speak of the free such object, etc.

Things get slightly more complicated if one adds variable binding to this picture. This has also been studied, though perhaps not in the same form as you are proposing.  Perhaps the closest to your approach is

(3) [Martin Hofmann, Semantical analysis of higher-order abstract syntax (1999)](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.4082)

Here one has $var : V \to T$, $app : T \times T \to T$, and $lam : (V \Rightarrow T) \to T$ (see top of p.9, with the obvious changes in notation). You will note the use of a function space $(V \Rightarrow T)$ in place of your cartesian product $V^* \times T$.

Another very similar paper is

(4) [M.P.Fiore, G.D.Plotkin and D.Turi. Abstract syntax and variable binding (1999)](http://citeseerx.ist.psu.edu/showciting?cid=198434) (Also available [here](http://www.cl.cam.ac.uk/~mpf23/papers/Types/Types.html).)

Here, one has $var : V \to T$, $app : T \times T \to T$, and $lam : \delta T \to T$ (as contained e.g. in the commutative diagram on p.6). Again, $\delta T$ is something akin to the function space $(V \Rightarrow T)$, but is also isomorphic, in a suitable sense, to $(T \Rightarrow T)$, as far as I remember (this is important for substitution, see below).

A third, technically slightly different (but conceptually similar) approach to abstract syntax with variable binders is:

(5) [Murdoch J. Gabbay, Andrew M. Pitts: A new approach to abstract syntax with variable binding (1999)](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.38.9383)

(6) [Murdoch J. Gabbay, Andrew M. Pitts: A new approach to abstract syntax with variable binding (2002)](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.62.9845)

See especially the second-to-last line of \[6, p.356\], and you immediately see the similarity. Their $[A]T$ operation is akin to $(V \Rightarrow T)$ in \[3\] and $\delta T$ in \[4\].

It is fun to note that \[3–5\] all appeared (independently) in the same conference in 1999. Semantics of variable binders was clearly a pressing problem that year.

I can comment briefly on the main difference between these works and what you are proposing. One thing that is important in syntax is substitution (of a term for a variable). In fact, in the usual abstract syntax (without binders), a term with $n$ variables is represented as a morphism $A^n \to A$. This is useful for substitution:
namely, if $f : A \times A^n \to A$ represents the term $t(x,y_1,...,y_n)$ and $g : A^n \to A$ represents $s(y_1,...,y_n)$, then $f \circ \langle g, id \rangle$ represents $t[s/x]$.

In the presence of variables, a term with $n$ variables is represented as $1 \to V^{*n} \times T$ (in your notation). However, for reasons of substitution, one would still like this hom-set to be in 1-1 correspondence with $T^n \to T$.  Somehow this is what the papers \[3–6\] manage to do, each in their own way.

I hope these references will be useful. It would be great if you had a more abstract monoidal framework of which the particular constructions in \[3–6\] are concrete examples. I have always wondered about the precise connection between \[3–6\], and whether there is a bigger picture.

[[!redirects sharing graphs]]


sharing graphs > history

< [[sharing graphs]]
[[!redirects sharing graphs -- history]]


Sharon Hollander

* [website](http://www.math.ist.utl.pt/~sjh/)

## Selected writings

On [[hypercovers]] in [[simplicial presheaves]]:

* {#DuggerHollanderIsaksen02} [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], _Hypercovers and simplicial presheaves_, Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 136. No. 1., 2004. ([arXiv:0205027](http://arxiv.org/abs/math/0205027), [K-theory archive](http://www.math.uiuc.edu/K-theory/0563/))

On the [[homotopy theory]] of [[stacks]] via [[Bousfield localization of model categories|localization]] of a [[model structure on simplicial presheaves]]:

* {#Hollander01} [[Sharon Hollander]], _A homotopy theory for stacks_, Israel Journal of Mathematics, 163(1), 93-124  (2008) ([arXiv:math.AT/0110247](http://arxiv.org/abs/math.AT/0110247), [doi:10.1007/s11856-008-0006-5](https://doi.org/10.1007/s11856-008-0006-5))

## Related $n$Lab entries

* [[presheaf of groupoids]], [[stack]]

* [[model structure on simplicial presheaves]]

category: people


sharp map

> This entry is about the concept related to [[homotopy pullbacks]]. For a different concept of the same name see at _[[sharp modality]]_.

#Contents#
* table of contents
{:toc}

## Idea

In a [[right proper model category]] a morphism is called _sharp_ if its [[pullback]] along any other morphism is already a [[homotopy pullback]]. In a general [[model category]] a morphism is sharp if all its pullbacks preserve weak equivalences under (further) pullback.

Other terms used for sharp morphisms include _right proper morphism_, _$h$-fibration_, and _$W$-fibration_; see the references.

## Definition

In a [[model category]] $\mathcal{M}$, a **sharp map** is a morphism $p : X \to Y$ satisfying the following condition: for any commutative diagram in $\mathcal{M}$ of the form below,
$$
  \array{
    X'' & \stackrel{f}{\longrightarrow} & X' & \longrightarrow & X \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{p}} \\
    Y'' & \stackrel{g}{\longrightarrow} & Y' & \longrightarrow & Y
  }
$$
if $g \colon Y'' \to Y'$ is a weak equivalence and both squares are pullback diagrams, then  $f \colon X'' \to X'$ is also a weak equivalence.

## Properties

A model category is [[right proper model category|right proper]] if and only if every fibration is sharp. ([Rezk 98, prop. 2.2](#Rezk98))

In a [[right proper model category]], the sharp maps in the full subcategory on sharp-fibrant objects form the fibrations of a [[category of fibrant objects]]. See there the section _[Examples -- Right proper model categories](category+of+fibrant+objects#RightProperModelCategories)_.

## Related concepts

* The dual notion is (most commonly) known as “[[h-cofibration]]”.

## References

The concept was introduced in

* {#Rezk98} [[Charles Rezk]], _Fibrations and homotopy colimits of simplicial sheaves_ ([arXiv:9811038](http://arxiv.org/abs/math/9811038)), 1998

The terminology arises by dualization of "flat morphism" which was used by Hopkins for the dual concept, which is presumably motivated by the fact that a [[ring homomorphism]] is flat if tensoring with it is exact, hence preserves weak equivalences of [[chain complexes]].

The notion was rediscovered and renamed by various other authors.  In

* [[Andrei Radulescu-Banu]], _Cofibrations in Homotopy Theory_, [arxiv](https://arxiv.org/abs/math/0610009), 2006

it was called a "right proper morphism" (with focus on the dual notion of "left proper morphism"), presumably due to the connection with right proper model categories.  In

* [[Denis-Charles Cisinski]], _Invariance de la K-théorie par équivalences dérivées_, 2010

sharp maps were renamed "weak fibrations".  The authors of

* [[Clark Barwick]] and [[Daniel Kan]], *Quillen Theorems Bn for homotopy pullbacks of (infinity, k)-categories*, [arxiv](https://arxiv.org/abs/1208.1777), 2012

chose instead to rename them to "fibrillations", because it sounds more like "fibration".  Whereas the authors of

* [[Michael Batanin]] and [[Clemens Berger]], _Homotopy theory for algebras over polynomial monads_, [arxiv](https://arxiv.org/abs/1305.0086), 2013

chose to rename the dual notion to "$h$-cofibrations", with reference to the use of that term for the related --- but nevertheless distinct --- notion of [[Hurewicz cofibration]] by [[Peter May]] and collaborators such as [[Johann Sigurdsson]] and [[Kate Ponto]].  In

* [[Dimitri Ara]] and [[Georges Maltsiniotis]], _Towards a Thomason model structure on the category of strict $n$-categories_, [arxiv](https://arxiv.org/abs/1305.5086), 2013

the dual notion was called a "$W$-cofibration", where $W$ is the [[category with weak equivalences|relevant class of weak equivalences]] (note that the definition only depends on the weak equivalences, not the whole model structure); apparently this terminology dates back to unpublished work of Grothendieck.

Some discussion about these various names was had at

* [[Mike Shulman]], _Pullbacks That Preserve Weak Equivalences_, [blog post](https://golem.ph.utexas.edu/category/2014/07/pullbacks_that_preserve_weak_e.html)

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sharp modality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
#### Type theory
+--{: .hide}
[[!include type theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

On a [[local topos]]/[[local (∞,1)-topos]] $\mathbf{H}$, hence equipped with a [[full and faithful (∞,1)-functor|fully faithful]] extra [[right adjoint]] $coDisc$ to the [[global section]] [[geometric morphism]] $(Disc \dashv \Gamma)$,  is induced an  [[idempotent monad]] $\sharp \coloneqq coDisc \circ \Gamma$, a [[modality]] which we call the _sharp modality_. This is itself the [[right adjoint]] in an [[adjoint modality]] with the [[flat modality]] $\flat \coloneqq Disc \circ \Gamma$.

### In type theory

We assume a [[dependent type theory]] with crisp term judgments $a::A$ in addition to the usual (cohesive) type and term judgments $A \; \mathrm{type}$ and $a:A$, as well as context judgments $\Xi \vert \Gamma \; \mathrm{ctx}$ where $\Xi$ is a list of crisp term judgments, and $\Gamma$ is a list of cohesive term judgments. A crisp type is a type in the context $\Xi \vert ()$, where $()$ is the empty list of cohesive term judgments. In addition, we also assume the dependent type theory has [[typal equality]] and [[judgmental equality]].

From here, there are two different notions of the sharp modality which could be defined in the type theory, the **strict sharp modality**, which uses [[judgmental equality]] in the [[computation rule]] and [[uniqueness rule]], and the **weak sharp modality**, which uses [[typal equality]] in the [[computation rule]] and [[uniqueness rule]]. When applied to a type they result in **strict sharp types** and **weak sharp types** respectively. The [[natural deduction]] rules for strict and weak sharp types are provided as follows:

* Formation rule for weak and strict sharp types:

$$\frac{\Xi, \Gamma \vert () \vdash A \; \mathrm{type}}{\Xi \vert \Gamma \vdash \sharp A \; \mathrm{type}}\sharp-\mathrm{form}$$

* Introduction rule for weak and strict sharp types:

$$\frac{\Xi, \Gamma \vert () \vdash a:A}{\Xi \vert \Gamma \vdash a^\sharp:\sharp A}\sharp-\mathrm{intro}$$

* Elimination rule for weak and strict sharp types:

$$\frac{\Xi \vert () \vdash a:\sharp A}{\Xi \vert \Gamma \vdash a_\sharp:A}\sharp-\mathrm{elim}$$

* Computation rule for weak and strict sharp types respectively:

$$\frac{\Xi \vert () \vdash a:A}{\Xi \vert \Gamma \vdash \beta_{\sharp A}(a):(a^\sharp)_\sharp =_A a}\sharp-\mathrm{comp}\mathrm{weak} \qquad \frac{\Xi \vert () \vdash a:A}{\Xi \vert \Gamma \vdash (a^\sharp)_\sharp \equiv a:A}\sharp-\mathrm{comp}\mathrm{strict}$$

* Uniqueness rule for weak and strict sharp types respectively:

$$\frac{\Xi \vert () \vdash a:\sharp A}{\Xi \vert \Gamma \vdash \eta_{\sharp A}(a):(a_\sharp)^\sharp =_{\sharp A} a}\sharp-\mathrm{uniq}{weak} \qquad \frac{\Xi \vert () \vdash a:\sharp A}{\Xi \vert \Gamma \vdash (a_\sharp)^\sharp \equiv a:\sharp A}\sharp-\mathrm{uniq}\mathrm{strict}$$

Weak sharp modalities are primarily used in cohesive [[weak type theories]], while strict sharp modalities are typically used in cohesive type theories which are not weak, such as cohesive [[Martin-Löf type theory]] ([[cohesive homotopy type theory]] or cohesive [[higher observational type theory]].

## Properties

### Relation to discrete and codiscrete objects

* The [[codiscrete objects]] in a local topos are the [[modal types]] for the sharp modality.

* The [[discrete objects]] in a local topos are the [[modal types]] for the [[flat modality]].

## Related concepts

* [[sharp excluded middle]]

[[!include cohesion - table]]

## References
 {#References}

The terminology of the sharp-modality in the above sense was introduced -- in the language of [[(infinity,1)-toposes|$(\infty,1)$-toposes]] and as part of the axioms on "[[cohesive (infinity,1)-toposes|cohesive $(\infty,1)$-toposes]]" -- in:

* [[Urs Schreiber]], *[[schreiber:differential cohomology in a cohesive topos|Differential cohomology in a cohesive $\infty$-topos]]* (2013)

See also the references at _[[local topos]]_.

Early discussion in view of [[homotopy type theory|homotopy]] [[type theory]] and as part of a set of  axioms for [[cohesive homotopy type theory]] is in

* [[Mike Shulman]], [[Urs Schreiber]], *[[schreiber:Quantum gauge field theory in Cohesive homotopy type theory]]* (2014)

The dedicated type theory formulation with "crisp" types, as part of the formulation of [[real cohesive homotopy type theory]], is due to:

* [[Mike Shulman]], *Brouwer's fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

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[[!redirects strict sharp types]]
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sharply smaller cardinal

# Sharply smaller cardinals

* table of contents
{: toc}

## Definition

+--{: .un_theorem}
###### Theorem
For [[regular cardinals]] $\lambda\le\mu$, the following are equivalent:

* Every $\lambda$-[[accessible category]] is $\mu$-accessible.
* For every $\mu'\lt\mu$, the set $P_\lambda(\mu')$ of subsets of $\mu'$ of [[cardinality]] $\lt\lambda$ has a [[cofinal subset]] of cardinality $\lt\mu$.
=--

For a proof, see Theorem 2.11 of [Adamek-Rosicky](#AR) or section 2.3 of [Makkai-Pare](#MP).

If these equivalent conditions hold, we write $\lambda\unlhd \mu$.  If $\lambda \unlhd \mu$ and $\lambda\lt\mu$, we write $\lambda\lhd \mu$ and say that $\lambda$ is **sharply smaller** than $\mu$.

### Examples

* For any uncountable regular cardinal $\lambda$ we have $\aleph_0\lhd \lambda$.  (In fact $\aleph_0$ is the only infinite regular cardinal with this property; see [this question](https://mathoverflow.net/q/150305).)

* For any regular cardinal $\lambda$ we have $\lambda\lhd \lambda^+$ (its [[successor cardinal]]).

* If $\lambda\le\mu$ then $\lambda \lhd (2^\mu)^+$.  Thus, for any set $S$ of regular cardinals there is a regular cardinal $\mu$ such that $\lambda\lhd \mu$ for all $\lambda\in S$.

* We write $\lambda\ll\mu$ if for every $\lambda'\lt\lambda$ and $\mu'\lt\mu$ we have $(\mu')^{\lambda'} \lt\mu$.  (This is [[Higher Topos Theory]], Definition A.2.6.3.)  Then if $\lambda\ll\mu$, then $\lambda\lhd \mu$.

  The converse claim ($\lambda \lhd \mu$ implies $\lambda\ll\mu$) is independent of [[ZFC]].  On one hand it implies the [[generalized continuum hypothesis]] (GCH) for regular cardinals (and in particular the ordinary [[continuum hypothesis]] (CH)), since if $\lambda^+ \lt 2^\lambda$ then we have $\lambda^+ \lhd \lambda^{++}$ but not $\lambda^+ \ll \lambda^{++}$.  Thus it is unprovable in ZFC (if ZFC is consistent), since CH is unprovable.  On the other hand, it is implied by the full GCH, as explained by [Goldberg](#Goldberg19), and is thus consistent with ZFC since GCH is.

* If $\kappa$ is an [[inaccessible cardinal]], then every $\lambda\lt\kappa$ satisfies $\lambda\lhd \kappa$.

* $\aleph_1 \lhd \aleph_{\omega+1}$ does not hold.

## References

* {#MP}  [[Michael Makkai]], [[Robert Paré]], _Accessible categories: The foundations of categorical model theory_ Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989.1989.
{#MakkaiPare}

* [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_, Cambridge University Press, (1994)
 {#AR}

* [[Jacob Lurie]], [[Higher Topos Theory]]

* {#Goldberg19} Gabe Goldberg, answer to MathOverflow question _Raising the index of accessibility_,  <https://mathoverflow.net/q/324492>

[[!redirects sharply smaller cardinal]]
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Shay Fuchs

* [webpage](http://www.utm.utoronto.ca/math-cs-stats/faculty-staff/fuchs-dr-shay)

category: people