slice of presheaves is presheaves on slice

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Let $\mathcal{C}$ be any [[small category]].

### Over a representable
 {#IdeaOverARepresentable}

For $X \,\in\, \mathcal{C}$ an [[object]], the [[category of presheaves]] $PSh\big( \mathcal{C}_{/X}\big)$ on the [[slice category]] $\mathcal{C}_{/X}$ is [[equivalence of categories|equivalent]] to the [[slice category|slice]] $PSh(\mathcal{C})_{/X}$ of the [[category of presheaves]] on $\mathcal{C}$ over the [[image]] of $X$ under the [[Yoneda embedding]].

The former [[presheaf topos]] is manifestly a [[Grothendieck topos]], whence this equivalence shows that also the slice $PSh(\mathcal{C})_{/X}$ is a Grothendieck topos. This is the archetypical special case of the *[[fundamental theorem of topos theory]]* which says that all [[slice category|slices]] of toposes are themselves toposes: *[[slice toposes]]*.

As shown in Prop. \ref{TheAdjointEquivalenceInOrdinaryCategoryTheory} below, this equivalence is canonically an [[adjoint equivalence]], where the [[right adjoint]] $R$ forms the [[hom-set]] in the [[slice category|slice]] over $y(X)$, hence is the functor which takes a [[bundle]] (in the broad sense) [[internalization|internal]] to [[presheaves]] to its system of sets $\Gamma_{(-)}(E)$ of [[local sections]]:

$$
  \array{
    PSh(\mathcal{C})_{/X}
    &
    \xrightarrow{\;\;\;\; \sim \;\;\;\;}
    &
    PSh
    \big(
      \mathcal{C}_{/X}
    \big)
    \\
    \left(
    \array{
      E
      \\
      \downarrow
      \\
      X
    }
    \right)
    &\mapsto&
    \big(
      (U \to X)
      \,\mapsto\,
      \Gamma_U(E)
    \big)
    \,.
  }
$$

If instead of presheaves of sets one considers [[simplicial presheaves]] then this adjoint equivalence becomes a [[Quillen equivalence]] with respect to the the [[projective model structure on simplicial presheaves]] and its [[slice model structure]] (Prop. \ref{SimplicialLocalSectionsIsRightQuillen} below).

As such this Quillen equivalence [[Ho(CombModCat)|models]] the analogous statement (Prop. \ref{EquivalenceOfInfinityCategories} below) for [[slice (infinity,1)-category|slice $\infty$-categories]] of [[(infinity,1)-category of (infinity,1)-presheaves|$\infty$-categories of $\infty$-presheaves]], which thus also are [[slice (infinity,1)-topos|slice $\infty$-toposes]]. This is the archetypical case of the *[[fundamental theorem of (infinity,1)-topos theory|fundamental theorem of $\infty$-topos theory]]*, see [there](slice+infinity1-topos#SheavesOnBigSite) for more.

### Over any presheaf
 {#IdeaOverAnyPresheaf}

More generally, the analogous statement remains true when $X \,\in\, PSh(\mathcal{C})$ is any [[presheaf]] (not necessarily a [[representable functor|representable]] in the [[image]] of the [[Yoneda embedding]]). In this more general case the equivalence reads just as before

$$
  \array{
    PSh(\mathcal{C})_{/X}
    &
    \xrightarrow{\;\;\;\; \sim \;\;\;\;}
    &
    PSh
    \big(
      \mathcal{C}_{/X}
    \big)
    \,,
  }
$$

only that now the [[site]] appearing on the right must be understood as a [[full subcategory]] of the [[slice category]] of the full [[category of presheaves]], on those objects whose [[domain]] is a [[representable functor|representable]]:

$$
  \mathcal{C}_{/X}
  \;\xhookrightarrow{\;\;}\;
  PSh(\mathcal{C})_{/X}
  \,.
$$

This may equivalently be understood as the [[Grothendieck construction]] on the functor $X$.

\linebreak

## Preliminaries

### Presheaves

Let $\mathcal{C}$ be a [[small category]], we write

$$
  PSh(\mathcal{C})
  \,\coloneqq\,
  Func(\mathcal{C}^{op} ,\, Set)
$$

for its [[category of presheaves]] and

\[
  \label{PlainYonedaEmbedding}
  y_{\mathcal{C}}
    \,\colon\,
  \mathcal{C}
    \xrightarrow{\;\;\;}
  PSh(\mathcal{C})
\]

for the [[Yoneda embedding]].

Recall (from [there](presheaf#PresheavesAreColimitsOfRepresentables)) that every [[presheaf]] $F \,\in\, PSh(\mathcal{C})$ is a [[colimit]] of [[representables]] $y_{\mathcal{C}}(c)$ indexed by the [[comma category]] of morphisms $y_{\mathcal{C}}(c) \to F$. We will denote this "[[co-Yoneda lemma]]" by

\[
  \label{PresheafAsColimitOfRepresentables}
  F
  \;\;
  \simeq
  \;\;
  \underset
    {\underset{ y_{\mathcal{C}}(c) \to F }{\longrightarrow}}
    {
      lim
    }
    \;
    y_{\mathcal{C}}(c)
  \,.
\]

### Slice categories

For $\mathcal{D}$ any [[category]] and $B \in \mathcal{D}$, the [[hom-object]] in the [[slice category]] is given by the following [[fiber product]] (e.g, [here](over-infinity1-category#HamSpacesInASlice)):

\[
  \label{HomInSliceCategoryAsFiberProduct}
  \mathcal{D}_{/B}
  \big(
    (U,p_U)
    \,\,
    (U',p_{U'})
  \big)
  \;\;
  \simeq
  \;\;
  \mathcal{D}(U, U')
  \underset{
    \mathcal{D}(U, B)
  }{\times}
  \{p_{U}\}
  \,.
\]

### Slices of presheaf categories

For any $X \,\in\, \mathcal{C}$ we denote the generic [[object]] of the [[slice category]] $\mathcal{C}_{/X}$ by

$$
  c_X
  \,=\,
  \left(
  \array{
    c
    \\
    \big\downarrow{\mathrlap{{}^{c_X}}}
    \\
    X
  }
  \;\;
  \right)
  \;\;
  \in
  \;
  \mathcal{C}_{/X}
  \,.
$$

Notice that the [[slice category]] $\mathcal{C}_{/X}$ has its own [[Yoneda embedding]]

$$
  y_{\mathcal{C}_{/X}}
  \;\colon\;
  \mathcal{C}_{/X}
  \xrightarrow{\;\;\;\;}
  PSh
  \big(
    \mathcal{C}_{/X}
  \big)
$$

but that it is also the [[source]] of the slicing of the plain  Yoneda embedding (eq:PlainYonedaEmbedding), which is still a [[fully faithful functor]]:

\[
  \label{SlicedYonedaEmbedding}
  \array{
    (y_{\mathcal{C}})_{/X}
    &\colon&
    \mathcal{C}_{/X}
    &\xhookrightarrow{\phantom{-----}}&
    PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)}
    \\
    &&
    c_X
    &\mapsto&
    \left(
    \array{
      y_{\mathcal{C}}(c)
      \\
      \downarrow^{\mathrlap{ y_{\mathcal{C}}(c_X) }}
      \\
      y_{\mathcal{C}}(X)
    }
    \;\;\;\;\;
    \right)
  }
\]

## Statement

### In plain category theory
 {#DetailsInPlainCategoryTheoryOverARepresentable}

We discuss the statement in plain [[category theory]] with general abstract proofs that will work in any other context (notably in [[enriched category theory]], see [below](#InEnrichedCategoryTheory), and [[(infinity,1)-category theory|$\infty$-category theory]], see [further below](#InInfinityCategoryTheory)) that satisfies the basic theorems of category theory (e.g. the natural [[hom-functor preserves limits|respect of hom-functors for (co-)limits]] etc.).

\linebreak

First we give a quick proof that (Prop. \ref{StatementInPlainCategoryTheoryOverAnyPresheaf} below) there is an [[equivalence of categories]] at all. Then we enhance this statement (in Prop. \ref{TheAdjointEquivalenceInOrdinaryCategoryTheory}) below to an [[adjoint equivalence]] whose [[right adjoint]] is concretely identified as forming [[sections]].

\linebreak

\begin{proposition}\label{StatementInPlainCategoryTheoryOverAnyPresheaf}
  For $X \,\in\, PSh(\mathcal{C})$ any presheaf, we have an [[equivalence of categories]]

$$
  PSh(\mathcal{C})_{/X}
  \;\simeq\;
  PSh
  \big(
    \mathcal{C}_{/X}
  \big)
  \,.
$$
\end{proposition}
\begin{proof}
Using that

1. every presheaf is a [[colimit]] of [[representables]] (the "[[co-Yoneda lemma]]");

1. colimits in slice categories are computed in the underlying categories (see [there](over+category#LimitsAndColimits))

$$
  \left(
  \array{
    U
    \\
    \big\downarrow
    \mathrlap{{}^{p_{U}}}
    \\
    X
  }
  \;\;\;
  \right)
  \;=\;
  \Big(
     U,
     \,
     p_U
  \Big)
  \;\simeq\;
  \Big(
  \underset{
    \underset{i \in \mathcal{I}}{\longrightarrow}
  }{\lim}
  \,
  c_U(i)
  ,\,
  (p_{c_U(i)})_{i \in \mathcal{I}}
  \big)
  \;\simeq\;
  \underset{
    \underset{i \in \mathcal{I}}{\longrightarrow}
  }{\lim}
  \big(
    c_U(i)
    ,\,
    p_{c_U(i)}
  \big)
$$
we have an evident functor
$$
  \array{
    PSh(\mathcal{C})_{/X}
    &\longrightarrow&
    PSh
    \big(
      \mathcal{C}_{/X}
    \big)
    \\
    (U,p_U)
    &\overset{\phantom{----}}{\mapsto}&
    \underset{\underset{
      i \in \mathcal{I}
    }{\longrightarrow}}{\lim}
    \big(
      c_U(i)
      ,\,
      p_{c_U(i)}
    \big)
  }
$$
which is clearly [[essentially surjective functor|essentially surjective]]. That it is also [[fully faithful functor|fully faithful]] is established by the following sequence of [[natural isomorphisms]]:

$$
  \begin{array}{l}
  PSh(\mathcal{C})_{/X}
  \Big(
    \big(
      U,
      \,
      p_U
    \big)
    \,,
    \big(
      U
      ,\,
      p_{U'}
    \big)
  \Big)
  \\
  \;\simeq\;
  PSh(\mathcal{C})
  \big(
    U
    \,,
    U'
  \big)
  \underset{
  PSh(\mathcal{C})
  \big(
    U
    \,,
    X
  \big)
  }{\times}
  \{p_U\}
  \\
  \;\simeq\;
  \underset{
    \underset{i \in \mathcal{I}}{\longleftarrow}
  }{\lim}
  \,
  PSh(\mathcal{C})
  \big(
    c_U(i)
    \,,
    U'
  \big)
  \underset{
  \underset{
    \underset{i \in \mathcal{I}}{\longleftarrow}
  }{\lim}
  \,
  PSh(\mathcal{C})
  \big(
    c_U(i)
    \,,
    X
  \big)
  }{\times}
  \Big\{
    \big(
      p_{c_U(i)}
    \big)_{i \in \mathcal{I}}
  \Big\}
  \\
  \;\simeq\;
  \underset{
    \underset{i \in \mathcal{I}}{\longleftarrow}
  }{\lim}
  \,
  \bigg(
  PSh(\mathcal{C})
  \big(
    c_U(i)
    \,,
    U'
  \big)
  \underset{
  PSh(\mathcal{C})
  \big(
    c_U(i)
    \,,
    X
  \big)
  }{\times}
  \big\{
    p_{c_U(i)}
  \big\}
  \bigg)
  \\
  \;\simeq\;
  \underset{
    \underset{i \in \mathcal{I}}{\longleftarrow}
  }{\lim}
  \,
  \bigg(
  \underset{
    \underset{i' \in \mathcal{I}'}{\longrightarrow}
  }{\lim}
  PSh(\mathcal{C})
  \big(
    c_U(i)
    \,,
    c_{U'}(i')
  \big)
  \underset{
  PSh(\mathcal{C})
  \big(
    c_U(i)
    \,,
    X
  \big)
  }{\times}
  \big\{
    p_{c_U(i)}
  \big\}
  \bigg)
  \\
  \;\simeq\;
  \underset{
    \underset{i \in \mathcal{I}}{\longleftarrow}
  }{\lim}
  \,
  \underset{
    \underset{i' \in \mathcal{I}'}{\longrightarrow}
  }{\lim}
  \,
  \bigg(
  PSh(\mathcal{C}_{/X})
  \Big(
    \big(
      c_U(i)
      ,\,
      p_{c_U(i)}
    \big)
    \,,
    \big(
      c_{U'}(i')
      ,\,
      p_{c_{U'}(i')}
    \big)
  \Big)
  \bigg)
  \\
  \;\simeq\;
  PSh(\mathcal{C}_{/X})
  \Big(
    \underset{
      \underset{i \in \mathcal{I}}{\longrightarrow}
    }{\lim}
    \big(
      c_U(i)
      ,\,
      p_{c_U(i)}
    \big)
    \,,
    \underset{
      \underset{i' \in \mathcal{I}'}{\longrightarrow}
    }{\lim}
    \big(
      c_{U'}(i')
      ,\,
      p_{c_{U'}(i')}
    \big)
  \Big)
  \bigg)
  \\
  \;\simeq\;
  PSh(\mathcal{C}_{/X})
  \Big(
    \big(
      U
      ,\,
      p_U
    \big)
    \,,
    \big(
      U'
      ,\,
      p_{U'}
    \big)
  \Big)
  \bigg)
  \mathrlap{\,.}
  \end{array}
$$

Here

* the first step is the definition (eq:HomInSliceCategoryAsFiberProduct) of the slice hom;

* the second step is the expression of $U$ as a colimit of representables ([[co-Yoneda lemma]]), together with the fact that any [[hom-functor preserves limits|hom functor sends colimits in the first argument to limits]];

* the third step is the the [[fiber product]] commutes with this limit (since [[limits commute with limits]]);

* the fourth step is the expression of $U'$ as a colimit of representables ([[co-Yoneda lemma]]), together with the [[Yoneda lemma]] and the fact that colimits of presheaves are computed objectwise ([here](category+of+presheaves#LimitsOfPresheavesAreComputedObjectwise));

* the fifth step is [[universal colimits]] in the ambient [[topos]] (of [[Sets]]);

* the sixths step recognizes that the slice hom (eq:HomInSliceCategoryAsFiberProduct) defining $\mathcal{C}_{/X}$ has appeared and moves the colimits back inside by the reverse of the previous arguments, now over the slice side.

\end{proof}

The following statement enhances this equivalence of categories to an [[adjoint equivalence]] and identifying its [[right adjoint]] as the [[functor]] of forming [[sections]]. This stronger version further enhances to a [[simplicial Quillen adjunction|simplicial]] [[Quillen equivalence]] below.

\begin{proposition}\label{TheAdjointEquivalenceInOrdinaryCategoryTheory}
For $X \,\in\, \mathcal{C} \xhookrightarrow{\;y\;} PSh(\mathcal{C})$.
the following anti-parallel functors constitute an [[adjoint equivalence]]
\begin{tikzcd}
    &
    \mathcal{C}_{/X}
    \ar[
      dl,
      hook',
      "{ (y_{\mathcal{C}})_{/X} }"{above, xshift=-8pt}
    ]
    \ar[
      dr,
      hook,
      "{ y_{(\mathcal{C}_{/X})} }"{above, xshift=4pt}
    ]
    \\
    \mathrm{PSh}
    (
      \mathcal{C}
    )_{/y(X)}
    \ar[
      rr,
      shift right=7pt,
      "{
        R
        \,\coloneqq\,
        \mathrm{PSh}(\mathcal{C})_{/y(X)}
        \big(
          (y_{\mathcal{C}})_{/X}(-)
          ,\,
          -
        \big)
      }"{below}
    ]
    &&
    \mathrm{PSh}
    \big(
      \mathcal{C}_{/X}
    \big)
    \ar[
      ll,
      shift right=7pt,
      "{L}"{above}
    ]
    \ar[
      ll,
      phantom,
      "\scalebox{.6}{$\bot$}"
    ]
\end{tikzcd}
\end{proposition}
Here:

1. the top functor $L$ is the [[cocontinuous functor|colimit-preserving functor]] that makes the top [[commuting diagram|triangle commute]], hence which takes representables over the slice site to the slicing of the underlying representables on the plain site. These two conditions fix the functor completely, by the fact (eq:PresheafAsColimitOfRepresentables) that every presheaf is a colimit of representables.

1. the bottom functor is the [[hom-functor]] of the slice category, which means (eq:HomInSliceCategoryAsFiberProduct) that it is given by a [[pullback]] of the hom-functor in $PSh(\mathcal{C})$ itself:

   \[
     \label{SliceHomOfPresheavesAsFiberOfPlainHom}
     PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)}
     \Big(
       \big(
         y_{\mathcal{C}})_{/X}(U \xrightarrow{\phi} X
       \big)
       ,\,
       \big(
         E \xrightarrow{p} y_{\mathcal{C}}(X)
       \big)
     \Big)
     \;=\;
     PSh(\mathcal{C})
     \big(
        y_{\mathcal{C}}(U)
        ,\,
        E
     \big)
     \underset
       {
         PSh(\mathcal{C})
         \big(
            y_{\mathcal{C}}(U)
            ,\,
            y_{\mathcal{C}}(X)
         \big)
       }
       {\times}
       \big\{
          y_{\mathcal{C}}(\phi)
       \big\}
   \]

\begin{tikzcd}[column sep=20pt]
     \mathrm{PSh}(\mathcal{C})_{/y_{\mathcal{C}}(X)}
     \Big(
       \big(
         y_{\mathcal{C}})_{/X}(U \xrightarrow{\phi} X
       \big)
       ,\,
       \big(
         E \xrightarrow{p} y_{\mathcal{C}}(X)
       \big)
     \Big)
     \ar[
       rr
     ]
     \ar[d]
     \ar[
        drr,
        phantom,
        "\mbox{\tiny\textrm(pb)}"
     ]
     &&
     \mathrm{PSh}(\mathcal{C})
     \big(
        y_{\mathcal{C}}(U)
        ,\,
        E
     \big)
     \ar[
       d,
       "{
         \mathrm{PSh}(\mathcal{C})
         \big(
           y_{\mathcal{C}}(U)
           ,\,
           p
         \big)
       }"
     ]
     \\
     \big\{
        y_{\mathcal{C}}(\phi)
     \big\}
     \ar[rr]
     &&
     \mathrm{PSh}(\mathcal{C})
     \big(
        y_{\mathcal{C}}(U)
        ,\,
        y_{\mathcal{C}}(X)
     \big)
\end{tikzcd}

\begin{proof}
First to see that the functors are [[adjoint functor|adjoint]], we check the required [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) by observing the following sequence of [[natural bijections]]:

$$
  \begin{aligned}
  PSh(\mathcal{C}_{/X})
  \big(
    A ,\, R(B)
  \big)
  &
  \;=\;
  PSh(\mathcal{C}_{/X})
  \Big(
    A
    ,\,
    PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)}
    \big(
      (y_{\mathcal{C}})_{/X}(-)
      ,\,
      B
    \big)
  \Big)
  \\
  & \;\simeq\;
  PSh
  \big(
    \mathcal{C}_{/X}
  \big)
  \Big(
    \underset
      {\underset{c_X \to A}{\longrightarrow}}
      {\mathrm{lim}}
    y_{(\mathcal{C}_{/X})}(c_{X})
    ,\,
    PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)}
    \big(
      (y_{\mathcal{C}})_{/X}(-)
      ,\,
      B
    \big)
  \Big)
  \\
  & \;\simeq\;
  \underset{
    \underset{c_X \to A}{\longleftarrow}
  }{\mathrm{lim}}
  PSh(\mathcal{C}_{/X})
  \Big(
    y_{(\mathcal{C}_{/X})}(c_{X})
    ,\,
    PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)}
    \big(
      (y_{\mathcal{C}})_{/X}(-)
      ,\,
      B
    \big)
  \Big)
  \\
  &
  \;
  \simeq
  \;
  \underset{
    \underset{c_X \to A}{\longleftarrow}
  }{\mathrm{lim}}
    PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)}
    \big(
      (y_{\mathcal{C}})_{/X}(c_X)
      ,\,
      B
    \big)
  \\
  &
  \;
  \simeq
  \;
    PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)}
    \Big(
      \underset{
        \underset{c_X \to A}{\longrightarrow}
      }{\mathrm{lim}}
      (y_{\mathcal{C}})_{/X}(c_X)
      ,\,
      B
    \Big)
  \\
  & \;=\;
    PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)}
    \big(
      L(A)
      ,\,
      B
    \big)
  \end{aligned}
$$
Here:

* the first step is the above definition of the right adjoint,

* the second step is (eq:PresheafAsColimitOfRepresentables),

* the third is that any [[hom-functor]] sends [[colimits]] in its first argument into [[limits]] ([[hom-functor preserves limits|here]]),

* the fourth step is the [[Yoneda lemma]] over the slice site,

* the fifth step takes the limit back into the hom-functor, but now that of the other category,

* the sixth step is the above definition of the would-be left adjoint, using again (eq:PresheafAsColimitOfRepresentables).

Now to see that these two functors are weak inverses of each other.

In one direction we have the following sequence of [[natural bijections]] for $A \,\in\, PSh\big( \mathcal{C}_{/X}\big)$:

$$
  \begin{aligned}
    R \circ L (A)
    &
    \;\simeq\;
    PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)}
    \big(
      (y_{\mathcal{C}})_{/X}(-)
      ,\,
      L(A)
    \big)
    \\
    & \;\simeq\;
    PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)}
    \Big(
      (y_{\mathcal{C}})_{/X}(-)
      ,\,
      \underset{\underset{c_X \to A}{\longrightarrow}}{\lim}
      (y_{\mathcal{C}})_{/X}(c_X)
    \Big)
    \\
    & \;\simeq\;
    \underset{\underset{c_X \to A}{\longrightarrow}}{\lim}
    y_{(\mathcal{C}_{/X})}(c_X)
    \\
    & \;\simeq\;
    A
  \end{aligned}
$$

Here:

* the first two steps just unwind again the above definitions of the functors;

* the third step follows by the [[Yoneda lemma]] over $\mathcal{C}$, to which applies by observing that that:

  1. colimits in slices are [[reflected colimit|reflected]] as colimits in the underlying category (by [this Prop](over+category#ColimitInSliceAreReflectedByColimitsInPlainCategory)),

  1. [[colimits of presheaves are computed objectwise]],

  1. the slice hom is a pullback of the plain hom (eq:SliceHomOfPresheavesAsFiberOfPlainHom),

  1. colimits in a topos such as $PSh(\mathcal{C})$ are [[pullback-stable colimit|pullback-stable]];

* the last step re-assembles the argument, by (eq:PresheafAsColimitOfRepresentables).

In the other direction we have the following sequence of natural bijections, for $B \,\in\, PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)}$:

$$
  \begin{aligned}
    L \circ R(B)
    & \;\simeq\;
    L
    \Big(
      \underset{
        \underset
        {
          y_{(\mathcal{C}_{/X})}(c_X) \to R(B)
        }
        {\longrightarrow}
      }
      {\lim}
      y_{(\mathcal{C}_{/X})}(c_X)
    \Big)
    \\
    & \;\simeq\;
    \underset{
      \underset
      {
        y_{(\mathcal{C}_{/X})}(c_X) \to R(B)
      }
      {\longrightarrow}
    }
    {\lim}
    (y_{\mathcal{C}})_{/X}(c_X)
    \\
    & \;\simeq\;
    \underset{
      \underset
      {
        L\big( y_{(\mathcal{C}_{/X})}(c_X) \big) \to B
      }
      {\longrightarrow}
    }
    {\lim}
    (y_{\mathcal{C}})_{/X}(c_X)
    \\
    & \;\simeq\;
    \underset{
      \underset
      {
        (y_{\mathcal{C}})_{/X}(c_X) \to B
      }
      {\longrightarrow}
    }
    {\lim}
    (y_{\mathcal{C}})_{/X}(c_X)
    \\
    & \;\simeq\;
    B
  \end{aligned}
$$

Here:

* the first step is the [[co-Yoneda lemma]] (eq:PresheafAsColimitOfRepresentables) for $R(B)$,

* the second step unwinds the definition of $L$ from above,

* the third step uses the [[adjoint functor|adjunction]] $L \dashv R$ established above on the indexing category of the colimit;

* the fourth step applies again the definition of $L$ from above,

* the last step is again the [[co-Yoneda lemma]] (eq:PresheafAsColimitOfRepresentables), now for $B$ itself.

\end{proof}

The above proof of Prop. \ref{TheAdjointEquivalenceInOrdinaryCategoryTheory} does not actually depend on the assumption that the base object is representable ($y(X)$).

\linebreak

### For simplicial presheaves on plain categories
 {#InSimplicialModelCategoryTheory}

We discuss generalizations of the above situation from [[presheaves]] to the [[homotopy theory]] of [[simplicial presheaves]].

#### Over 0-truncated objects
 {#SimplicialPresheavesOverZeroTruncatedObjects}

For

* $\mathcal{C}$ an [[category]],

* $X \,\in\, \mathcal{C}$ an object,

* $\mathcal{C}_{/X}$ the [[sSet]]-[[enriched slice category]]

\begin{lemma}\label{ForSimplicialPresheaves}
With [[categories of simplicial presheaves]] denoted $sPSh(-)$, Prop. \ref{TheAdjointEquivalenceInOrdinaryCategoryTheory} generalizes to:

\begin{tikzcd}
    \mathrm{sPSh}(\mathcal{C})_{/y_{\mathcal{C}}(X)}
    \ar[
      rrr,
      "{
        \mathrm{PSh}(\mathcal{C})_{/y(X)}
        \big(
          (y_{\mathcal{C}})_{/X}(-)
          ,\,
          -
        \big)
      }"{above},
      "{ \sim }"{below}
    ]
    &&&
    \mathrm{sPSh}
    \big(
      \mathcal{C}_{/X}
    \big)
\end{tikzcd}
\end{lemma}

##### Globally

Write:

* $sPSh\big(\mathcal{C}_{/X}\big)_{proj}$ for the projective [[model structure on sSet-enriched presheaves]] (relative to the [[classical model structure on simplicial sets]] $sSet_{Qu}$) on the [[enriched slice category]],

  notice that this is an [[classical model structure on simplicial sets|$sSet_{Qu}$]]-[[enriched model category]], hence a [[simplicial model category]];

* $\big(sPSh(\mathcal{C})_{/y_{\mathcal{C}(X)}}\big)_{proj}$ for the [[slice model structure]] over the [[representable functor|representable]] $y_{\mathcal{C}}(X)$ of the projective [[model structure on sSet-enriched presheaves]] on $\mathcal{C}$ itself.

\begin{proposition}\label{SimplicialLocalSectionsIsRightQuillen}
  Relative to these projective (slice) model structures, the comparison functor from Exp. \ref{ForSimplicialPresheaves} is a [[right Quillen functor]], hence the [[right adjoint]] in a [[simplicial Quillen adjunction]], which is a [[Quillen equivalence]]:

\begin{tikzcd}
   \big(
    \mathrm{sPSh}(\mathcal{C})_{/y_{\mathcal{C}}(X)}
  \big)_{\mathrm{proj}}
    \ar[
      rrr,
      shift right=8pt,
      "{
        \mathrm{PSh}(\mathcal{C})_{/y(X)}
        \big(
          (y_{\mathcal{C}})_{/X}(-)
          ,\,
          -
        \big)
      }"{below}
    ]
    &&&
    \mathrm{sPSh}
    \big(
      \mathcal{C}_{/X}
    \big)_{\mathrm{proj}}
    \ar[
       lll,
       shift right=8pt
    ]
    \ar[
      lll,
      phantom,
      "{ \scalebox{.6}{$\simeq_{\mathrlap{\mathrm{Qu}}}$} }"
    ]
\end{tikzcd}

\end{proposition}
\begin{proof}
Observe that:

1. Since representables are [[cofibrant objects|cofibrant]] (evidently so in the projective model structure, since [[acyclic Kan fibrations]]  [are surjective](acyclic+Kan+fibration#AcyclicKanFibrationsAreSurjective)), the unsliced simplicial hom out of a representable is a right Quillen functor by the [pullback-power axiom](enriched+model+category#PullbackPowerAxiom) in the $sSet_{Qu}$-[[enriched model category]] $sSh(\mathcal{C})$.

1. The [[base change]]-functor by [[pullback]] is a right Quillen functor on [[slice model categories]] of $sSet_{Qu}$  (by [this Prop.](slice+model+structure#LeftBaseChangeQuillenAdjunction)).

Together this implies that their composite (eq:SliceHomOfPresheavesAsFiberOfPlainHom) is a [[right Quillen functor]].

By [[Ken Brown's lemma]] ([here](factorization+lemma#KenBrownLemma)) it follows that the right adjoint preserves weak equivalences between [[fibrant objects]]. We claim that it also *reflects* weak equivalences between fibrant objects, in that a morphism between fibrant objects on the left is a weak equivalence if and only if its image under the right adjoint functor is a weak equivalences.
Since the functor is also an [[equivalence of categories]], by Prop. \ref{TheAdjointEquivalenceInOrdinaryCategoryTheory}, this immediately implies that the [[derived adjunction]] is an [[equivalence of categories|equivalence]] of [[homotopy category of a model category|homotopy categories]], and hence that we have a [[Quillen adjunction]].

To see this remaining claim (that the right adjoint reflects weak equivalences between fibrant objects) consider a morphism $f \,\in\, sPsh(\mathcal{C})_{/y_{\mathcal{C}}(X)}$ between fibrations such that for all $U \xrightarrow{\phi} X$ in $\mathcal{C}_{/X}$ the [[base change]] (eq:SliceHomOfPresheavesAsFiberOfPlainHom) of its values on $U \,\in\, \mathcal{C}$ is a weak equivalence:

\begin{tikzcd}[column sep=20pt]
   &[30pt]
   \mathllap{
      \mathrm{W}
      \,\ni\,
   }
   \big(y_{\mathcal{C}}(\phi)\big)^\ast
   \Big(
     \mathrm{PSh}(\mathcal{C})
     \big(
        y_{\mathcal{C}}(U)
        ,\,
        f
     \big)
   \Big)
     \ar[
       rr
     ]
     \ar[d]
     \ar[
        drr,
        phantom,
        "\mbox{\tiny\textrm(pb)}"
     ]
     &&
     \mathrm{PSh}(\mathcal{C})
     \big(
        y_{\mathcal{C}}(U)
        ,\,
        f
     \big)
     \ar[
       d
     ]
     \\
     &
     \big\{
        y_{\mathcal{C}}(\phi)
     \big\}
     \ar[rr]
     &&
     \mathrm{PSh}(\mathcal{C})
     \big(
        y_{\mathcal{C}}(U)
        ,\,
        y_{\mathcal{C}}(X)
     \big)
\end{tikzcd}

Here the right vertical morphisms are [[Kan fibrations]] by the fact that $sPSh(\mathcal{C})\big( y_{\mathcal{C}}(U),\, - \big)$ is a [[right Quillen functor]] as in the first item above.
Therefore -- since this holds for all $\phi$, by assumption -- [this Prop.](slice+model+structure#SimplicialWeakEquivalencesInSliceDetectedFiberwise) implies that
$
  f(U)
  \,\simeq\,
  \mathrm{PSh}(\mathcal{C})
     \big(
        y_{\mathcal{C}}(U)
        ,\,
        f
     \big)
$
is a weak equivalence.  And since this holds for all $U \,\in\, \mathcal{C}$, this means that $f$ is a weak equivalence in the slice of the projective model structure.
\end{proof}

##### Locally

We now promote the [[Quillen equivalence]] in the previous section to the case of Čech-[[local model structures on simplicial presheaves]].

Recall that these are obtained as a [[left Bousfield localization]] of the (say) [[projective model structure]] on [[simplicial presheaves]] with respect to [[Čech nerves]] of [[covering families]].

We reuse the notation of the previous section.

\begin{proposition}
The [[Quillen equivalence]] of \ref{SimplicialLocalSectionsIsRightQuillen}
descends to a [[Quillen equivalence]] of the corresponding Čech-local projective model structures.
\begin{tikzcd}
   \big(
    \mathrm{sPSh}(\mathcal{C})_{/y_{\mathcal{C}}(X)}
  \big)_{\mathrm{\check Cech,proj}}
    \ar[
      rrr,
      shift right=8pt,
      "{
        \mathrm{PSh}(\mathcal{C})_{/y(X)}
        \big(
          (y_{\mathcal{C}})_{/X}(-)
          ,\,
          -
        \big)
      }"{below}
    ]
    &&&
    \mathrm{sPSh}
    \big(
      \mathcal{C}_{/X}
    \big)_{\mathrm{\check Cech,proj}}
    \ar[
       lll,
       shift right=8pt
    ]
    \ar[
      lll,
      phantom,
      "{ \scalebox{.6}{$\simeq_{\mathrlap{\mathrm{Qu}}}$} }"
    ]
\end{tikzcd}
\end{proposition}

\begin{proof}
Both [[model categories]] are [[left proper]] and [[combinatorial]].
Therefore we can take [[left Bousfield localizations]] with respect to arbitrary sets of morphisms.

We localize both sides with respect to [[Čech nerves]] of respective covering families.
Observe that [[Čech nerves]] of covering families in $\mathcal{C}_{/X}$ are mapped to [[Čech nerves]] of covering families in $sPSh(\mathcal{C})$ and therefore also in the [[slice category]] $sPSh(\mathcal{C})_{y_{\mathcal{C}}(X)}$.
Thus, we have an induced [[Quillen adjunction]] between localized [[model categories]].

It remains to show that this [[Quillen adjunction]] is a [[Quillen equivalence]].

It suffices to show that the right adjoint reflects [[weak equivalences]] between [[fibrant objects]].
Here [[fibrant objects]] are objectwise [[Kan complexes]] that satisfy the appropriate variant of the [[homotopy descent property]].
Local weak equivalences between locally fibrant objects coincide with objectwise weak equivalences.
As established in the previous section, the right adjoint functor reflects objectwise weak equivalences between objectwise fibrant presheaves, which completes the proof.
\end{proof}

\linebreak

#### Over 1-truncated objects
 {#SimplicialpresheavesSlicedOverOneTruncatedObjects}

Over a 1-truncated object (hence over a simplicial presheaf with values in the [[simplicial nerve]] of a [[groupoid]]):

* the slice category needs to be generalized to the homotopy slice, now given by the [[Grothendieck construction]] (in groupoids);

* the right adjoint needs to be generalized from forming [[fibers]] (eq:SliceHomOfPresheavesAsFiberOfPlainHom) to forming [[homotopy fibers]].

This happens in [Hollander 2008, Def. 5.1](#Hollander08).

The resulting Quillen equivalences for the global [[model structure on simplicial presheaves]] is [Hollander 2008, Them. 5.2](#Hollander08), and that for any local model structure is [Hollander 2008, Thm. 6.2](#Hollander08).

\linebreak

### In $\infty$-category theory
 {#InInfinityCategoryTheory}

All the natural equivalences used in a category-theoretic proof such as of Prop. \ref{StatementInPlainCategoryTheoryOverAnyPresheaf} above also hold in [[(infinity,1)-category theory|$\infty$-category theory]].

More explicitly,
since [[simplicial localization]] at the [[Quillen equivalences]] [[Ho(CombModCat)|identifies]]  the [[homotopy theory]] ([[(infinity,1)-category|$\infty$-category]]) of [[combinatorial model categories]] (such as [[model categories of simplicial presheaves]] and their [[slice model structures]]) with that of [[presentable (infinity,1)-categories|presentable $\infty$-categories]], Prop. \ref{SimplicialLocalSectionsIsRightQuillen} immediately [[Ho(CombModCat)|implies]] [[simplicial Quillen adjunction|simplicial]] [[Quillen equivalences]] which [[presentable (infinity,1)-category|present]] the equivalence of $\infty$-categories.

Either way, we obtain the following conclusion (and again this hold verbatim also over non-representable base presheaves):

\begin{prop}\label{EquivalenceOfInfinityCategories}
**(fundamental theorem of $\infty$-presheaf $\infty$-topos theory)**
\linebreak
  For $\mathbf{C}$ a [[small (infinity,1)-category|small $\infty$-category]] and $X \,\in\, \mathbf{S}$ an [[object]], the operation of forming systems of [[local sections]] of [[bundles]] of [[(infinity,1)-presheaves|$\infty$-presheaves]] over $y(X)$ is an [[equivalence of (infinity,1)-categories|equivalence of $\infty$-categories]]:

\begin{tikzcd}
    \mathrm{PSh}_\infty(\mathbf{C})_{/y_{\mathbf{C}}(X)}
    \ar[
      rrr,
      "{
        \mathrm{PSh}_\infty(\mathbf{C})_{/y_{\mathbf{C}}(X)}
        \big(
          (y_{\mathbf{C}})_{/X}(-)
          ,\,
          (-)
        \big)
     }"
    ]
    &&&
    \mathrm{PSh}_\infty
    \big(
      \mathbf{C}_{/X}
    \big)
\end{tikzcd}

from the [[slice (infinity,1)-category|slice $\infty$-category]] of the [[(infinity,1)-category of (infinity,1)-presheaves|$\infty$-category of $\infty$-presheaves]] over $\mathbf{C}$ to the [[(infinity,1)-category of (infinity,1)-presheaves|$\infty$-category of $\infty$-presheaves]] over the [[slice (infinity,1)-category|slice $\infty$-category]] of $\mathbf{C}$.
\end{prop}

An alternative proof of this statement in terms of [[quasi-categories]] is in [Lurie 2009, Prop. 5.1.6.12](#Lurie09). (See also [here](slice+infinity1-topos#SheavesOnBigSite) at *[[slice (infinity,1)-topos|slice $\infty$-topos]]*.)

A(nother) [[model category]]-theory argument for the statement over any [[1-truncated]] simplicial presheaf is in ([Hollander 2008](#Hollander08)).

A [[quasi-category]]-proof of this statement over representables is in [Lurie 2009, Cor. 5.1.6.12](#Lurie09). (This states a more general theorem which superficially looks like it may cover the case of non-representable base presheaves, but at least not directly so.)

\begin{example}\label{InfinityActionsAsSlice}
**($\mathcal{G}$-[[infinity-action|$\infty$-actions]] as slice over $B\mathcal{G}$)**
\linebreak
  Consider
$$
  \mathcal{G}
    \,\simeq\,
  \Omega B \mathcal{G}
    \,\in\,
  Grp\big(Grpd_\infty\big)
$$
an [[infinity-group|$\infty$-group]] with [[delooping]]/[[classifying space]] $B \mathcal{G} \,\in\, Grpd_\infty^{\geq 1}$. Observe that the external slice of the point by any $\mathcal{X} \,\in\, Grpd_\infty$ is that $\infty$-groupoid:
$$
  \ast_{/\mathcal{X}}
  \;\simeq\;
  \mathcal{X}
  \,,
$$
so that Prop. \ref{EquivalenceOfInfinityCategories} here gives the following sequence of equivalences:
$$
  \array{
    \big(Grpd_\infty\big)_{/\mathbf{B}\mathcal{G}}
    \,\simeq\,
    \big(
      PSh_\infty(\ast)
    \big)_{/\mathbf{B}\mathcal{G}}
    &
    \underoverset
      {\phantom{--}\sim\phantom{--}}
      {\Gamma}
      {\longrightarrow}
    &
    PSh_\infty
    \big(
      \ast_{/\mathbf{B}\mathcal{G}}
    \big)
    \;\simeq\;
    PSh_\infty(
      \mathbf{B}\mathcal{G}
    \big)
    \;\simeq\;
    \mathcal{G} Act
    \big(
      Grpd_\infty
    \big)
    \,.
  }
$$
This is the identification of the slice over $B \mathcal{G}$ with [[infinity-action|$\infty$-actions]] of $\mathcal{G}$. For a [[model category]]-presentation see at *[Borel model structure -- Relation to the slice over the simplicial classifying space](Borel+model+structure#RelationToSliceOverSimplicialClassifyingSpace)*.
\end{example}

\begin{example}\label{CohesionOfGlobalOverGEquivariantHomotopyTheory}
**([[cohesion of global- over G-equivariant homotopy theory]])**
\linebreak

In the case that $\mathbf{C} \,=\, Snglrt \,\coloneqq\, Grpd^{fin}_{1,\geq 1}$ is the [[global orbit category]] (a [[(2,1)-category]]) the equivalence of Prop. \ref{EquivalenceOfInfinityCategories} extracts the system of [[fixed loci]] of an object in [[global equivariant homotopy theory]] sliced over the archetypical $G$-[[orbi-singularity]], for some [[equivariance group]] $G$.
Together with the [[adjoint quadruple]] that is induced (see [here](adjoint+quadruple#ViaKanExtensionOfAdjointPairs)) via [[(infinity,1)-Kan extension|$\infty$-Kan extension]] from the [[reflective sub-(infinity,1)-category|reflection]] onto the $G$-[[orbit category]],  this implies the [[cohesion of global- over G-equivariant homotopy theory]]. See there for more.
\end{example}

\linebreak

## References

Textbook accounts for the statement in plain [[category theory]]:

* {#SGA41}[[Michael Artin]], [[Alexander Grothendieck]], [[Jean-Louis Verdier]], Ex. 1 Prop. 5.11, p. [27](http://www.cmls.polytechnique.fr/perso/laszlo/sga4/SGA4-1/sga41.pdf#page=27) in:  _Théorie des Topos et Cohomologie Etale des Schémas_ ([[SGA4]]) Tome 1: *Théorie des Topos* Springer **LNM** **269** (1972) ([doi:10.1007/BFb0081551](https://link.springer.com/book/10.1007/BFb0081551), [pdf](http://www.cmls.polytechnique.fr/perso/laszlo/sga4/SGA4-1/sga41.pdf))

* {#KS06} [[Masaki Kashiwara]], [[Pierre Schapira]], Lemma 1.4.12 in: *[[Categories and Sheaves]]*, Grundlehren der Mathematischen Wissenschaften __332__, Springer 2006 ([doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf))

Discussion in [[(infinity,1)-category theory|$\infty$-category theory]]:

via [[model categories]]:

* {#Hollander08} [[Sharon Hollander]], Thm. 6.1(a) in : *Diagrams indexed by Grothendieck constructions*, Homology Homotopy Appl. 10(3): 193-221 (2008) ([doi:10.4310/HHA.2008.v10.n3.a10](https://dx.doi.org/10.4310/HHA.2008.v10.n3.a10),  [euclid:1251832473](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-10/issue-3/Diagrams-indexed-by-Grothendieck-constructions/hha/1251832473.full))

and via [[quasi-categories]]:

* {#Lurie09} [[Jacob Lurie]], Prop. 5.1.6.12 in: *[[Higher Topos Theory]]*, Annals of Mathematics Studies 170, Princeton University Press, 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))


slice spectral sequence

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

[[!include Lurie spectral sequences -- table]]

## References

On slice spectral sequence in [[motivic homotopy theory]]:

* [[Vladimir Voevodsky]], _Open problems in the motivic stable homotopy theory. I, Motives, polylogarithms and Hodge theory, Part I_ (Irvine, CA, 1998), Int. Press Lect. Ser., vol. 3, Int. Press, Somerville, MA, 2002, pp. 3–34. MR MR1977582 (2005e:14030)

In [[equivariant stable homotopy theory]]:

* John Ullman, _On the Slice Spectral Sequence_ ([arXiv:1206.0058](http://arxiv.org/abs/1206.0058))

[[!redirects slice spectral sequences]]


slice theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[group]] $G$ equipped with an [[action]] on some [[space]] $X$, a _slice through the $G$-orbits in $X$_ is a subspace $S\hookrightarrow X$ such that $X$ is exactly exhausted by the $G$-[[orbits]] of $S$.

More generally, if $H \subset G$ is a [[subgroup]], then a _slice through $G$-orbits modulo $H$_, or simply an *$H$-slice*, is a subspace $S \subset G$ to which the $H$-[[action]] on $X$ restricts, and such that the $G$-orbits of $S$ modulo this $H$-action on $S$ exactly exhaust $X$.

In [[mathematics]], specifically in [[equivariant differential topology]], the terminology is traditionally used by default to refer to slices through [[open subset|open sub]] [[G-spaces]] $U \subset X$ of a given ambient [[topological G-space]] $X$.

Here by a *slice theorem* is traditionally meant a list of sufficient conditions which guarantee that suitable slices do exist in a given situation. This plays a central role for instance in the [[local triviality]] of [[equivariant bundles]].

In [[physics]] one typically considers this for $G$ a [[gauge group]], in which case one speaks of *gauge slices* and thinks of these as a choice of *[[gauge fixing]]* (see there for more).

## Slices of group action

There is the following [[general abstract]] definition of slices via [[adjunctions]] (which however seems not to be made explicit in existing literature):

\begin{defn}\label{HSlice}
**($H$-slice)**
\linebreak
Let $H \subset G$ be a [[subgroup]] inclusion such that restricting given [[actions]] ([[internalization|internal]] to a given ambient [[category]]) of $G$ to $H$ has a [[left adjoint]] $G \times_{H} (-)$ (e.g. for [[topological G-spaces]] the [topological induced action](topological+G-space#induced_action)).

Then an *$H$-slice* in a $G$-action $U$ is an $H$-subaction inclusion $S \overset{i}{\hookrightarrow} U$ whose induction/restriction-[[adjunct]] is an [[isomorphism]]:

$$
  G \times_ H S
  \underoverset
    {\simeq}
    {
      \;\;
      \tilde i
      \;\;
    }
    {\longrightarrow}
  U
  \,.
$$
\end{defn}

Here are more traditional ways to say this:

\begin{defn}\label{SlicesDef}
  Let $G$ be a [[topological group]] and $X$ a [[topological G-space]].

For $H \subset G$ a [[closed subspace|closed]] [[subgroup]],
a [[topological subspace]] $S \subset X$ is called:

* an *$H$-slice* if

  1. $S$ is an $H$-subspace;

  1. and $H$ is maximal with this property, in that the following map is an [[isomorphism]] (i.e. [[homeomorphism]]):

     \[
       \label{IsomorphismDefiningASlice}
       \array{
         G \times_H S
         &
         \overset
         {\simeq}
         {
           \longrightarrow
         }
         &
         G \cdot S
         \\
         [g, s]
         &
         \mapsto
         &
         g \cdot s
       }
     \]

  1. whose image $G\cdot S \subset X$ is [[open subset|open]];

* a *slice through $x \in X$* if

  1. $x \in S \subset X$;

  1. $S$ is a $G_x$-slice in the above sense,

     for $G_x \coloneqq Stab_G(x)$ the [[stabilizer group]] of $x$.

\end{defn}

(e.g. [Bredon 72, Ch. II, Def. 4.1](#Bredon72))

Alternatively but equivalently (as in [Bredon 72, Ch. II, Thm. 4.4 (iv)](#Bredon72)):

\begin{defn}\label{PalaisSlicesDef}
  Let $G$ be a [[topological group]] and $X$ a [[topological G-space]].

For $H \subset G$ a [[closed subspace|closed]] [[subgroup]],
a [[topological subspace]] $S \subset X$ is called:

* an *$H$-kernel* if it is the [[preimage]] of the base point $[H] \in G/H$ in the [[coset space]] under a $G$-[[equivariant function|equivariant]] [[continuous function]] $f$ from the $G$-[[orbit]] of $S$:

  $$
    \underset{
      G\cdot S
      \overset{f}{\to}
      G/H
    }
    {
      \exists
    }
    \;
    S
      \;\simeq\;
    f^{-1}
    \big(
      [H]
    \big)
  $$

* an *$H$-slice* if it is an $H$-kernel and its [[orbit]] is an [[open subspace]]:

  $$
    G \cdot S
    \underset{open}
      {\subset}
    X
  $$

* a *slice through $x$* if it is a $G_x$-slice for some $x \in S \subset X$ with [[stabilizer group]] $G_x \coloneqq Stab_G(x) \simeq H$.

\end{defn}

([Palais 61, Def. 2.1.1](#Palais61), recalled as [Karppinen 16, Def. 6.1.1](#Karppinen16))

## Properties

### Existence of slices

A _slice theorem_ is a statement of sufficient conditions such that there is a slice through each point of a given [[topological G-space]].

\begin{prop}\label{ExistenceOfSlicesInProperActionsOnLocallyCompactSpaces}
**(existence of local slices for proper actions on locally compact spaces)** \linebreak
Let

* $G$ be the [[topological group]] underlying a [[Lie group]],

* $X$ be a [[locally compact Hausdorff space]],

* $G \times X \overset{\rho}{\to} X$ be a [[proper action]].

Then for every point $x \in X$ there exists a slice through $x$ (Def. \ref{SlicesDef}).
\end{prop}

This is due to [Palais 61, Prop. 2.3.1](#Palais61), recalled as [Karppinen 2016, Thm. 6.2.7](#Karppinen16).

\begin{remark}\label{PalaisProofBeyondLocalCompactness}
  The thrust of [Palais 61](#Palais61) is to state Prop. \ref{ExistenceOfSlicesInProperActionsOnLocallyCompactSpaces} without the assumption that $X$ be [[locally compact topological space|locally compact]], in which case the definition of "[[proper action]]" needs to be strengthened ("[Palais proper action](proper+action#PalaisProperness)", [Palais 61, Def. 1.2.2](#Palais61)). Under the assumption of local compactness, Palais' more general statement reduces as above, see [Karppinen 2016, Rem. 5.2.4](#Karppinen16).
\end{remark}

When the group $G$ is [[compact topological group|compact]] then the condition on the $G$-space $X$ may be relaxed:

\begin{prop}\label{ExistenceOfSlicesInCompactGroupActionsOnCompletelyRegularSpaces}
**(existence of local slices for [[compact topological group|compact group]] actions on [[completely regular spaces]])**
\linebreak

Let

* $G$ be a [[compact topological group]]

* $X$ a [[completely regular topological space]].

and $G \times X \overset{\rho}{\to} X$ any [[continuous function|continuous]] [[group action|action]].

Then for every point $x \in X$ there exists a slice through $x$ (Def. \ref{SlicesDef}).
\end{prop}
This is due to [Mostow 1957, Thm. 2.1](#Mostow57), reproved as [Palais 60b, Cor. 1.7.19](#Palais60b). It is also a special case of [Palais 61, Prop. 2.3.1](#Palais61), using that compact group actions on completely regular spaces are "Cartan actions" in the sense of Palais (by [Karppinen 2016, Prop. 5.1.7](#Karppinen16)).

For [[smooth manifold|smooth]] [[G-manifolds]] the $H$-space $S$ may be taken to be a [[linear representation]] (e.g. [tomDieck 87, Thm. 5.6](#tomDieck87)).

## Examples and applications

### General

\begin{example}\label{GSliceThroughGFixedPoint}
**($G$-slice through $G$-fixed point)**
\linebreak
  If a point $x \in X$ in a [[topological G-space]] $X$ is [[fixed point|fixed]] by all of $G$, so that $Stab_G(x) \,=\, G$, then $X$ itself is a $G$-slice through $x$ (Def. \ref{SlicesDef}), since we trivially have
$G \times_G X \,\simeq\, X$ and $X \underset{open}{\subset} X$.
\end{example}

\begin{example}\label{SlicesThroughPointsInPuncturedOrthogonalRepresentation}
**(slices through points in orthogonal representation)**
\linebreak
  For $n \in \mathbb{N}$ consider the defining [[group action]] of the [[orthogonal group]] $G \coloneqq O(n+1)$ on the [[Cartesian space]] $\mathbb{R}^{n+1}$.

Then two cases of [[stabilizer groups]] appear:

1. the origin $0 \,\in\, \mathbb{R}^{n+1}$ is [[fixed point|fixed]] by all of $O(n+1)$, and a $Stab_G(0) = O(n+1)$-slice through origin is given by all of $\mathbb{R}^{n+1}$ (by Ex. \ref{GSliceThroughGFixedPoint}) or by any [[open ball]] around it;

1. for every other point $x \in \mathbb{R}^{n+1} \setminus \{0\}$ the [[stabilizer subgroup]] is $G_x = O(n)$ and the [[coset space]] $G/G_x = S^n$ is the [[n-sphere]] (see [there](sphere#LabelCosetSpaceStructure)).

The ray $ R_X \,\coloneqq\, \{c \cdot x \vert c \in \mathbb{R}_{\gt 0}\} \subset \mathbb{R}^{n+1} \setminus \{0\} $ has [[orbit]] the [[complement]] $\mathbb{R}^{n+1} \setminus \{0\}$ and is thus an $O(n)$-slice through $x$ (Def. \ref{SlicesDef}) as exhibited by the radial quotient map
$$
  \mathbb{R}^{n+1}
    \setminus
  \{0\}
    \longrightarrow
  S^n
   =
  O(n+1)/O(n)
  \,.
$$

$$
  R_x \times_{O(n)} O(n+1)
  \;\simeq\;
  \mathbb{R}^{n+1} \setminus \{0\}
  \,.
$$
\end{example}
(also  [Karppinen 2016, Ex. 6.1.10](#Karppinen16))

\begin{example}
If time evolution on some [[Lorentzian manifold]] is given as an $\mathbb{R}^1$-action with [[timelike]] flow lines, then slices ("1-slices") for this action are known as *[[Cauchy surfaces]]*.
\end{example}

### Principal bundles
 {#PrincipalBundles}

\begin{proposition}\label{SlicesOfFreeActionsThroughEachPointImplyPrincipalQuotient}
**([[slice theorem]] implies that [[free action|free]] [[quotient space|quotient]] is [[principal bundle]])**
\linebreak
Consider a [[topological G-space]] $P$ such that

1. the [[group action|action]] by $G$ is [[free action|free]],

1. there exist slices *through* each $p \in P$ (Def. \ref{SlicesDef}).

Then the [[quotient space]] [[coprojection]] $P \xrightarrow{\;q\;} P/G$ is a [[locally trivial fiber bundle]], in fact a $G$-[[principal bundle]].
\end{proposition}
([Palais 1961, p. 315, Sec. 4.1](#Palais61))
\begin{proof}
  We need to show that for each point $x \in P/G$ there exists an [[open neighbourhood]] $x \,\in\, U_x \,\subset\, P/G$ such that its [[preimage]] under the [[coprojection]] is [[equivariant function|equivariantly]] [[homeomorphism|homeomorphic]] to its [[product space]] with $G$:
\[
  \label{ConditionToBeShownInProofOfPrincipalityFromSliceTheorem}
  G \times U _x
    \;\simeq\;
  q^{-1}(U_x)
  \;\;\;\;\;\;
  \in
  \;
  G Act(TopSp)
  \,.
\]

Now, picking any [[preimage]] $\hat x \,\in\, q^{-1}\big(\{x\}\big) \,\subset\, P$,
there exists, by assumption, a $G_{\hat x}$-slice $S_{\hat x} \,\subset\, P$ through an [[open neighbourhood]] $\hat x \in \widehat{U}_{\hat x} \,\subset\, P$. But since the $G$-action is assumed to be [[free action|free]], the [[stabilizer subgroup]] $G_{\hat x} \,\simeq\, 1$ is necessarily the [[trivial group]], so that the defining property (eq:IsomorphismDefiningASlice) of the slice is:
\[
  \label{LocalTrivializationInProofOfPrincipalityFromSliceTheorem}
  \array{
    G \times S_{\hat x}
    &
    \xrightarrow{ \;\; \sim \;\; }
    &
    \widehat{U}_{\hat x}
    &
    \underset{open}{\subset}
    \;
    P
    \\
    (g, p)
    &\mapsto&
    g \cdot p
    \mathrlap{\,.}
  }
\]

Setting
\[
  \label{OpenChartInProofOfPrincipalityFromSliceTheorem}
  U_x
  \;\coloneqq\;
  \widehat{U}_{\hat x}/G
  \;\subset\;
  P/G
  \,,
\]
observe that this implies:
$$
  q^{-1}(U_x)
  \;\simeq\;
  \widehat{U}_{\hat x}
  \,.
$$
and this implies the claim:

1. $U_x \,\subset\, P/G$ is an [[open subset]]

   (since $\widehat U_{\hat x} \,\subset\, P$ is [[open subset|open]]  and using  the definition of the [quotient topology](quotient+space#QuotientTopologicalSpace));

1. $U_x$ satisfies the required condition (eq:ConditionToBeShownInProofOfPrincipalityFromSliceTheorem)

   (since this is now the slice condition (eq:LocalTrivializationInProofOfPrincipalityFromSliceTheorem)),

(In fact, the slice $S_x$ we used in this argument gives the function
$
  U_x \xrightarrow{\;\sim\;} S_x \xhookrightarrow{\;} P
$
which is the [[local section]] that exhibits the [[local trivialization]].)
\end{proof}

As a corollary:

\begin{prop}\label{FreeAndProperLieGroupActionsOnLCHSpacesArePrincipal}
**([[free action|free]] and [[proper action|proper]] [[Lie group]] [[group action|actions]] on [[locally compact Hausdorff spaces]] are [[locally trivial bundle|locally trivial]])**
\linebreak
Consider a [[topological G-space]] $P$ such that

1. $G$ carries the structure of a [[Lie group]],

1. $P$ is a [[locally compact Hausdorff space]],

1. the [[group action|action]] by $G$ is [[free action|free]] and [[proper action|proper]].

Then the [[quotient space]] [[coprojection]] $P \xrightarrow{q} P/G$ is a $G$-[[principal bundle]].
\end{prop}
\begin{proof}
  By Prop. \ref{ExistenceOfSlicesInProperActionsOnLocallyCompactSpaces} the assumptions imply that through each point there exists a slice, so that the claim follows by Prop. \ref{SlicesOfFreeActionsThroughEachPointImplyPrincipalQuotient}.
\end{proof}

\begin{remark}
  Due to Prop. \ref{FreeAndProperLieGroupActionsOnLCHSpacesArePrincipal}, some authors *define* a $G$-[[principal bundle]] to be a free and proper action on a locally compact Hausdorff space, without mentioning local trivializability (e.g. [Raeburn & Williams 1991, Def. 2.1](https://www.jstor.org/stable/24896249 )).
\end{remark}

## Related concepts

* [[topological G-space]], [[G-manifold]]

* [[compact Lie group]]

* [[equivariant bundle]]

* [[equivariant differential topology]]

* [[equivariant Tietze extension theorem]]

## References

Original discussion:

* [[Andrew Gleason]], _Spaces With a Compact Lie Group of Transformations_, Proceedings of the American Mathematical Society Proceeding, Vol. 1, No. 1 (Feb., 1950), pp. 35-43 ([jstor:2032430](https://www.jstor.org/stable/2032430), [doi:10.2307/2032430](https://doi.org/10.2307/2032430))

* [[Deane Montgomery]], [[Chung Tao Yang]], _The existence of a slice_, Ann. of Math. 65 (1957), 108-116 ([jstor:1969667](https://www.jstor.org/stable/1969667), [doi:10.2307/1969667](https://doi.org/10.2307/1969667))

* {#Mostow57} [[George Mostow]], _Equivariant embeddings in Euclidean space_,  Ann. of Math. 65 (1957), 432-446 ([jhir:1774.2/46183](http://jhir.library.jhu.edu/handle/1774.2/46183), [pdf scan](https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/46183/Mostow_Mar57.pdf?sequence=1&isAllowed=y))

* {#Palais60} [[Richard Palais]], _Slices and equivariant embeddings_, chapter VIII in: [[Armand Borel]] (ed.), _Seminar on Transformation Groups_, Annals of Mathematics Studies 46, Princeton University Press 1960 ([jstor:j.ctt1bd6jxd](https://www.jstor.org/stable/j.ctt1bd6jxd))

* {#Palais60b} [[Richard Palais]], Section 1.7 of: *The classification of $G$-spaces*, Memoirs of the AMS, **36**, 1960 ([ISBN:978-0-8218-9979-3](https://bookstore.ams.org/memo-1-36) [pdf](http://vmm.math.uci.edu/PalaisPapers/ClassificationOfG-Spaces.pdf), [[Palais_ClassificationOfGSpaces.pdf:file]])

* {#Palais61} [[Richard Palais]], _On the Existence of Slices for Actions of Non-Compact Lie Groups_, Annals of Mathematics Second Series, Vol. 73, No. 2 (Mar., 1961), pp. 295-323 ([jstor:1970335](https://www.jstor.org/stable/1970335), [doi:10.2307/1970335](https://doi.org/10.2307/1970335), [pdf](http://vmm.math.uci.edu/ExistenceOfSlices.pdf))

* {#Bredon72} [[Glen Bredon]], Section II.4 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN:9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

Review:

* {#Karppinen16} [[Sini Karppinen]], _The existence of slices in $G$-spaces, when $G$ is a Lie group_, Helsinki 2016 ([hdl:10138/190707](http://hdl.handle.net/10138/190707))

See also:

* {#tomDieck87} [[Tammo tom Dieck]], Section I.5 of: _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

* [[Sergey Antonyan]], *Characterizing slices for proper actions of locally compact groups*, Topology and its Applications Volume 239, 15 April 2018, Pages 152-159 ([arXiv:1702.08093](https://arxiv.org/abs/1702.08093), [doi:10.1016/j.topol.2018.02.026](https://doi.org/10.1016/j.topol.2018.02.026))

Discussion for [[G-CW complexes]]:

* {#UribeLück14} [[Bernardo Uribe]], [[Wolfgang Lück]], Section 7 of: *Equivariant principal bundles and their classifying spaces*, Algebr. Geom. Topol. **14** (2014) 1925-1995 [[arXiv:1304.4862](https://arxiv.org/abs/1304.4862), [doi:10.2140/agt.2014.14.1925](http://dx.doi.org/10.2140/agt.2014.14.1925)]

[[!redirects slice theorems]]


sliced adjoint functors -- section

### Slicing of adjoint functors
 {#OnSlices}

\begin{proposition}\label{SliceAdjoints}
**(sliced adjoints)** \linebreak
Let

$$
   \mathcal{D}
   \underoverset
    {\underset{\;\;\;\;R\;\;\;\;}{\longrightarrow}}
    {\overset{\;\;\;\;L\;\;\;\;}{\longleftarrow}}
    {\bot}
  \mathcal{C}
$$

be a pair of [[adjoint functors]] ([[adjoint (∞,1)-functors|adjoint ∞-functors]]), where the [[category]] ([[(∞,1)-category|∞-category]]) $\mathcal{C}$ has all [[pullbacks]] ([[homotopy pullbacks]]).

Then:

1. For every [[object]] $b \in \mathcal{C}$ there is induced a pair of [[adjoint functors]] between the [[slice categories]] ([[slice (∞,1)-categories|slice ∞-categories]]) of the form

   \[
     \label{SlicedAdjointFunctorsOverLb}
     \mathcal{D}_{/L(b)}
     \underoverset
       {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}}
       {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}}
       {\bot}
     \mathcal{C}_{/b}
     \mathrlap{\,,}
   \]

   where:

   * $L_{/b}$ is the evident induced functor (applying $L$ to the entire triangle [[diagrams]] in $\mathcal{C}$ which represent the morphisms in $\mathcal{C}_{/b}$);

   * $R_{/b}$ is the [[composition|composite]]

     $$
       R_{/b}
         \;\colon\;
       \mathcal{D}_{/{L(b)}}
         \overset{\;\;R\;\;}{\longrightarrow}
       \mathcal{C}_{/{(R \circ L(b))}}
         \overset{\;\;(\eta_{b})^*\;\;}{\longrightarrow}
       \mathcal{C}_{/b}
     $$

     of

     1. the evident functor induced by $R$;

     1. the ([[homotopy pullback|homotopy]]) [[pullback]] along the $(L \dashv R)$-[[unit of an adjunction|unit]] at $b$ (i.e. the [[base change]] along $\eta_b$).

1. For every [[object]] $b \in \mathcal{D}$ there is induced a pair of [[adjoint functors]] between the [[slice categories]] of the form

   \[
     \label{SlicedAdjointFunctorsOverRb}
     \mathcal{D}_{/b}
     \underoverset
       {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}}
       {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}}
       {\bot}
     \mathcal{C}_{/R(b)}
     \mathrlap{\,,}
   \]

   where:

   * $R_{/b}$ is the evident induced functor (applying $R$ to the entire triangle [[diagrams]] in $\mathcal{D}$ which represent the morphisms in $\mathcal{D}_{/b}$);

   * $L_{/b}$ is the [[composition|composite]]

     $$
       L_{/b}
         \;\colon\;
       \mathcal{D}_{/{R(b)}}
         \overset{\;\;L\;\;}{\longrightarrow}
       \mathcal{C}_{/{(L \circ R(b))}}
         \overset{\;\;(\epsilon_{b})_!\;\;}{\longrightarrow}
       \mathcal{C}_{/b}
     $$

     of

     1. the evident functor induced by $L$;

     1. the [[composition]] with the $(L \dashv R)$-[[counit of an adjunction|counit]] at $b$ (i.e. the left [[base change]] along $\epsilon_b$).

\end{proposition}
The first statement appears, in the generality of [[(∞,1)-category theory]], as [[Higher Topos Theory|HTT, prop. 5.2.5.1]]. For discussion in [[model category|model category theory]] see at *[sliced Quillen adjunctions](slice+model+structure#SlicedQuillenAdjunction)*.
\begin{proof}
(in [[1-category|1-]][[category theory]])

Recall that ([this Prop.](adjoint+functor#GeneralAdjunctsInTermsOfAdjunctionUnitCounit)) the hom-isomorphism that defines an adjunction of functors ([this Def.](adjoint+functor#AdjointFunctorsInTermsOfNaturalBijectionOfHomSets)) is equivalently given in terms of [[composition]] with

* the [[adjunction unit]] $\;\;\eta_c \colon c \xrightarrow{\;} R \circ L(c)$

* the [[adjunction counit]] $\;\;\epsilon_d \colon L \circ R(d) \xrightarrow{\;} d$

as follows:

\begin{tikzcd}
    L(c)
    \ar[rr, "f"]
    &&
    d
    &{\phantom{AAA}}\leftrightarrow{\phantom{AAA}}&
    c
    \ar[rr, "\eta_c"]
    \ar[
      rrrr,
      rounded corners,
      to path={
           -- ([yshift=+12pt]\tikztostart.north)
           --node[above]{
               \scalebox{.7}{$
                 \widetilde{f}
               $}
             } ([yshift=+8.5pt]\tikztotarget.north)
           -- (\tikztotarget.north)}
    ]
    &&
    R \circ L(c)
    \ar[rr, "R(f)"]
    &&
    R(d)
\end{tikzcd}

\begin{tikzcd}
    c
    \ar[rr, "\widetilde f"]
    &&
    R(d)
    &{\phantom{AAA}}\leftrightarrow{\phantom{AAA}}&
    L(c)
    \ar[rr, "L(\widetilde{f})"]
    \ar[
      rrrr,
      rounded corners,
      to path={
           -- ([yshift=+8pt]\tikztostart.north)
           --node[above]{
               \scalebox{.7}{$
                 f
               $}
             } ([yshift=+8.5pt]\tikztotarget.north)
           -- (\tikztotarget.north)}
    ]
    &&
    L \circ R(d)
    \ar[
      rr,
      "\epsilon_d"
    ]
    &&
    d
\end{tikzcd}

Using this, consider the following transformations of morphisms in slice categories, for the **first case**:

**(1a)**

\begin{tikzcd}
    L(c)
    \ar[rr, "f", dashed]
    \ar[dr]
      &&
    d
    \ar[dl, "p"]
    \\
    &
    L(b)
\end{tikzcd}

**(2a)**

\begin{tikzcd}
    c
    \ar[
      rr,
      "{\eta_c}"
    ]
    \ar[dr]
    \ar[
      rrrr,
      rounded corners,
      to path={
           -- ([yshift=+30pt]\tikztostart.north)
           --node[above]{
               \scalebox{.7}{$
                 \widetilde{f}
               $}
             } ([yshift=+27pt]\tikztotarget.north)
           -- (\tikztotarget.north)}
    ]
    &&
    R \circ L(c)
    \ar[rr, "{R(f)}", dashed]
    \ar[dr]
      &&
    R(d)
    \ar[dl, "R(p)"]
    \\
    &
    b
    \ar[
      rr,
      "\eta_b"{below}
    ]
    &
    &
    R \circ L(b)
\end{tikzcd}

**(2b)**

\begin{tikzcd}
    c
    \ar[
      rr,
      dashed
    ]
    \ar[dr]
      \ar[
      rrrr,
      rounded corners,
      to path={
           -- ([yshift=+12pt]\tikztostart.north)
           --node[above]{
               \scalebox{.7}{$
                 \widetilde{f}
               $}
             } ([yshift=+8pt]\tikztotarget.north)
           -- (\tikztotarget.north)}
      ]
    &&
    \eta_b^\ast\big(R(d)\big)
    \ar[rr]
    \ar[dl]
    \ar[
      dr,
      phantom,
      "\mbox{\tiny\rmfamily(pb)}"
    ]
      &&
    R(d)
    \ar[dl, "R(p)"]
    \\
    &
    b
    \ar[
      rr,
      "\eta_b"{below}
    ]
    &
    &
    R \circ L(b)
\end{tikzcd}

**(1b)**

\begin{tikzcd}
    L(c)
    \ar[dr]
    \ar[rrrr, "L(\widetilde{f})"]
    \ar[
      rrrrrr,
      rounded corners,
      to path={
           -- ([yshift=+8pt]\tikztostart.north)
           --node[above]{
               \scalebox{.7}{$
                 f
               $}
             } ([yshift=+8pt]\tikztotarget.north)
           -- (\tikztotarget.north)}
    ]
    &&
    &&
    L \circ R(d)
    \ar[rr, "{\epsilon_d}"]
    \ar[dl]
    &&
    d
    \ar[dl, "p"]
    \\
    &
    L(b)
    \ar[
      rr,
      "L(\eta_b)"{below}
    ]
    \ar[
      rrrr,
      rounded corners,
      to path={
           -- ([yshift=-8pt]\tikztostart.south)
           --node[below]{
               \scalebox{.7}{$
                 \mathrm{id}
               $}
             } ([yshift=-8pt]\tikztotarget.south)
           -- (\tikztotarget.south)}
    ]
    &
    &
    L \circ R \circ L(b)
    \ar[rr, "\epsilon_{L(b)}"{below}]
    &&
    L(b)
\end{tikzcd}

Here:

* (1a) and (1b) are equivalent expressions of the same morphism $f$ in $\mathcal{D}_{/L(b)}$, by (at the top of the diagrams) the above expression of [[adjuncts]] between $\mathcal{C}$ and $\mathcal{D}$ and (at the bottom) by the [[triangle identity]].

* (2a) and (2b) are equivalent expression of the same morphism $\tilde f$ in $\mathcal{C}_{/b}$, by the [[universal property]] of the [[pullback]].

Hence:

* starting with a morphism as in (1a) and transforming it to $(2)$ and then to (1b) is the identity operation;

* starting with a morphism as in (2b) and transforming it to (1) and then to (2a) is the identity operation.

In conclusion, the transformations (1) $\leftrightarrow$ (2) consitute a [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) that witnesses an adjunction of the first claimed form (eq:SlicedAdjointFunctorsOverLb).

\linebreak

The **second case** follows analogously, but a little more directly since no pullback is involved:

**(1a)**

\begin{tikzcd}
  c
  \ar[rr, dashed, "f"]
  \ar[dr]
  &&
  R(d)
  \ar[dl]
  \\
  &
  R(b)
\end{tikzcd}

**(2)**

\begin{tikzcd}
  L(c)
  \ar[rr, dashed, "L(f)"]
  \ar[dr]
  \ar[
      rrrr,
      rounded corners,
      to path={
           -- ([yshift=+8pt]\tikztostart.north)
           --node[above]{
               \scalebox{.7}{$
                 \widetilde{f}
               $}
             } ([yshift=+8pt]\tikztotarget.north)
           -- (\tikztotarget.north)}
  ]
  &&
  L \circ R(d)
  \ar[rr, "\epsilon_d"{above}]
  \ar[dl]
  &&
  d
  \ar[dl]
  \\
  &
  L \circ R(b)
  \ar[rr, "\epsilon_b"{below}]
  &&
  b
\end{tikzcd}

**(1b)**

\begin{tikzcd}
  c
  \ar[rr, "\eta_c"{above}]
  \ar[dr]
  \ar[
      rrrrrr,
      rounded corners,
      to path={
           -- ([yshift=+12pt]\tikztostart.north)
           --node[above]{
               \scalebox{.7}{$
                 f
               $}
             } ([yshift=+8pt]\tikztotarget.north)
           -- (\tikztotarget.north)}
  ]
  &&
  R \circ L(c)
  \ar[rrrr,"R(\widetilde{f})"]
  \ar[dr]
  &&
  %R \circ L \circ R(d)
  %\ar[rr, "R(\epsilon_d)"{above}]
  %\ar[dl]
  &&
  R(d)
  \ar[dl]
  \\
  &
  R(b)
  \ar[rr, "\eta_{R(b)}"{below}]
    \ar[
      rrrr,
      rounded corners,
      to path={
           -- ([yshift=-8pt]\tikztostart.south)
           --node[below]{
               \scalebox{.7}{$
                 \mathrm{id}
               $}
             } ([yshift=-8pt]\tikztotarget.south)
           -- (\tikztotarget.south)}
    ]
  &&
  R \circ L \circ R(b)
  \ar[rr, "R(\epsilon_b)"{below}]
  &&
  R(b)
\end{tikzcd}

In conclusion, the transformations (1) $\leftrightarrow$ (2) consitute a [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) that witnesses an adjunction of the second claimed form (eq:SlicedAdjointFunctorsOverRb).
\end{proof}

\begin{remark}
  \label{LeftAdjointOfSlicedAdjunctionFormsAdjuncts}
**(left adjoint of sliced adjunction forms adjuncts)** \linebreak
  The sliced adjunction (Prop. \ref{SliceAdjoints}) in the second form (eq:SlicedAdjointFunctorsOverRb) is such that the sliced [[left adjoint]]  sends slicing morphism $\tau$ to their [[adjuncts]] $\widetilde{\tau}$, in that (again by [this Prop.](adjoint+functor#GeneralAdjunctsInTermsOfAdjunctionUnitCounit)):

$$
  L_{/d}
  \,
  \left(
    \array{
      c
      \\
      \big\downarrow {}^{\mathrlap{\tau}}
      \\
      R(b)
    }
  \right)
  \;\;
  =
  \;\;
  \left(
  \array{
    L(c)
    \\
    \big\downarrow {}^{\mathrlap{\widetilde{\tau}}}
    \\
    b
  }
  \right)
  \;\;\;
  \in
  \;
  \mathcal{D}_{/b}
$$
\end{remark}

The two adjunctions in \ref{SliceAdjoints} admit the following joint generalisation, which is proven [[Higher Topos Theory|HTT, lem. 5.2.5.2]].
(Note that the statement there is even more general and here we only use the case where $K = \Delta^0$.)

\begin{proposition}\label{SliceAdjointsGeneralized}
**(sliced adjoints)** \linebreak
Let

$$
   \mathcal{C}
   \underoverset
    {\underset{\;\;\;\;R\;\;\;\;}{\longleftarrow}}
    {\overset{\;\;\;\;L\;\;\;\;}{\longrightarrow}}
    {\bot}
  \mathcal{D}
$$

be a pair of [[adjoint (∞,1)-functors|adjoint ∞-functors]], where the [[(∞,1)-category|∞-category]] $\mathcal{C}$ has all [[homotopy pullbacks]].
Suppose further we are given objects $c \in \mathcal{C}$ and $d \in \mathcal{D}$ together with a morphism $\alpha: c \to R(d)$ and its adjunct $\beta:L(c) \to d$.

Then there is an induced a pair of [[adjoint (∞,1)-functors|adjoint ∞-functors]] between the [[slice (∞,1)-categories|slice ∞-categories]] of the form

   \[
     \label{SlicedAdjointFunctorsGeneral}
     \mathcal{C}_{/c}
     \underoverset
       {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longleftarrow}}
       {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longrightarrow}}
       {\bot}
     \mathcal{D}_{/d}
     \mathrlap{\,,}
   \]

   where:

   * $L_{/c}$ is the [[composition|composite]]

     $$
       L_{/c}
         \;\colon\;
       \mathcal{C}_{/{c}}
         \overset{\;\;L\;\;}{\longrightarrow}
       \mathcal{D}_{/{L(c)}}
         \overset{\;\;\beta_!\;\;}{\longrightarrow}
       \mathcal{D}_{/d}
     $$

     of

     1. the evident functor induced by $L$;

     1. the [[composition]] with $\beta:L(c) \to d$ (i.e. the left [[base change]] along $\beta$).

   * $R_{/d}$ is the [[composition|composite]]

     $$
       R_{/d}
         \;\colon\;
       \mathcal{D}_{/{d}}
         \overset{\;\;R\;\;}{\longrightarrow}
       \mathcal{C}_{/{R(d)}}
         \overset{\;\;(\alpha^*\;\;}{\longrightarrow}
       \mathcal{C}_{/c}
     $$

     of

     1. the evident functor induced by $R$;

     1. the [[homotopy pullback|homotopy]] along $\alpha:c \to R(d)$ (i.e. the [[base change]] along $\alpha$).

\end{proposition}


slideshow sandbox

author: AnonymousCoward
company:
title: pdf books
subtitle:
slide_theme: default
slide_footer:
slide_subfooter:

pdf books
==============

My First Slide
-----------------

set theory

[[!redirects pdf books]]
[[!redirects Slideshow sandbox]]


slope

In the contents of elementary [[geometry]]

* [[slope of a line]] (cf. *[[tangent]]*)

In the context of [[stability conditions]]:

* [[slope of a coherent sheaf]]

* [[slope filtration]]

[[!redirects slopes]]


slope filtration

#Contents#
* table of contents
{:toc}

## Overview

There are many examples that the objects of [[additive categories]] (abelian, abelian tensor categories, triangulated etc.) are equipped with **slope filtrations**, which are labelled by **real** numbers.

Slope is here related to [[Bridgeland stability]]. For instance [[stable coherent sheaf|stability of coherent sheaves]] in the case of [[slope of coherent sheaves]]

The _Stokes phenomenon_ in the study of asymptotics of differential equations may be regarded as a similar structure on the [[D-modules]] (or by the [[Riemann-Hilbert correspondence]], on the object in certain category  of monodromy data), which correspond to the solutions of differential equations.

See also [[wall crossing]], [[Stokes phenomenon]], [[Newton polygon]].

## Survey by Andre

For the slopes in algebraic and arithmetic geometry there is a recent survey

* Yves Andre, _Slope filtrations_, [arxiv/0812.3921](http://arxiv.org/abs/0812.3921)

Here is an introductory passage:

> Slope filtrations occur in algebraic and analytic geometry, in asymptotic analysis, in ramification theory, in p-adic theories, in geometry of numbers...Five basic examples are the Harder-Narasimhan filtration of vector bundles over a smooth projective curve, the Dieudonne-Manin filtration of F-isocrystals over the p-adic point, the Turrittin-Levelt filtration of formal differential modules, the Hasse-Arf filtration of finite Galois representations of local fields, and the Grayson-Stuhler filtration of euclidean lattices. Despite the variety of their origins, these filtrations share a lot of similar features.

## Other aspects and references

The story of slope filtrations is important for the construction of various moduli spaces (esp. in [[algebraic geometry]] and in mathematical physics), for [[Hodge theory]], theory of [[motive]]s, study of motivic properties of Feynman integrals...

* L. Katzarkov, [[Maxim Kontsevich|M. Kontsevich]], T. Pantev, _Hodge theoretic aspects of mirror symmetry_, [arxiv/0806.0107](http://arxiv.org/abs/0806.0107)

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Motivic Donaldson-Thomas invariants: summary of results_, [0910.4315](http://arxiv.org/abs/0910.4315)


slope of a coherent sheaf

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[rational number]] associated with a [[coherent sheaf]] $E$ over a [[curve]]: the [[degree of a coherent sheaf|degree]] divided by the [[rank of a coherent sheaf|rank]]:

$$
  \mu(E)
  \;\coloneqq\;
  \frac{deg(E)}{rank(E)}
  \,.
$$

For [[vector bundles]]: "normalized [[Chern character]]" ([Shatz 77](#Shatz77))

e.g. ([Huybrechts-Lehn 96, def. 1.2.11](#HuybrechtsLehn96))

This is what enters the condition on _[[stable coherent sheaves]]_.

## Related concepts

* [[slope filtration]]

* [[degree of a coherent sheaf]]

* [[rank of a coherent sheaf]]

* [[stable coherent sheaf]]

* [[stable vector bundle]]

* [[Bridgeland stability condition]]

## References

* {#Shatz77} Shatz 1977

* {#AtiyahBott83} [[Michael Atiyah]], [[Raoul Bott]], section 7 of _The Yang-Mills equations over Riemann surfaces_, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 ([jstor](http://www.jstor.org/stable/37156), [lighning summary](http://math.stackexchange.com/a/295505/58526))

* {#HuybrechtsLehn96} [[Daniel Huybrechts]], [[Manfred Lehn]], _The Geometry of the Moduli Spaces of Sheaves_, 1996 ([[HuybrechtsLehn.pdf:file]])

[[!redirects slope of a vector bundle]]


slope of a line

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

> This entry is about the concept in elementary [[geometry]]. For other notions of _[[slope]]_ see there.

***

#Contents#
* table of contents
{:toc}

## Idea

The _slope_ of a line in the [[Cartesian space|Cartesian]] [[plane]] $\mathbb{R}^2$ given, in the canonical [[coordinates]] $(x,y \;\colon\; \mathbb{R}^2 \to \mathbb{R})$ by the [[equation]]

$$
  y
    \,=\,
  y_0 + s \cdot x
$$

is the factor $s$.

## Related concepts

* [[tangent]]

* [[direction of a vector]]

* [[gradient]]

## References

See also

* Wikipedia, _[Slope](https://en.wikipedia.org/wiki/Slope)_

[[!redirects slope of line]]

[[!redirects slope of line]]
[[!redirects slopes of lines]]


small cardinality selection axiom

# The small cardinality selection axiom
* table of contents
{: toc}

## Idea

The small cardinality selection axiom (SCSA) is a weak form of the [[axiom of choice]], which asserts in a certain precise way that choice fails "in at most a [[small set|small]] way".  It was introduced by [[Michael Makkai]] for the study of [[anafunctors]], and thus it also has consequences for the existence of [[stack]] completions.

## Definition

The "global version" of SCSA states:

* The ([[large category|large]]) [[category]] [[Set]] of [[sets]] is [strongly equivalent](/nlab/show/equivalence+of+categories#StrongEquivalence) to a full subcategory of itself, each of whose isomorphism classes contains only a set of objects.

Therefore, we have a [[class]] [[function]] assigning to each set $A$, another set ${\Vert A\Vert}$ and a [[bijection]] $A \cong {\Vert A \Vert}$, in such a way that ${\Vert{-}\Vert}$ takes only set-many values on each isomorphism class in [[Set]].

There is also a "local version" of SCSA which can be stated as follows.  Given a function $f:A\to I$, say that a function $g:B\to J$ equipped with $\phi:J\to I$ is a *local pullback* of $f$ if there is a pair of pullback squares
\begin{tikzcd}
B \ar[d,"g"'] & C \ar[d] \ar[l] \ar[r] \ar[dl,phantom,near start,"\llcorner"] \ar[dr,phantom,near start,"\lrcorner"] & A \ar[d,"f"]\\
J & K \ar[l,"p"] \ar[r,"\phi p"'] & I
\end{tikzcd}
in which $p$ is surjective.  Then the local SCSA states:

* For any $f:A\to I$, the category whose objects are local pullbacks of $f$ and whose morphisms are pullback squares over $I$ has a [[weak limit|weakly terminal object]].

In more set-theoretic language, given a family of sets $(A_i)_{i\in I}$, define a family of sets $(B_j)_{j\in J}$ equipped with $\phi:J\to I$ to be a *local reindexing* of $(A_i)$ if for each $j\in J$ there exists an isomorphism $B_j \cong A_{\phi(j)}$ (but there is not necessarily a function picking out such an isomorphism).  Then the local SCSA states that for any $(A_i)_{i\in I}$ has a local reindexing $(B_j)_{j\in J}$ with $\phi:J\to I$ such that for any other local reindexing $(C_k)_{k\in K}$ with $\psi:K\to I$, there exists a function $\chi : K\to J$ such that $\phi\chi = \psi$ and a specified family of isomorphisms $C_k \cong B_{\chi(k)}$.

To deduce the local version from the global version, given $(A_i)_{i\in I}$ let $J$ be the set of pairs $(i, Y)$ such that $Y = \Vert X\Vert$ for some $X\cong A_i$, with $\phi$ the first projection and $B_{(i,Y)} = Y$.  Then given $(C_k)_{k\in K}$ with $\psi:K\to I$ such that $C_k \cong A_{\psi(k)}$, define $\chi(k) = (\psi(k),\Vert C_k\Vert)$.

## Relationship to other forms of choice

* In the presence of the [[axiom of global choice]] in [[material set theory]], the category [[Set]] has a [[skeleton]], namely the category of [[von Neumann ordinals|von Neumann cardinals]], which of course implies the global SCSA.  Similarly, the ordinary axiom of choice implies that every set is bijective to a unique von Neumann cardinal, and we can choose such a bijection uniformly for any small family of sets, which implies the local SCSA.

* Global SCSA also follows from the "global" version of the axiom of [[small violations of choice]], as proven in [Makkai's paper](#Makkai) (attributed to the referee).  Makkai claims there is a local form of SCSA that follows from the local SVC.  I have not attempted to show that this is true for the above local SCSA, although it is a guess at the sort of axiom Makkai might have had in mind.

## Consequences

* SCSA implies that the [[anabicategory]] of categories and [[anafunctors]] is [[locally small category|locally essentially small]] and [[cartesian closed category|cartesian closed]].  Specifically, local SCSA implies that for any two categories $X,A$ there is a small set of anafunctors from $X$ to $A$ such that any such anafunctor is isomorphic to one in this set.  The idea is that we can replace any anafunctor by a saturated one, in which case the set of "specifications that $F(x)=a$" for any $x\in X, a\in A$ is, if nonempty, a [[torsor]] over $Iso(a,a)$, and hence isomorphic to it.  Thus, applying local SCSA to the family $(Iso(a,a))_{a\in A}$, we obtain a "universal" set of specifications for saturated anafunctors.

  This implies that there exists a full subcategory of $Cat_{ana}(X,A)$ whose inclusion is a weak equivalence, and that an exponential $A^X$ exists.  If local SCSA holds in "functional" form, meaning there is a class-function specifying weakly terminal local reindexings, then this implies that $Cat_{ana}$ is weakly equivalent (i.e. ana-equivalent) to a locally small bicategory and that it admits a functor $(X,A) \mapsto A^X$.  (Of course, global SCSA implies functional local SCSA.)

## Models

[Makkai](#Makkai) states only the global SCSA, but claims that there is a local version of it that holds in all Grothendieck toposes.  The local form of SCSA stated above is a guess at the one Makkai might have had in mind.  I have not attempted to verify that it is true in all Grothendieck toposes.

## References

* {#Makkai} [[Michael Makkai]], _Avoiding the axiom of choice in general category theory_, Journal of Pure and Applied Algebra **108** isse 2 (1996) pp 109-173, doi:[10.1016/0022-4049(95)00029-1](https://doi.org/10.1016/0022-4049%2895%2900029-1), ([author's page](http://www.math.mcgill.ca/makkai/anafun/))

[[!redirects SCSA]]

category: foundational axiom

[[!redirects small cardinality selection axiom]]
[[!redirects axiom of small cardinality selection]]
[[!redirects small cardinality selection]]
[[!redirects SCSA]]


small category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Definition

A [[category]] is **small** if it has a [[small set]] of [[object]]s and a [[small set]] of [[morphism]]s.

In other words, a small category is an [[internal category]] in the category [[Set]].

A category which is not small may be called [[large category|large]], especially when it is not *essentially* small (see below).

## Properties

Small categories are free of some of the subtleties that apply to large categories.

A category is said to be **essentially small** if it is [[equivalence of categories|equivalent]] to a small category.  Assuming the [[axiom of choice]], this is the same as saying that it has a small [[skeleton]], or equivalently that it is [[locally small category|locally small]] and has a [[small set|small number]] of isomorphism classes of objects.

A **small category structure** on a [[locally small category]] $C$ is an [[essentially surjective functor]] from a set (as a [[discrete category]]) to $C$.  A category is essentially small iff it is locally small and has a small category structure; unlike the previous paragraph, this result does not require the axiom of choice.

## Smallness in the context of universes
 {#SmallnessInTheContextOfUniverses}

If [[Grothendieck universe]]s are being used, then for $U$ a fixed Grothendieck universe, a category $C$ is **$U$-small** if its collection of objects and collection of morphisms are both elements of $U$. Thus,

* a $U$-small category is a category [[internal category|internal to]] $U Set$.

This of course is a [[material set theory|material]] formulation. We may call $C$ **structurally $U$-small** if there is a [[bijection]] from its set of morphisms to an element of $U$ (the same for the set of objects follows). This gives an up-to-isomorphism version of $U$-smallness (see [[universe in a topos]] for an alternative structural formulation). Such structural $U$-smallness may be substituted in the discussion below.

Let $U\Set$ be the category of $U$-small sets. Similar considerations lead us to say

* a $U$-category (a [[locally small category|locally U-small]])-category is a category [[enriched category|enriched over]] $U Set$,

and that a category $C$ **essentially $U$-small** if it is locally $U$-small and admits an essentially surjective functor from a discrete $U$-small category.

A category is **$U$-moderate** if its set of objects and set of morphisms are both subsets of $U$.  However, some categories (such as the category of $U$-moderate categories!) are larger yet.

## Related concepts

* **small category**, [[locally small category]], [[complete small category]]

* [[essentially small (∞,1)-category]]

* [[small site]]

* [[large category]]

[[!redirects small]]
[[!redirects small category]]
[[!redirects small categories]]
[[!redirects essentially small category]]
[[!redirects essentially small categories]]
[[!redirects moderate category]]
[[!redirects moderate categories]]
[[!redirects essentially moderate category]]
[[!redirects essentially moderate categories]]

[[!redirects essentially small]]


small fibration

A [[fibered category]] is a **small fibration** if it is equivalent to a fibered category which is obtained via a specific construction of the [[externalization]] of an internal category. The input for externalization is an [[internal category]] $C = (C_0,C_1,s,t,c,i)$ in a category $E$ with pullbacks, and the output is a fibered category $P:F\to E$, whose fiber over an object $I$ ("index set") in $C$ is the small category $C^I = \hom_E(I,C)$ whose object part is $C_0^I$ and morphism part is $C_1^I$; and the structure maps of $C^I$ are obtained by postcomposing with structure maps of $C$.

Conventions for the internal category $C$: the composition $c$ takes arguments in Leibniz convention, and the source map is the right leg of the span $C_0\leftarrow C_1\to C_0)$; the projection $p_1,p_2:C_1\times_{C_0} C_1$ have $t\circ p_1 = s\circ p_2$.

Consider objects $m:I\to C_0$ and $n:J\to C_0$ in fibers over $I$ and $J$. Morphisms in $\hom_F(m,n)$ are pairs $(\alpha,f):m\to n$ where $\alpha:I\to J$ and $f:J\to C_1$ are morphisms in $E$ satisfying $s\circ f = m$, $t\circ f\circ\alpha = n$. The interesting part of the construction is the composition: given also $p:K\to C_0$ and $m\stackrel{(\alpha, f)}\to n\stackrel{(\beta,g)}\to p$, their composition is $(\beta\circ\alpha,c\circ h)$ where $c:C_1\times_{C_0} C_1\to C_1$ is the composition and $h=(f,g\circ\alpha):I\to C_1\times_{C_0} C_1$ is the map given by the universal property of the fibered product and $f =p_1\circ h$ and $g\circ\alpha = p_2\circ h$. One checks that so defined composition is associative.

Finally, the projection $P:F\to C$ is rather obvious: $P(m:I\to C_0) = I$, and in notation from above, $P(\alpha,f)=\alpha$. One checks that $P$ is a fibered category with a canonical [[cleavage]]: the chosen [[cartesian morphism]] over $\alpha : I\to J$ with given target $n:J\to C_0$ is $(\alpha,i\circ n\circ\alpha):(I,n\circ\alpha)\to(J,n)$. Indeed, for any object $l:L\to C_0$ over $L$ and a morphism $(\lambda,g):l\to n$ with $\lambda = \alpha\circ\beta$, one can decompose $(\lambda,g)=(\alpha,i\circ n\circ\alpha)\circ(\beta,g)$ where $(\beta,g)$ is a morphism in $F$ with $P(\beta,g)=\beta$.

The basic example is the small fibration $\mathrm{Set}(C)\to\mathrm{Set}$ of *set-indexed families of objects in $C$*, for a [[small category]] $C$. The objects in the fiber $C^I$, that is the functions $m:I\to C_0$, $m:i\mapsto m_i\in C_0$ are the $I$-indexed families $(m_i)_{i\in I}$ of objects in $C$. A map $(\alpha,f):m\to n$ is a map $\alpha:I\to J$ together with an $I$-indexed family $(f_i)_{i\in I}$ of morphisms $f_i:m_i\to n_{\alpha(i)}$ in $C$. The composition is $(\beta,(g_j)_{j\in J})\circ(\alpha,(f_i)_{i\in I})= (\beta\circ\alpha,(g_{\alpha(i)}\circ f_i)_{i\in I})$.

* [[Francis Borceux]], [[Handbook of Categorical Algebra]] vol.2, ch. 8


small groupoid

A _small groupoid_ is a [[groupoid]] with a [[small set]] of [[objects]], instead of a [[proper class]]. Hence a small groupoid is a [[small category]] that is also a groupoid

In plain groupoid theory ([[homotopy theory]]) smallness is usually the default meaning of "groupoid". But if one thinks of groupoids as special cases of [[categories]], and since for categories the default definition allows a proper class of objects, while otherwise one explicitly speaks of a _[[small category]]_, it does make sense to speak of "small groupoids" at times.

[[!redirects small groupoids]]


small instanton

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[instanton]] in plain [[Yang-Mills theory]] on a 4-[[dimension|dimensional]] [[manifold]] $K^{(4)}$ necessarily has a [[positive number|positive]] characteristic [[radius]]. Indeed, in the [[moduli space]] of instantons the would-be point of vanishing characteristc radius corresponds to a [[singularity]] which is unphysical, in that it is not part of the [[phase space]] of the [[theory (physics)|theory]].

But also the [[gauge field]] of [[heterotic string theory]] admits [[instanton]]-configurations on a transversal 4-space, and these are pullbacks of ordinary [[Yang-Mills instantons]] on the [[Euclidean space|Euclidean]] 4-manifold $K^{(4)}$ to a $(6+4)$-dimensional [[product manifold]]-[[spacetimes]]

\[
  \label{ProductSpacetime}
  X^{(9+1)} \;\simeq\; Q^{(5+1)} \times K^{(4)}
\]

along the [[projection map]] $Q^{(5+1)} \times K^{(4)} \overset{pr_2}{\longrightarrow} K^{(4)}$.

Therefore it makes sense to ask if, after embedding [[Yang-Mills instantons]] into [[heterotic string theory]] this way, the [[singularity]] in the moduli space of instantons becomes part of the phase space of the theory (now that [[quantum gravity]]-effects are supposedly included) so that instantons of vanishing radius do acquire "stringy" physical meaning after all -- then to be called _small instantons_.

**In HET-O.** A circumstantial but widely accepted argument ([[folklore]]) due to [Witten 96](#Witten96) says that this is indeed the case, and that these _small instantons_ in [[SemiSpin(32)]]-[[heterotic string theory]] are identified with heterotic [[NS5-branes]] [[wrapped brane|wrapped]] on the $Q^{(5+1)}$-factor in (eq:ProductSpacetime) and carrying an exotic [[Sp(1)]]-[[gauge field]] on their worldvolume, identified with the [[SU(2)]]-structure group of a single instanton.

Indeed, solutions of [[heterotic supergravity]] involving [[positive number|positive]]-[[radius]] [[Yang-Mills instantons]] had been interpreted as [[black brane|black]] [[NS5-branes]] of non-vanishing "thickness" in [Strominger 90](#Strominger90). This makes it plausible that these vanishing-radius small instantons correspond to fundamental hererotic 5-branes of vanishing thickness ([Witten 96, top of p. 9](#Witten96))

More precisely, this $Sp(1)$-gauge field is argued to be a [[non-perturbative effect]] invisible in [[perturbative string theory]], hence distinct from the [[SemiSpin(32)]]- or [[E8]]$\times$[[E8]]-gauge field famously seen in [[perturbative string theory|perturbative]] [[heterotic string theory]].

There this interpretation of small instantons as [[wrapped brane|wrapped]] 5-branes really pertains to [[M5-branes]] in [[heterotic M-theory]].

**In F-theory**. Under [[duality between F-theory and heterotic string theory]] the argument of [Witten 96](#Witten96) was claimed to be confirmed in [Aspinwall-Gross 96](#AspinwallGross96), [Aspinwall-Morrison 97](#AspinwallMorrison97).

**In Type I.** On the other hand, under [[duality between type I and heterotic string theory]] one expects to see [[duality in string theory|dually]] as a phenomenon visible in perturbative [[type I string theory]], now pertaining to [[D5-D9-brane bound states]]. Indeed, in general [[Dp-D(p+4)-brane bound states]] are argued to exhibit instantons in the [[worldvolume]] gauge theory of the $D(p+4)$-brane, and specifically for [[D5-D9-brane bound states]] in [[type I string theory]] one finds ([Witten 96. Section 3](#Witten96)) from [[orientifold]]-analysis ([Gimon-Polchinski 96, p. 7](#GimonPolchinski96)) that the [[gauge group]] on a single D5 is $Sp(2)$.

**In HET-E.** Similarly under [[T-duality]] of the [[SemiSpin(32)]]- to the [[E8]]$\times$[[E8]]-[[heterotic string]] the small instantons should also correspond to heterotic 5-branes there. That this is the case was argued, indirectly, in [Ganor-Hanany 96](#GanorHanany96), appealing now to [[orientifold]] image pairs under the [[Horava-Witten theory|Horava-Witten]] $\mathbb{Z}_2$-action.

## Related concepts

[[!include brane bound states -- table]]

## References

### General

* {#Strominger90} [[Andrew Strominger]], _Heterotic solitons_, _Nuclear Physics B, Volume 343, Issue 1, Pages 167-184, 1990 (doi:10.1016/0550-3213(90)90599-9)

* {#Witten96} [[Edward Witten]], _Small Instantons in String Theory_, Nucl. Phys. B460:541-559, 1996 ([arXiv:hep-th/9511030](https://arxiv.org/abs/hep-th/9511030))

* {#GimonPolchinski96} [[Eric Gimon]], [[Joseph Polchinski]], _Consistency Conditions for Orientifolds and D-Manifolds_, Phys. Rev. D54:1667-1676, 1996 ([arxiv:hep-th/9601038](https://arxiv.org/abs/hep-th/9601038))

* {#GanorHanany96} [[Ori Ganor]], [[Amihay Hanany]], _Small $E_8$ Instantons and Tensionless Non-critical Strings_, Nucl. Phys. B474 (1996) 122-140 ([arXiv:hep-th/9602120](https://arxiv.org/abs/hep-th/9602120))

* {#AspinwallGross96} [[Paul Aspinwall]], [[Mark Gross]], _The $SO(32)$ Heterotic String on a K3 Surface_, Phys. Lett. B387 (1996) 735-742 ([arXiv:hep-th/9605131](https://arxiv.org/abs/hep-th/9605131))

* {#AspinwallMorrison97} [[Paul Aspinwall]], [[David Morrison]], _Point-like Instantons on K3 Orbifolds_, Nucl. Phys. B503 (1997) 533-564 ([arXiv:hep-th/9705104](https://arxiv.org/abs/hep-th/9705104))

* {#Johnson98} [[Clifford Johnson]], _Études on D-Branes_, in: [[Mike Duff]] et. al. (eds.) _Nonperturbative aspects of strings, branes and supersymmetry_, Proceedings, Trieste, Italy, March 23-April 3, 1998 ([arXiv:hep-th/9812196](https://arxiv.org/abs/hep-th/9812196), [spire:481393](https://inspirehep.net/literature/481393))

Discussion within [[AdS-CFT duality]]:

* Tony Gherghetta, Alex Pomarol, *Small Instantons in Weakly-Gauged Holographic Models* ([arXiv:2110.01762](https://arxiv.org/abs/2110.01762))

[[!include flavor on heterotic M5-branes -- references]]

[[!redirects small instantons]]


small limit

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

A [[diagram]] $D \to C$ in a [[category]] $C$ is called _small_ if $D$ is a [[small category]].

A [[limit]] or [[colimit]] over a small diagram is a **small limit** or **small colimit**.
A [[product]] or [[coproduct]] over a small diagram is a **small product** or **small coproduct**.

[[!redirects small colimit]]

[[!redirects small limits]]
[[!redirects small colimits]]

[[!redirects small product]]
[[!redirects small products]]
[[!redirects small coproduct]]
[[!redirects small coproducts]]


small object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Compact objects
+--{: .hide}
[[!include compact object - contents]]
=--
=--
=--

#Small objects#
* table of contents
{:toc}

## Definition ##

An [[object]] $X$ of a [[category]] is **small** if it is $\kappa$-[[compact object|compact]] for some [[regular cardinal]] $\kappa$ (and therefore also for all greater regular cardinals).

Here, $X$ is called $\kappa$-compact if the [[representable functor|corepresentable functor]] $hom(X,\cdot)$ preserves $\kappa$-[[directed colimit]]s.

## Details ##

We unwrap the definition further. Let $J$ be a $\kappa$-filtered [[poset]], i.e. one in which every sub-poset $J' \subset J$ of cardinality $|J'| \lt \kappa$ has an upper bound in $J$.

Let $C$ be a category and $F : J \to C$ a diagram, called a $\kappa$-filtered diagram. Let $X \in C$ be any [[object]].

Then the condition that $X$ commutes with the colimit over $F$ means that the map of [[hom-set]]s

$$
  \lim_{\to^j} Hom_C(X, F(j))
  \to
  Hom_C(X,\lim_{\to^j} F(j))
$$

is an [[isomorphism]], i.e. a [[bijection]].

By the general properties of [[colimit]] (recalled at [[limits and colimits by example]]), the colimit

$$
  \lim_{\to^j} Hom_C(X,F(j))
$$

may be expressed as a [[coequalizer]]

$$
  \stackrel{\to}{\to} \coprod_{j \in J} Hom_C(X,F(j))
  \to
  \lim_{\to^j} Hom_C(X,F(j))
$$

hence as a [[quotient set]] of the the set of morphism in $C$ from $X$ into one of the objects $F(j)$. Being a quotient set, every element of it is _represented_ by one of the original elements in $\coprod_j Hom_C(X,F(j))$.

This means that we have

**Restatement**

The map of [[hom-set]]s

$$
  \lim_{\to^j} Hom_C(X, F(j))
  \to
  Hom_C(X,\lim_{\to^j} F(j))
$$
is onto precisely if every morphism $X \to \lim_\to F$ lifts to a morphism $X \to F(j)$ into one of the $F(j)$, schematically:

$$
  \array{
    \cdots&\to&F(j-1) &\to& F(j) &\to& F(j+1) &\to& \cdots
    \\
    &&&{}^{\mathllap{\exists \hat f}}\nearrow&\downarrow & \swarrow
    \\
    &&X& \stackrel{f}{\to} &\lim_\to F
  }
  \,.
$$

## Properties ##

Let $\lambda \gt \kappa$ be a regular cardinal greater than $\kappa$.  Then any $\lambda$-[[filtered category]] $D$ is also $\kappa$-filtered.  For being $\lambda$-filtered means that any diagram in $D$ of size $\lt\lambda$ has a cocone; but any diagram of size $\lt\kappa$ is of course also $\lt\lambda$.  Thus, any $\lambda$-[[filtered colimit]] is also a $\kappa$-filtered colimit, so any functor which preserves $\kappa$-filtered colimits must in particular preserve $\lambda$-filtered colimits.  It follows that any $\kappa$-compact object is also $\lambda$-compact.

## Examples and applications ##

* By definition, in a [[locally presentable category]] every object is a [[colimit]] over small objects.

* Smallness of objects plays a crucial role in the [[small object argument]].

## Related concepts

* [[cosmall object]], which is just the dual concept, but is interesting in its own right.

* [[object classifier]]

[[!redirects small objects]]