Postnikov system

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###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
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#### Factorization systems
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[[!include factorization systems - contents]]
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#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _Postnikov system_ is the [[infinitary factorization system]] of [[(n-epi, n-mono) factorization system|(n-epi, n-mono)-factorizations]] through [[n-images]] in an [[(∞,1)-topos]].

The basic (and historically first) example is the factorization in [[∞Grpd]]/[[Top]] of morphisms to the point: here the Postnikov system assigns to a [[homotopy type]] $X$ a [[tower]]

$$
  X \to \cdots \to X_3 \to X_2 \to X_1 \to X_0  \simeq *
$$

where each $X_k$ is a [[homotopy n-type|homotopy (k-2)-type]]/[[n-truncated object in an (infinity,1)-category|(n-2)-truncated object]] and such that each morphism $X_{k+1} \to X_k$ induces an [[isomorphism]] on [[homotopy groups]] in degree $\leq k-2$.
This may be thought of as decomposing $X$ into its "layers" as seen by the degree of [[homotopy groups]]. The [[characteristic classes]] which give the [[∞-group extension]] in each layer are called the _[[Postnikov invariants]]_ or _[[k-invariants]]_ of $X$.

More generally, in any [[(∞,1)-topos]] and for any [[morphism]] $f \colon X \to Y$ the corresponding Postnikov system is a [[tower]]

$$
  f \colon X \simeq im_\infty(f) \to \cdots \to im_2(f) \to im_1(f) \to im_0(f) \simeq Y
$$

of [[n-images]] of $f$ which interpolates between $X$ and $Y$ along $f$: the object $im_n(f)$ is a [[homotopy type]] that looks like $X$ in low degrees (below $n-1$), looks like $Y$ in high degrees (above degree $n$) and looks like the ordinary (1-[[topos theory|topos theoretic]]) [[image]] of $f$ _on homotopy groups_ in degree $n-1$ itself.

## Definition

In full generality, the Postnikov system is the [[infinitary factorization system]] of [[(n-epi, n-mono) factorization system|(n-epi, n-mono)-factorizations]] through [[n-images]] in an [[(∞,1)-topos]].

The following spells this out explicitly for default [[homotopy theory]], hence in [[∞Grpd]] and in terms of the [[presentable (∞,1)-category|presentation]] by the [[model structure on topological spaces]] and the [[model structure on simplicial sets]]:

* _[For topological spaces](#DefinitionForTopologicalSpaces)_

* _[For simplicial sets](#DefinitionForSimplicialSets)_.

With a little bit of care this induces corresponding presentations of Postnikov systems in general [[(∞,1)-topos]] by prolonging to suitable [[model structures on simplicial presheaves]]:

* _[For simplicial presheaves](#DefinitionForSimplicialPresheaves)_.

For more discussion of the general abstract situation see at

* _[[Postnikov tower in an (∞,1)-category]]_

### For topological spaces
 {#DefinitionForTopologicalSpaces}

Historically, Postnikov systems were first described on the model of [[homotopy types]] constituted by [[topological spaces]].

A __Postnikov system__ or __Postnikov [[tower]]__ or **Moore-Postnikov tower/system** of [[topological spaces]], or rather of their [[homotopy types]], is a sequence of [[path connected topological space|path connected]], [[pointed object|pointed]] [[topological spaces]] $X^{(n)}$, $n \geq 1$,
such that $\pi_r(X^{(n)})=0$ for $r \gt n$,
together with a sequence $\pi_n$ of [[module]]s of the [[fundamental group]] $\pi_1(X^{(1)})$ of $X^{(1)}$ and
[[fibration]]s $p_n : X^{(n+1)} \to X^{(n)}$ classified up to [[homotopy type]] by a specified cohomology class $k^{n+1} \in H^{n+1}(X^{(n)}, \pi_n)$.

### For simplicial sets
 {#DefinitionForSimplicialSets}

We discuss the realization of Postnikov systems on [[simplicial sets]]/[[Kan complexes]], first in the

* _[Absolute version](#DefinitionForSimplicialSetsAbsoluteVersion)_

and then more generally for the

* _[Relative version](#ForSimplicialSetsRelativeVersion)_

#### Absolute version
 {#DefinitionForSimplicialSetsAbsoluteVersion}

+-- {: .num_defn #SimplicialPostnikovTower}
###### Definition

Let $X$ be a [[simplicial set]]. A **Postnikov tower** for $X$ is

1. a sequence

   $$
     \cdots \to X_2
       \stackrel{q_1}{\to}
     X_1
       \stackrel{q_0}{\to}
     X_0
   $$

   with maps $i_n : X \to X_n$ such that all [[diagram]]s

   $$
     \array{
        && X
        \\
        & {}^{\mathllap{i_n}}\swarrow &&
         \searrow^{\mathrlap{i_{n-1}}}
        \\
        X_n && \stackrel{q_{n-1}}{\to} && X_{n-1}
     }
   $$

   [[commuting diagram|commute]];

1. such that for all vertices $v \in X_0$ we have for the [[homotopy group]]s

   $$
     \pi_{\gt n}( X_n, v) = 0
   $$

   and

   $$
     (i_n)_* : \pi_i (X,v) \stackrel{\simeq}{\to} \pi_i X_n
   $$

   for $i \leq n$.

=--

This appears for instance as ([Goerss-Jardine, def VI 3.1](#Goerss Jardine)).

#### Relative version
 {#ForSimplicialSetsRelativeVersion}

+-- {: .num_defn #SimplicialPostnikovTowerRelative}
###### Definition

Let $f : X \to Y$ be a homomorphism of [[simplicial sets]].
A (relative) **Postnikov tower** for $f$ is a [[tower]]

$$
  \array{
    X
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    im_\infty(f)
    \\
    \downarrow
    \\
    \vdots
    \\
    \downarrow
    \\
    im_2(f)
    \\
    \downarrow
    \\
    im_1(f)
    \\
    \downarrow
    \\
    im_0(f)
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    Y
  }
$$

that factors $f : X \to Y$ such that for all $n \in \mathbb{N}$

1. $X \to im_n(f)$

   1. induces an [[epimorphism]] on [[homotopy groups]] in degree $n-1$;

   1. induces an [[isomorphism]] on [[homotopy groups]] in degree $\lt n-1$

1. $im_n(f) \to Y$

   1. induces a [[monomorphism]] on [[homotopy groups]] in degree $n-1$;

   1. induces an [[isomorphism]] on [[homotopy groups]] in degree $\geq n$.

=--

This appears for instance as ([Goerss-Jardine, def. VI 2.9](#GoerssJardine)).

+-- {: .num_remark }
###### Remark

By the [[long exact sequence of homotopy groups]], the relative Postnikov tower is the tower of the [[(n-connected, n-truncated) factorization system]] of $f$ regarded as a morphism in the [[(∞,1)-category]] [[∞Grpd]]: is is the [[n-epimorphism]],[[n-monomorphism]] factorization through the  _[[n-image]]_ of a morphism.

=--

### For simplicial presheaves
 {#DefinitionForSimplicialPresheaves}

## Constructions

We discuss explicit constructions/presentations of Postnikov systems.

* _[For simplicial sets](#ConStructionForSimplicialSets)_

* _[For strict ω-groupoids](#ForStrctOmegaGroupoids)_.

### For simplicial sets
 {#ConStructionForSimplicialSets}

There are three main functorial models for the Postnikov tower of a [[simplicial set]]:

* _[Coskeleton tower](#CoskeletonTower)_;

* _[Identification relative to skeleta](#IdentificationRelativeSkeleta)_;

* _[Homotopy classes relative to skeleta](#HomotopyClassesRelativeSkeleta)_

#### Coskeleton tower
 {#CoskeletonTower}

+-- {: .num_prop}
###### Proposition

If $X$ is regarded as an [[∞-groupoid]] modeled as a [[Kan complex]], then the [[coskeleton]] sequence

$$
  X
  \;\simeq\;
  \lim_n
  \big(
     cosk_n X \to \cdots \to cosk_{n+1} X \to cosk_{n} X \to \cdots \to *
  \big)
$$

exhibits a Postnikov tower for $X$.

=--

This is observed for instance in ([ArtinMazur](#ArtinMazur)) or ([DwyerKan](#DwyerKan)). Also see [[coskeleton]] for more details.

#### Identification relative to skeleta
 {#IdentificationRelativeSkeleta}

The following construction quotients out the relations encoded by the cells that are thrown in in the above construction, such as to make the maps in the Postnikov tower into [[Kan fibrations]].

We first discuss the absolute tower  and then the relative version.

##### Absolute Postnikov tower

+-- {: .num_defn}
###### Definition

Let $X$ be a [[Kan complex]]. Define for each $n \in \mathbb{N}$  an [[equivalence relation]]  $\sim_n$ on the simplices of $X$ as follows: two $q$-simplices

$$
  \alpha, \beta : \Delta^q \to X
$$

are equivalent if their restriction to the $n$-[[simplicial skeleton|skeleton]] coincides

$$
  sk_n(\alpha) = sk_n(\beta)
  :
  sk_n(\Delta^q)
  \hookrightarrow
  \Delta^q
  \to X
  \,.
$$

Write

$$
  X(n) := X/_{\sim_n}
$$

for the [[quotient]] [[simplicial set]].

=--

There are evident morphisms

$$
  X(n) \to X(n-1).
$$

+-- {: .num_prop #SimplPostnikovByCoskeletonQuotient}
###### Proposition

This is a Postnikov tower, def. \ref{SimplicialPostnikovTower}, and all morphisms are [[Kan fibrations]].

Moreover the canonical morphism

$$
  X \to \lim_{\leftarrow_n} X(n)
$$

is an [[isomorphism]], exhibiting $X$ as the [[limit]] ("[[inverse limit]]") over this tower diagram.

=--

This appears for instance as ([Goerss-Jardine, theorem Vi 2.5](#GoerssJardine)).

##### Relative Postnikov tower
 {#RelativePostnikovTowerConstruction}

We discuss a model for the relative Postnikov tower, def. \ref{SimplicialPostnikovTowerRelative}.

+-- {: .num_defn}
###### Definition

For $f : X \to Y$ a [[Kan fibration]] between [[Kan complexes]],
define for each $n$ and each $k$ an [[equivalence relation]] $\sim_n$ on $k$-[[simplices]]

$$
  \alpha, \beta \colon \Delta^k \to X
$$

such that $\alpha \sim_n \beta$ if

1. the $n$-[[skeleta]] $sk_n \Delta^k \to \Delta^k \stackrel{\alpha, \beta}{\to} X$ are equal;

1. the [[images]] $f(\alpha), f(\beta)\in Y$ are equal.

Let

$$
  im_{n+1}(f) \coloneqq X/\sim_n
$$

be the simplicial set of [[equivalence classes]] under this equivalence relation.

This canonically comes with morphisms $im_{n_1}(f) \to im_{n_2}(f)$ for $\infty \geq n_1 \gt n_2 \geq 0$.

=--

For instance ([Goerss-Jardine, def. VI 2.9](#GoerssJardine)).

+-- {: .num_prop}
###### Proposition

This construction gives indeed a relative Postnikov tower for $f$.

=--

For instance ([Goerss-Jardine, theorem VI 2.11](#GoerssJardine)).

#### Homotopy classes relative to skeleta
 {#HomotopyClassesRelativeSkeleta}

For $X$ a [[Kan complex]] and $n \in \mathbb{N}$,
let $\tau_{\lt n}X$ be the simplicial set defined as
the [[quotient]]

$$
  \tau_{\lt m}X : k \mapsto X_k / homotopy-rel-(n-1)-skeleton
  \,,
$$

where two $k$-cells are identified if there is a simplicial [[homotopy]] between them that fixes their $(n-1)$-skeleton.

This is due to [[John Duskin]]. See for instance ([Beke, pages 302-305](#Beke)).

### For strict $\omega$-groupoids
 {#ForStrictOmegaGroupoids}

+-- {: .num_prop}
###### Proposition

Consider an object in [[sSet]]/[[∞Grpd]] that is in the image of $Str\omega Grpd$, hence given by a morphism $f \colon X \to Y$ of [[strict ∞-groupoids]].

Then for $n \in \mathbb{N}$ the $(n-1)$-Postnikov stage of $f$ is given by the [[strict ∞-groupoid]] $im_n(f)$ with

$$
  \left(im_n\left(f\right)\right)_k
  =
  \left\{
    \array{
      X_k & \forall  k \lt n-1
      \\
      im(X_{n-1}) \subset Y_{n-1} & \forall \; k = n-1
      \\
      \ast & \forall  k \geq n
    }
  \right.
$$

equipped with the evident [[composition]] operations induced from those of $X$ and $Y$, and equipped with the canonical morphisms of strict $\omega$-groupoids

$$
  X \to im_n(f) \to \ast
$$

(the left one being the identity in degree $k \lt n-1$, the quotent projection in degree $n-1$ and $f$ in degree $k \geq n$, and the right one being $f$ in degree $k \lt n-1$, the image inclusion in degree $n-1$ and the identity in degree $k \geq n$).
=--

This  is discussed in ([BFGM](#BFGM)).

+-- {: .proof}
###### Proof

The [[homotopy groups]] of a strict $\omega$-groupoid in any degree $k$ are simply given by the groups of $k$-[[automorphisms]] of the identity $(k-1)$-morphism on a given baspoint modulo $(k+1)$-morphisms (hence the homology of the corresponding [[crossed complex]] in that degree). Therefore it is clear from the construction of $im_n(f)$ above that $X \to im_n(f)$ is surjective on $\pi_0$ and an isomorphism on $\pi_{k \lt n-1}$, and that $im_n(f)$ is a monomorphism on $\pi_{n-1}$ and an isomorphism on $\pi_{k \geq n}$.

=--

### For chain complexes
 {#ExamplesForChainComplexes}

The following gives a model for the $(n-1)$-stage of a relative Postnikov tower for the special case that the the morphism of Kan complexes is the image under the [[Dold-Kan correspondence]] of a [[chain map]] between [[chain complexes]].

Before we give the explicit formule below as prop. \ref{ImageFactorizationForChainComplexes}, the following remark \ref{ChainComplexnPlusOneImageInDegreen} motivates the formula by regarding chain complexes as models for [[strict ∞-groupoids]] (by [this prop.](Dold-Kan correspondence#GlobularAndCubical))

+-- {: .num_remark #ChainComplexnPlusOneImageInDegreen}
###### Remark

Let $f_\bullet \colon V_\bullet \longrightarrow W_\bullet$
be a [[chain map]] between [[chain complexes]]

For $n \in \mathbb{N}$, consider the abelian group

$$
  (im_{n+1}(f))_n
    \;\coloneqq\;
  coker(\,
     ker(\partial_V) \cap ker(f_n)
     \to
     V_n
  \,)
$$

For the following it is helpful to think of this abelian group in the following equivalent ways.

Define an [[equivalence relation]] on $V_n$ by

$$
  \left(
     v_n \sim v'_n
  \right)
  \;\Leftrightarrow\;
  \left(
    (\partial_V v_n = \partial_V v'_n)
    \;\text{and}\;
    (f_n(v_n) = f_n(v'_n))
  \right)
  \,.
$$

Then

$$
  (im_{n+1}(f))_n \simeq V_n/_\sim
$$

is equivalently the set of [[equivalence classes]] of this equivalence relation, which inherits abelian group structure since the eqivalence relation is linear.

This is because the equivalence relation says equivalently that

$$
  \left(
     v_n \sim v'_n
  \right)
  \;\Leftrightarrow\;
  \left(
    v_n - v'_n \;\in\; ker(\partial_V) \cap ker(f_n)
  \right)
$$

and hence is generated under linearity by

$$
  \left(
     v_n \sim 0
  \right)
  \;\Leftrightarrow\;
  \left(
    v_n  \in ker(\partial_V) \cap ker(f_n)
  \right)
  \,.
$$

Moreover, notice that the [[Dold-Kan correspondence]]

$$
  DK
  \;\colon\;
  Ch_{\bullet \geq 0}
    \longrightarrow
  KanCplx
$$

factors through [[globe|globular]] [[strict omega-groupoids]] ([here](Dold-Kan+correspondence#GlobularAndCubical)). An [[n-morphism]] in the [[strict omega-groupoid]] $DK(V_\bullet)$ is of the form

$$
  (v_{n-1})
    \overset{\phantom{AA}v_n\phantom{AA}}{\longrightarrow}
  (v_{n-1} + \partial v_n)
  \,.
$$

In terms of these morphisms the [[equivalence relation]] above says that two of them are equivalent precisely if

1. they are "[[parallel morphisms]]" in that they have the same [[source]] and [[target]];

1. they have the same image under $f$ in the [[n-morphisms]] of  $DK(W_\bullet)$.

This suggests yet another equivalent way to think of $(im_{n+1}(f))_n$: it is the [[disjoint union]] over the [[target]] $(n-1)$-cells in $V_{n-1}$ of the images under $f$ of the sets of $n$-cells from zero to that target:

$$
  (im_{n+1}(f))_n
   \simeq
   \underset{v_{n-1} \in V_{n-1}}{\sqcup}
    \left\{
      f_n(v_n) \vert v_n \in V_n \,\text{and}\,\partial v_n = v_{n-1}
    \right\}
  \,.
$$

=--

+-- {: .num_prop #ImageFactorizationForChainComplexes}
###### Proposition

Let $f_\bullet \colon V_\bullet \longrightarrow W_\bullet$
be a [[chain map]] between [[chain complexes]] and let
$n \in \mathbb{N}$. Recall the abelian group
 $\underset{v_{n-1}}{\sqcup}\{f_n(v_n) \vert \partial v_n = v_{n-1}\}$ from remark \ref{ChainComplexnPlusOneImageInDegreen}.

The following [[diagram]] of [[abelian groups]]
[[commuting diagram|commutes]]:

$$
  \array{
    \vdots && \vdots && \vdots
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_{n+3}
      &\overset{f_{n+3}}{\longrightarrow}&
    W_{n+3}
      &\overset{=}{\longrightarrow}&
    W_{n+3}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_{n+2}
      &\overset{f_{n+2}}{\longrightarrow}&
    W_{n+2}
      &\overset{=}{\longrightarrow}&
    W_{n+2}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\mathrlap{ \partial_W  } }
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_{n+1}
      &\overset{f_{n+1}}{\longrightarrow}&
    \left\{
      w_{n+1} | \exists v_n :  \partial_W w_{n+1} = f_n(v_n), \partial_V v_n = 0,
    \right\}
      &\overset{}{\longrightarrow}&
    W_{n+1}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\partial_W}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_n
      &\overset{ (f_n, \partial_V) }{\longrightarrow}&
    \underset{v_{n-1}}{\sqcup}
    \left\{
      f_n(v_n) \vert \partial_V v_n = v_{n-1}
    \right\}
     &\overset{  }{\longrightarrow}&
    W_n
    \\
    \downarrow^{\mathrlap{\partial_V}}
    &&
    \downarrow^{\mathrlap{(f_n(v_n),\partial_V v_n) \mapsto \partial_V v_n}}
    &&
    \downarrow^{\mathrlap{\partial_W}}
    \\
    V_{n-1}
      &\overset{=}{\longrightarrow}&
    V_{n-1}
      &\overset{f_{n-1}}{\longrightarrow}&
    W_{n-1}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_W}}
    \\
    V_{n-2}
      &\overset{=}{\longrightarrow}&
    V_{n-2}
      &\overset{f_{n-2}}{\longrightarrow}&
    W_{n-2}
    \\
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_W}}
    \\
    \vdots && \vdots && \vdots
  }
$$

Moreover, the middle vertical sequence is a chain complex $im_{n+1}(f)_\bullet$, and hence the diagram gives a factorization of $f_\bullet$ into two chain maps

$$
  f_\bullet
   \;\colon\;
  V_\bullet \longrightarrow im_{n+1}(f)_\bullet \longrightarrow W_\bullet
  \,.
$$

Finally, this is a model for the [[n-image|(n+1)-image factorization]] of $f$ in that on [[homology groups]] the following holds:

1. $H_{\bullet \lt n}(V) \overset{\simeq}{\to} H_{\bullet \lt n}(im_{n+1}(f))$ are [[isomorphisms]];

1. $H_n(V) \to H_n(im_{n+1}(f)) \hookrightarrow H_n(W)$ is the [[image|image factorization]] of $H_n(f)$;

1. $H_{\bullet \gt n}(im_{n+1}(f)) \overset{\simeq}{\to} H_{\bullet \gt n}(W)$   are [[isomorphisms]].

=--

+-- {: .proof}
###### Proof

This follows by elementary and straightforward direct inspection.

=--

### In simplicial (pre-)sheaves

The following gives a sufficient condition for modeling [[n-image]] factorizations in some [[(∞,1)-toposes]] with particularly convenient presentation.

+-- {: .num_prop #nImageFactroizationModeledOnSimplicialPrsheaves}
###### Proposition

Let $C$ be a site with [[point of a topos|enough points]], so that the weak equivalences in $sPSh(C)_{\mathrm{loc}}$ are detected on [[stalks]] ([this prop.](model+structure+on+simplicial+presheaves#OverSiteWithEnoughPointsWeakEquivalencesDetectedOnStalks)). Then given a morphism of [[Kan complex]]-valued [[simplicial presheaves]]

$$
  f \colon X \longrightarrow Y$ in $sPSh(C)
$$

such that both  $X$ and $Y$ are [[homotopy n-types|homotopy k-types]] for some finite $k \in \mathbb{N}$, then its [[n-image]] factorization in the [[(∞,1)-topos]] $L_{lwhe} sPSh(C)_{loc}$ for any $n \in \mathbb{N}$ is presented by any factorization $X \longrightarrow im_{n}(f) \longrightarrow Y$ in $sPSh(C)$ through some Kan-complex valued simplicial presheaf $im_n(f)$ such that for each object $U \in C$ the [[simplicial homotopy groups]] satisfy the following conditions:

1. $\pi_{\bullet \lt n}\left(X(U) \to (im_{n}(f))(U)\right)$ are [[isomorphisms]];

1. $\pi_n\left(X(U) \to (im_{n}(f))(U)\to Y(U)\right)$ is the [[(epi,mono) factorization]] of $\pi_n(f(U))$;

1. $\pi_{\bullet \gt n}\left((im_{n}(f))(U) \to Y(U)\right)$ are [[isomorphisms]].

=--

+-- {: .proof}
###### Proof

Evalutation on [[stalks]] is a [[filtered colimit]] which preserves the [[finite limits]] and [[finite colimits]] that go into the definition of [[simplicial homotopy groups]]. Therefore the global conditions assumed on the simplicial homotopy groups imply that the same kind of conditions holds for the stalkwise homotopy groups. These are the [[categorical homotopy groups in an infinity-topos|categorical homotopy groups]] in $L_{lwhe} sPSh(C)_{loc}$. By [this prop.](n-truncated+object+of+an+infinity-category#RecognizngnTuncationOnSimplicialHomotopyGroups) and [this def.](n-connected+object+of+an+infinity-topos#Connectedness) we may recognize $n$-truncation of morphisms on categorical homotopy groups (using the assumption that $X$ and $Y$ are $k$-truncated for some $k$). Therefore the claim now follows from the stalkwise [[long exact sequence of homotopy groups]].

=--

## Properties

### General

It is known that Postnikov systems classify all weak, pointed [[connected space|connected]] [[homotopy type]]s. In particular, if $X$ satisfies $\pi_r(X)=0 $ for $1 \lt r \lt n$ then the first non trivial Postnikov invariant is an  element $k^{n+1}$ of [[group cohomology]] (with twisted coefficients of course). Such elements are also determined by $n$-fold crossed extensions of $\pi_n$ by $\pi_1$,
which are exact [[crossed complex]]es of the form
$$ 0 \to \pi_n \to C_n \to C_{n-1} \to \cdots \to C_2 \to C_1 $$
together with an isomorphism $Coker(C_2 \to C_1) \cong \pi_1$. This
gives an algebraic model of such an $n$-[[homotopy n-type|type]]. Advantages of
algebraic models are that algebraic constructions can be made on
them, such as forming [[limits]] or colimits. The various [[higher
homotopy van Kampen theorem]]s are useful in the latter case. For
example, it may be difficult or well nigh impossible to write down a
determination of the Postnikov invariant of a pushout of crossed
modules, even if the pushout consists of finite groups.

A Postnikov system is easiest to understand in the 2-stage case,
i.e. two non vanishing homotopy groups, and focuses attention on the
cohomology of [[Eilenberg-Mac Lane space]]s, which also determine all
[[cohomology operation]]s. Basic work on this area was done by Eilenberg
and Mac Lane, and by H. Cartan, while the theory of cohomology
operations, including [[Steenrod operation]]s, is itself a large area.

The reference below shows the problems in the 3-stage systems.

For homotopy 3-types, the algebraic model of [[crossed square]]s is more explicit than the corresponding Postnikov system, and more calculable. However, not much work has been done on, say, [[cohomology operation]]s using the algebraic model of $n$-fold groupoids, and it is not clear if that would help.

### For simplicial sets

+-- {: .num_prop }
###### Proposition

Let $X$ be a [[Kan complex]] and $\{X(n)\}$ the model for its Postnikov tower from prop. \ref{SimplPostnikovByCoskeletonQuotient}. For any vertex $v \in X_0$ write $K(n)$ for the [[pullback]]

$$
  \array{
    K(n) &\to& *
    \\
    \downarrow && \downarrow^{\mathrlap{b}}
    \\
    X(n) &\to& X(n-1)
  }
  \,.
$$

Let $K(\pi_n (X,v), n)$ be the [[Eilenberg-MacLane object]] on the $n$-th [[homotopy group]] of $X$. Then there is a [[weak homotopy equivalence]]

$$
  K(n) \stackrel{\simeq}{\to} K(\pi_n(X,v),n)
  \,.
$$

=--

This appears for instance as [GoerssJardine, corollary VI 3.7](#GoerssJardine).

+-- {: .proof}
###### Proof

Since $K(n) \to K(n-1)$ is a [[Kan fibration]] by prop.
\ref{SimplPostnikovByCoskeletonQuotient} the pullback $K(n)$ is the [[homotopy fiber]] of $X(n) \to X(n-1)$.

=--

## Generalizations

There are analogues in other setups, e.g.

* [[Postnikov system in triangulated category|Postnikov systems in triangulated categories]]

* [[motivic homotopy theory]] (M. Levine, [The Postnikov tower in motivic stable homotopy theory](http://www.math.uiuc.edu/K-theory/0692)).

### Postnikov tower in an $(\infty,1)$-category

We may think of [[Top]] as being the archetypical [[(∞,1)-category]].

In every [[(∞,1)-category]] there is a notion of [[n-truncated object in an (∞,1)-category|n-truncated object]] and accordingly a notion of

* [[Postnikov tower in an (∞,1)-category]].

The traditional case of Postnikov towers in [[Top]] is a special case of this more general concept.

## Related concepts

* [[tower of homotopy fibers]], [[Adams tower]]

* [[n-image]], [[n-epimorphism]], [[n-monomorphism]]

* [[Postnikov invariant]]/[[k-invariant]]

* [[k-ary factorization system]]

* [[chromatic tower]], [[Taylor tower]]

## References

Textbook accounts:

* [[George Whitehead]], Chapter IV: _Elements of homotopy theory_, Springer 1978 ([doi:10.1007/978-1-4612-6318-0](https://link.springer.com/book/10.1007/978-1-4612-6318-0))

* [[Peter May]], Section 8 of:  _Simplicial methods in algebraic topology_  ([djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu), 1967)

* {#GoerssJardine} [[Paul Goerss]], [[J. F. Jardine]], Section VI of: _[[Simplicial homotopy theory]]_, 1999

The original articles:

* [[M. M. Postnikov]], _Determination of the homology groups of a space by means of the homotopy invariants_, Doklady Akad. Nauk SSSR (N.S.) 76: 359–362 (1951)

* [[M. M. Postnikov]], _Issledovaniya po gomotopičeskoĭ teorii nepreryvnyh otobraženiĭ. I. Algebraičeskaya teoriya sistem. II. Naturalʹnaya sistema i gomotopičeskiĭ tip_. (Russian) $[$_Investigations in homotopy theory of continuous mappings. I. The algebraic theory of systems. II. The natural system and homotopy type._$]$ Trudy Mat. Inst. Steklov. no. 46. Izdat. Akad. Nauk SSSR, Moscow, 1955. ([mathnet:tm1182](http://mi.mathnet.ru/eng/tm1182))

Early development:

* Donald W. Kahn, _The spectral sequence of a Postnikov system_, Comm. Math. Helv. 40, n.1, 169--198, 1965 [doi](http://dx.doi.org/10.1007/BF02564370)

* P. I. Booth, _An explicit classification of three-stage Postnikov towers_, Homology, homotopy and applications 8 (2006), No. 2, 133--155
* [[G. J. Ellis]] and R. Mikhailov, *A colimit of classifying spaces* [[arXiv:0804.3581](http://www.arxiv.org/abs/0804.3581)]

Probably the earliest treatment of Postnikov systems for [[simplicial sets]] is in

* [[John Moore|J. C. Moore]], _Semisimplicial complexes and Postnikov systems_, Symposium Internacional de Topologia Algebraica, Mexico City, 1958, pp. 232-247,

and as a result, in that context, they are sometimes referred to as **Moore-Postnikov systems**

The coskeleton construction for the Postnikov tower of a Kan complex is already in

* {#ArtinMazur}[[M. Artin]], [[B. Mazur]], _Étale homotopy_, (1969)(Lecture Notes in Maths. 100).

Another  classical article that amplifies the expression of Postnikov towers in terms of [[coskeleta]] is

* {#DwyerKan} [[William Dwyer]], [[Dan Kan]], _An obstruction theory for diagrams of simplicial sets_ ([pdf](http://www.nd.edu/~wgd/Dvi/ObstructionTheoryForDiagrams.pdf)), from 1984.

Analogous remarks are also in

*  [[John Duskin]] _Simplicial matrices and the nerves of weak $n$-categories I: Nerves of bicategories_ , TAC **9** no. 2, (2002). ([web](http://www.emis.de/journals/TAC/volumes/9/n10/9-10abs.html))

reviewed around page 302 in

* {#Beke} [[Tibor Beke]], _Higher Čech theory_, K-Theory, 32(4):293–322 (2004) ([web](http://www.math.uiuc.edu/K-theory/0646/))

Discussion for [[spectra]] includes

* {#Schwede12} [[Stefan Schwede]], chapter II, section 8 of _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

Discussion in [[homotopy type theory]] is in

* [[Univalent Foundations Project]], section 7.3 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

Discussion for [[homotopy types]] modeled by [[crossed complexes]]/[[strict ∞-groupoids]] is in

* {#BFGM} M. Bullejos, E. Faro, and M. A. García-Munoz, _Postnikov Invariants of Crossed Complexes_, Journal of Algebra Volume 285, Issue 1, 1 March 2005, Pages 238–291 ([arXiv:math/0409339](http://arxiv.org/abs/math/0409339)).

and for $n$-[[hypergroupoids]] in the thesis,

* M. A. García-Munoz,  2003, _Un aceramiento algebraico a la theoría de
Torres de Postnikov_, [thesis](https://digibug.ugr.es/handle/10481/1419),  Universidad de Granada.

This also contains a good discussion of the link with twisted cohomology and homotopy colimits.

A pedagogical introduction to Postnikov systems with an eye towards their generalization from [[homotopy types]] to [[n-categories]] is in

* [[John Baez]], [[Mike Shulman]], _[[Lectures on n-Categories and Cohomology]]_

[[!redirects Postnikov tower]]
[[!redirects Postnikov towers]]

[[!redirects Postnikov decomposition]]
[[!redirects Postnikov systems]]

[[!redirects Postnikov approximation]]
[[!redirects Postnikov approximations]]

[[!redirects Moore-Postnikov system]]
[[!redirects Moore-Postnikov systems]]


Postnikov system in a triangulated category

# Idea #

A [[triangulated category]], or at least an [[enhanced triangulated category]], is a model for a [[stable (infinity,1)-category]]. The archetypical example of these is the [[stable (infinity,1)-category of spectra]] of [[topological space]]s. For topological spaces there is the classical notion of [[Postnikov system]].

A Postnikov system in a general triangulated category is an abstraction of that topological construction to the general case.

# Definition #

Given a triangulated category $D$ a finite complex over $D$ is any sequence of objects and morphisms in $D$

$$
X^\bullet = \{ X^0 \stackrel{d^0}\to X^1 \stackrel{d^1}\to X^2 \stackrel{d^2}\to\ldots \stackrel{d^n}\to X^n\}
$$

in which $d_k\circ d_{k-1} = 0$.

A __right Postnikov system subordinated to__ $X^\bullet$ is given by the following data

* a sequence of objects and morphisms

$$Y^0\stackrel{f_1}\leftarrow Y^1\stackrel{f_2}\leftarrow Y^2 \stackrel{f_3}\leftarrow\ldots\stackrel{f_{n-1}}\leftarrow Y^{n-1}\stackrel{f_n}\leftarrow Y^n= X^n[1]$$

* morphisms $j_{k-1}: Y^k\to X^k[n-k+1]$, $k = 0,\ldots,n-1$ with $j_{n-1} = id : Y^n = X^n[1]\to X^n[1]$

* morphisms $i_k : X^k[n-k]\to Y^{k+1}$, $k = 0,\ldots,n-1$

such that

$$ j_k\circ i_k = d^k[n-k]: X^k[n-k]\to X^{k+1}[n-k], \,\,\,\,\,\,\,for\,\,\,all\,\,\,\,\,\,\,\,  0\lt k \lt n-1 $$

(in particular, $i_{n-1} = d^{n-1}$) and for all $0\leq k\leq n$ the triangles

$$
 X^k[n-k]\stackrel{i_k}\to Y^{k+1}\stackrel{f_k}\to Y^k
\stackrel{j_{k-1}}\to X^k[n-k+1]
$$

are distinguished.

A __right convolution__ of complex $X^\bullet$ is any object in $D$ of the form $Y^0[n-1]$ where $Y^0$ is the
target end of the sequence as above in some right Postnikov system subordinated to $X^\bullet$.

There is also a left-hand version of Postnikov systems and convolutons; however the classes of left and right convolutions of complex $X^\bullet$ coincide; denote that class $Tot(X^\bullet)$. This class in general may be empty, but it may also contain many nonisomorphic objects. D. Orlov (see below) has proved some criteria for existence and uniquenss of (classes of isomorphic) objects in $Tot(X^\bullet)$.

If $D = K^b(A)$ is the triangulated category of bounded complexes in an additive category $A$ then many examples can be obtained via [[twisted complex]]es in that setup, as introduced by Kapranov. Postnikov systems are also used to define generalizations of stability conditions in triangulated setup.

The standard reference (beware typoi!) is chapter 4 (exercise section after 4.2) of

* S. I. Gel'fand, Yu. I. Manin, _Methods of homological algebra_, Russian Наука 1988, English Springer (2 editions).

Other references

* D. Orlov, _Equivalences of derived categories and K3 surfaces_, ([alg-geom/9606006](http://arxiv.org/abs/alg-geom/9606006))

* A. Beilinson, J. Bernstein, P. Deligne, _Faisceaux pervers_, Astérisque 100 (1982). MR 86g:32015

* G. Hein, D. Ploog [Postnikov stability for complexes](http://www.uni-due.de/~mat903/preprints/hein-ploog-postnikov-stability.pdf), [Postnikov-stability vs semistability of sheaves](http://www.uni-due.de/~mat903/preprints/hein_ploog_P-stability.pdf)

* M. Kapranov, _On the derived categories of coherent sheaves on some homogeneous spaces_, Inventiones Mathematicae, vol. 92, n. 3, 479--508, 1988 [doi:10.1007/BF01393744](http://dx.doi.org/10.1007/BF01393744)

* A. L. Gorodentsev, S. A. Kuleshov, A. N. Rudakov, _t-stabilities and t-structures on triangulated categories_,
Izv. Math. 68, 749--781, 2004 [doi](http://dx.doi.org/10.1070/IM2004v068n04ABEH000497)

* A. L. Gorodentsev, S. A. Kuleshov, _On finest and modular t-stabilities_, MPIM Bonn [preprint](http://www.mpim-bonn.mpg.de/preprints/send?bid=2649)

[[!redirects Postnikov system in triangulated category]]


Postnikov tower > history

< [[Postnikov tower]]
[[!redirects Postnikov tower -- history]]


Postnikov tower in an (infinity,1)-category

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#### $(\infty,1)$-Category theory
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#### $(\infty,1)$-Topos theory
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[[!include (infinity,1)-topos - contents]]
=--
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of **Postnikov tower in an $(\infty,1)$-category** is the  generalization of the notion of [[Postnikov tower]] from the archetypical [[(∞,1)-category]] [[Top]]$\simeq$ [[∞Grpd]] to more general $(\infty,1)$-categories.

## Definition

+-- {: .num_prop}
###### Proposition

For $C$ a [[presentable (∞,1)-category]] the subcategory $C_{\leq n}$ of [[n-truncated object in an (∞,1)-category|n-truncated objects]] is a [[reflective (∞,1)-subcategory]]

$$
  C_{\leq n} \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow}
  C
  \,.
$$

=--

This is ([Lurie, prop. 5.5.6.18](#Lurie)).

We write

$$
  \mathbf{\tau}_{\leq n} : C \stackrel{\tau_{\leq n}}{\to} C_{\leq n} \hookrightarrow C
$$

for the corresponding localization. For $X \in C$, we say that $\mathbf{\tau}_{\leq n} X$ is the **$n$-truncation of $X$**.

The reflector of the reflective embedding provides morphisms

$$
  X \to \mathbf{\tau}_{\leq n} X
$$

from each object to its $n$-truncation.

+-- {: .num_defn}
###### Definition

A **Postnikov tower** for $X \in C$ is a diagram

$$
  X \to \cdots \to X_2 \to X_1 \to X_0
$$

such that each $X \to X_n$ exhibits $X_n$ as [[generalized the|the]] $n$-truncation of $X$.

=--

This is [[Higher Topos Theory|HTT, def. 5.5.6.23]].

+-- {: .num_defn}
###### Definition

A **Postnikov pretower** is a pre-tower

$$
  \cdots \to X_2 \to X_1 \to X_0
$$

(no initial $X$ on the left!) which exhibits each $X_n$ as the $n$-truncation of $X_{n+1}$.

We say **Postnikov towers converge** in the ambient [[(∞,1)-category]] if the [[forgetful functor|forgetful]] [[(∞,1)-functor]] from Postnikov towers to Postnikov pretowers is an [[equivalence of (∞,1)-categories]].

=--

This is ([Lurie, def. 5.5.6.23](#Lurie)).

## Examples

### In $\infty Grpd$

When the archetypical [[(∞,1)-topos]] [[∞Grpd]] is [[locally presentable (∞,1)-category|presented]] by the [[model structure on simplicial sets]], truncation is given by the the [[coskeleton]] endofunctor
$\mathbf{cosk}_{n+1}$ on [[sSet]].

The unit of the adjunction $(tr_n \dashv cosk_n)$

$$
  \mathbf{\tau}_n : \mathbf{}X \to \mathbf{cosk}_n(X)
$$

sends an $\infty$-groupoid modeled as a [[Kan complex]] [[simplicial set]] to its $n$-[[truncated|truncation]].

Discussion of this can be found for instance in

* [[William Dwyer]], [[Dan Kan]], _An obstruction theory for diagrams of simplicial sets_ ([pdf](http://www.nd.edu/~wgd/Dvi/ObstructionTheoryForDiagrams.pdf))

*  [[John Duskin]] _Simplicial matrices and the nerves of weak $n$-categories I: Nerves of bicategories_ , TAC **9** no. 2, (2002). ([web](http://www.emis.de/journals/TAC/volumes/9/n10/9-10abs.html))

### In $\infty$-Lie groupoids

...

### In $E_\infty$-rings

The Postnikov tower of a connective [[E-∞ ring]] is a sequence of [[square-zero extensions]]. See [Basterra 99](#Basterra99) and [Lurie "Higher Algebra", section 8.4](#LurieAlgebra) (the result is due to [Kriz](#Kriz93)).

### In simplicial commutative rings

(A special case of the above:) The Postnikov tower of a [[simplicial commutative ring]] is a sequence of [[square-zero extensions]]. See [Toen-Vezzosi](#TVHAGII).

## Properties

### Criteria for convergence

We discuss conditions that ensure that Postnikov towers converge.

+-- {: .num_prop}
###### Proposition

In an [[(∞,1)-topos]] which is _locally_ of finite [[homotopy dimension]], Postnikov towers converge.

=--

This is ([Lurie, prop. 7.2.1.10](#Lurie)).

## Relation to other concepts

* At least if the ambient $(\infty,1)$-category is a [[locally contractible (∞,1)-topos]] $\mathbf{H}$, so that there is a notion of structured [[schreiber:path ∞-groupoid]]-functor $\mathbf{\Pi} : \mathbf{H} \to \mathbf{H}$, the [[homotopy fiber]]s of the morphisms $X \to \mathbf{\Pi}_n(X)$ into the Postnikov tower of $\mathbf{\Pi}(X)$ form the

  * [[Whitehead tower in an (∞,1)-topos|Whitehead tower in the (∞,1)-topos]] $\mathbf{H}$ of the object $X$.

* In the context of [[nonabelian cohomology]] in [[(∞,1)-topos]]es the fact that we have Postnikov towers has been called the [[Whitehead principle of nonabelian cohomology]].

## References

Section 6.5...

* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_

For $E_\infty$-rings:  section 7.4 of

* {#LurieAlgebra} [[Jacob Lurie]], _[[Higher Algebra]]_

and in more classical language, section 8 of

* {#Basterra99} [[M. Basterra]], _Andre-Quillen cohomology of commutative S-algebras_, J. Pure Appl. Algebra 144 (1999), no. 2, 111–143.

* {#Kriz93} [[Igor Kriz]], _Towers of $E_\infty$-ring spectra with an application to BP_, preprint, 1993.

For [[simplicial commutative rings]],

* {#TVHAGII} [[Bertrand Toen]], [[Gabriele Vezzosi]], _Homotopical Algebraic Geometry II: geometric stacks and applications_, [arXiv:math/0404373](http://arxiv.org/abs/math/0404373).

[[!redirects Postnikov tower in an (∞,1)-category]]
[[!redirects Postnikov system in an (∞,1)-category]]
[[!redirects Postnikov system in an (infinity,1)-category]]


postselection

\tableofcontents

\section{Related concepts}

* [[PostBQP]]

\section{References}

* [[Scott Aaronson]], *Quantum computing, postselection, and probabilistic polynomial-time*, Proceedings of the Royal Society A **461** 2063 (2005) 3473–3482 [[doi:10.1098/rspa.2005.1546](https://doi.org/10.1098%2Frspa.2005.1546), [arXiv:quant-ph/0412187](https://arxiv.org/abs/quant-ph/0412187)]

Wikipedia:

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

[[!redirects postselect]]
[[!redirects postselected]]


potential

* in [[physics]]: a [[energy]] different from [[kinetic energy]];

* in [[de Rham complex|de Rham calculus]]: a [[cochain]] whose differential is a given exact form,

* in [[philosophy]]: the [[modality|modal]] [[category]] of *[[potentia]]*

[[!redirects potentials]]

category: disambiguation


potential energy

#Contents#
* table of contents
{:toc}

## Idea

That part of the [[energy]] of a [[mechanical system]] which arises (via [[Legendre transform]]) from the non-[[kinetic action]], hence the [[interaction]].

## Related concepts

* [[potential]]

[[!redirects potential energies]]


potential theory

**Potential theory** is a branch of mathematical analysis (and mathematical physics) studying the functions which are solutions of Laplace equations (harmonic functions), and some generalizations in the setting of more general elliptic partial differential equations.

* [wikipedia](http://en.wikipedia.org/wiki/Potential_theory)


potentiality and actuality

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#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

### In classical physics
 {#InClassicalPhysics}

> The solution of the difficulty that is raised about the motion --- whether it is in the movable --- is plain. It is the fulfilment of this **potentiality**, and by the action of that which has the power of causing motion; and the **actuality** of that which has the power of causing motion is not other than the actuality of the movable, for it must be the fulfilment of both. A thing is capable of causing motion because it can do this, it is a mover because it actually does it. But it is on the movable that it is capable of acting. Hence there is a single actuality of both alike... [[Aristotle](#Aristotle), as reproduced in [Jaeger (2017), p. 7](#Jaeger17), our emphasis]

### In quantum physics
 {#InQuantumPhysics}

Regarding the [[modality|modal]] nature of [[quantum states]] in [[quantum physics]]:

> In throwing dice we do not know the fine details of the motion of our hands which determine the fall of the dice and therefore we say that the probability for throwing a special number is just one in six. The probability wave of Bohr, Kramers, Slater, however, meant more than that; it meant a tendency for something. It was a quantitative version of the old concept of "**potentia**" in Aristotelian philosophy. It introduced something standing in the middle between the idea of an event and the **actual** event, a strange kind of physical reality just in the middle between possibility and reality. [[Heisenberg (1958)](#Heisenberg1958), our emphasis]

## Related entries

* [[Aristotle]]

## References

[[Aristotle]]'s discussion of potentiality and actuality permeates his text *Physics*:

* {#Aristotle} [[Aristotle]], *Physics*, in: Jonathan Barnes (ed.), *The Complete Works of Aristotle* Vol 1 (1984) [[pdf](https://sites.unimi.it/zucchi/NuoviFile/Barnes%20%20-%20Physics.pdf), [ISBN:9780691016504](https://press.princeton.edu/books/hardcover/9780691016504/complete-works-of-aristotle-volume-1)]

* {#FerroniGili16} Lorenzo Ferroni, Luca Gili, *Non-existent but potentially actual. Aristotle on plenitude (Met. Θ 3-4, 1047b1-6)*,  Revue de philologie, de littérature et d'histoire anciennes, Tome XC (2016) 81-114 [[doi:10.3917/phil.901.0081](https://www.cairn.info/revue-de-philologie-litterature-et-histoire-anciennes-2016-1-page-81.htm)]

See also:

* Stanford Encyclopedia of Philosophy, *[Aristotle's Metaphysics: Actuality and Potentiality](https://plato.stanford.edu/entries/aristotle-metaphysics/#ActuPote)*

* Wikipedia, *[Potentiality and actuality](https://en.wikipedia.org/wiki/Potentiality_and_actuality#Potentiality)*

With respect to [[quantum states]] in [[quantum physics]]:

* {#Heisenberg1958} [[Werner Heisenberg]], *Physics and Philosophy -- The Revolution in Modern Science*, Harper & Brothers (1958) [[ark:13960/t3c050g7k](https://archive.org/details/physics-and-philosophy-the-revolution-in-modern-scirnce-werner-heisenberg-f.-s.-c.-northrop)]

further discussed in:

* {#Jaeger17} [[Gregg Jaeger]], *Quantum potentiality revisited*, Phil. Trans. R. Soc. A **375** 20160390 (2017) [[doi:10.1098/rsta.2016.0390](https://doi.org/10.1098/rsta.2016.0390)]

* [socraticdictum.com/between-possibility-and-reality-heisenbergs-appeal-to-aristotelian-metaphysics-part-two/](https://socraticdictum.com/between-possibility-and-reality-heisenbergs-appeal-to-aristotelian-metaphysics-part-two/)

[[!redirects potentia]]
[[!redirects potentiality]]

[[!redirects actuality]]


Poul Olesen

* [webpage](https://www.nbi.ku.dk/english/staff/?pure=en/persons/157669)

## Selected writings

Introducing [[Nielsen-Olesen vortex strings]] inspired from [[Abrikosov vortices]] in [[type II superconductors]]:

* {#NielsenOlesen73} [[Holger Bech Nielsen]], [[Poul Olesen]], _Vortex-line models for dual strings_, Nuclear Physics B Volume 61, 24 September 1973, Pages 45-61 (doi:10.1016/0550-3213(73)90350-7)

category: people


power dagger 2-poset

[[!redirects power dagger 2-posets]]

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## Contents ##
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{:toc}

## Definition ##

A power dagger 2-poset is a [[dagger 2-poset]] $C$ such that for every object $A \in Ob(C)$ there exists an object $\mathcal{P}(A)$ called the power object of $A$ and a morphism $\in_A \in Hom_C(A, \mathcal{P}(A))$ called subobject membership in $A$, such that for each morphism $R \in Hom_C(A,B)$, there exists a [[map in a dagger 2-poset|map]] $\chi_R \in Map_C(A,P(B))$ called the characteristic map such that $R = (\in_B^\dagger) \circ \chi_R$.

## Examples ##

The dagger 2-poset [[Rel]] of [[sets]] and [[relations]] is a power dagger 2-poset.

## See also ##

* [[dagger 2-poset]]
* [[elementarily topical dagger 2-poset]]


power object

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#### Category theory
+-- {: .hide}
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=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _power object_ generalizes the notion of [[power set]] from the [[category]] [[Set]] to an arbitrary category with [[finite limit]]s.

## Definition

Let $C$ be a [[category]] with [[finitely complete category|finite limits]]. A **power object** of an object $c \in C$ is

* an [[object]] $\Omega^c$

* a [[monomorphism]] $\in_c \hookrightarrow c \times \Omega^c$

such that

* for every other object $d$ and every [[monomorphism]] $r \hookrightarrow c \times d$ there is a unique morphism
$\chi_r : d \to \Omega^c$ such that $r$ is the [[pullback]]

$$
  \array{
    r &\to& \in_c
    \\
    \downarrow && \downarrow
    \\
    c \times d
    &\stackrel{Id_c \times \chi_r}{\to}&
    c \times \Omega^c
  }
$$

If $C$ may lack some finite limits, then we may weaken that condition as follows:

*  If $C$ has all [[pullbacks]] (but may lack products), then equip each of $\in_c$ and $r$ with a jointly monic pair of morphisms, one to $c$ and one to $\Omega^c$ or $d$, in place of the single monomorphism to the product of these targets; $r$ must then be the joint pullback

   $$ \array {
   & c          & \leftarrow    & r & \rightarrow & d &  \\
   Id_c & \downarrow & & \downarrow & & \downarrow & \chi_r \\
   & c          & \leftarrow      & \in_c      & \rightarrow       & \Omega^c & \\
   } $$

*  If $C$ may lack some pullbacks, then we simply require that the pullback that $r$ is to equal must exist.  But arguably we should require, if $\Omega^c$ is to be a power object, that this pullback exists for any given map $\chi: d \to \Omega^c$.

## Examples

* If $1$ is a [[terminal object]], then $\Omega^1$ is precisely a [[subobject classifier]].

* A power object in [[Set]] is precisely a [[power set]].

* A category with finite [[limits]] and power objects for all objects is precisely a [[topos]]. The power object $P A $ of any object $A$ in the topos is the [[exponential object]] $P A = \Omega^A$ into the [[subobject classifier]].

* See [[Trimble on ETCS I]] for the axiom of power sets in the elementary theory of the category of sets.

[[!redirects power objects]]


power operation

> under construction

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=--

#Contents#
* table of contents
{:toc}

## Idea

In [[algebraic topology]], _power operations_ are [[cohomology operations]] in [[multiplicative cohomology theory]] which are higher-degree analogs of [[cup product]]-squares [[symmetric algebra|symmetrized]] in the appropriate [[homotopy theory|homotopy-theoretic sense]].

+-- {: .num_remark #AsPDerivations}
###### Remark

At least to some extent, power operations may be understood as the [[higher algebra]]-generalization of the ordinary $p$-power map $(-)^p$ on a [[commutative ring]], the one that appears in the definition of [[Fermat quotients]], [[p-derivations]] and [[Frobenius morphisms]].

See for instance [Lurie, from remark 2.2.7 on](#Lurie) for relation to the [[Frobenius homomorphism]] and see the example [below](#OnK1LocalKUAlgebras). See [Guillot 06](#Guillot06), [Morava-Santhanam 12](#MoravaSanthanam) for further discussion and speculation in this direction.

=--

For $E$ an [[E-∞ ring]] and $X$ a [[topological space]] ([[∞-groupoid]], [[homotopy type]]), a map $a\;\colon\;X \to E$ is a [[cocycle]] in the [[Whitehead-generalized cohomology]] of $X$ with [[coefficients]] in $E$.

The $n$-th [[cup product]] power of this $a$ is the composite

$$
  a^n
    \;\colon\;
  X^{\times n}
    \overset{\;\;\;
      (a,\cdots,a)
    \;\;\;}{\longrightarrow}
  E^{\times n}
    \overset{\;\; \mu \;\;}{\longrightarrow}
  E
  \,,
$$

where the second map is given the [[multiplication]] operation in the [[ring spectrum]] $E$. Since this is, by assumption, commutative up to coherent higher homotopy, this map factors through the [[homotopy quotient]] by the [[∞-action]] of the [[symmetric group]] $\Sigma_n$:

$$
  a^n
    \;\colon\;
  X \times \ast \sslash \Sigma_n
   \longrightarrow
  X^n \sslash \Sigma_n
    \longrightarrow
  E
  \,.
$$

The cohomology class of this $E$-cocycle on $X \times B \Sigma_n$ is the $n$-th (symmetric) power of $a$.

## Examples

### Steenrod squares and Steenrod power operations
 {#SteenrodPowerOperations}

On [[ordinary cohomology]] over a [[topological space]], the power operations are the [[Steenrod operations]];

Specifically for $n = 2$ and $E = H \mathbb{Z}_2$, the second (symmetric) power of $a \in H(X,\mathbb{Z}_2)$ is an element in $H^\bullet(\mathbb{R}P^\infty \times X, \mathbb{Z}_2) \simeq H^\bullet(X,\mathbb{Z}_2)[x]$ and the [[coefficients]] of this [[polynomial]] in $x$ are the [[Steenrod operations]] on $a$.

For $p \gt 2$ there are the Steenrod power operations (e.g. [Rognes 12, around theorem 3.3](#Rognes12), quick exposition [here](https://www.math.purdue.edu/~pvankoug/talks/steenrod16.pdf#page=3)).

### Kudo-Araki-Dyer-Lashof operations

On an [[infinite loop space]] the power operations are the [[Kudo-Araki-Dyer-Lashof operations]]

### Adams operations

In the context of  [[complex K-theory]] power operations are the [[Adams operations]].

### On $K(1)$-local $KU$-algebras
 {#OnK1LocalKUAlgebras}

> From [this MO comment](http://mathoverflow.net/a/178627/381) by [[Akhil Mathew]]:

Let $R$ be a [[K(n)-local stable homotopy theory|K(1)-local]] [[E-∞ ring]] under ([[p-completion|p-adic]]) [[complex K-theory]] [[KU]]. Then there exists a basic power operation $\theta \colon \pi_0 R \to \pi_0 R$ (see [Hopkins](#Hopkins)) such that :

*  $\psi(x) \stackrel{\mathrm{def}}{=} x^p + p \theta(x)$ defines a [[ring homomorphism]] from $\pi_0 R \to \pi_0 R$.

* $\theta$ satisfies all the identities needed to make $\psi$ a ring-homomorphism after "division by $p$." For instance $\psi(x+y) = \psi(x) + \psi(y)$ implies that

  $$
    \theta(x+y) = \theta(x) + \theta(y) + \frac{x^p - y^p - (x+y)^p}{p}
    \,,
  $$

  where the last term is an integral polynomial in $x,y$ and is interpreted as such.

(see also [Rezk 09, example 1.3](#Rezk09))

This is a "$\theta$-algebra."/[[p-derivation]] as in remark \ref{AsPDerivations} above.

Notice that $\psi$ is, in particular, a lift of the [[Frobenius homomorphism]]. There are generalizations of $\psi, \theta$ at higher [[chromatic levels]], too, and there is a modular interpretation of the resulting algebraic structure in ([Rezk 09](#Rezk09)).

By ([Strickland 98](#Strickland98)) we have that if $G$ is the [[formal group]] associated to a [[Morava E-theory]], then Frobenius lifts (twhich corresponds to degree $p^k$ subgroups of $G$) are classified by maps into $E^0(B \Sigma_{p^r})/I_{t r}$ where $I_{t r}$ is the transfer ideal. So, for example, the map $\psi$ above corresponds to a universal map $KU^0 \to KU^0(B \Sigma_p)/I_{t r} \simeq KU^0$.

## Related concepts

* [[Adams operation]]

* [[logarithmic cohomology operation]]

* [[spectral symmetric algebra]]

## References

The basic idea is nicely described in

* [[Charles Rezk]], [MO comment](http://mathoverflow.net/a/6384/381) Nov 09

(from which some of the above text is adapted).

More technical surveys include

* [[Charles Rezk]], _Lectures on power operations_ ([dvi](http://www.math.uiuc.edu/~rezk/power-operation-lectures.dvi)) (2006)

* [[Charles Rezk]], _Power operations in Morava E-theory --  a survey_ (2009) ([pdf](http://www.math.uiuc.edu/~rezk/midwest-2009-power-ops.pdf))

* {#Rezk14} [[Charles Rezk]], _Isogenies, power operations, and homotopy theory_, article ([pdf](http://www.math.uiuc.edu/~rezk/rezk-icm-talk-posted.pdf)) and talk at ICM 2014 ([pdf](http://www.math.uiuc.edu/~rezk/rezk-icm-2014-slides.pdf))

Lecture notes on the Steenrod squares and power operations include

* {#Rognes12} [[John Rognes]], section 3 of _The Adams spectral sequence_, 2012 ([pdf](http://folk.uio.no/rognes/papers/notes.050612.pdf))

The original articles are

* {#Rezk06} [[Charles Rezk]], _The units of a ring spectrum and a logarithmic cohomology operation_, J. Amer. Math. Soc. 19 (2006), 969-1014 ([arXiv:math/0407022](http://arxiv.org/abs/math/0407022))

* [[Charles Rezk]], _Power operations for Morava E-theory of height 2 at the prime 2_ ([arXiv:0812.1320](http://arxiv.org/abs/0812.1320))

* {#Rezk09} [[Charles Rezk]], _The congruence criterion for power operations in Morava E-theory_, Homology, Homotopy and Applications, 11(2), 327-379 ([arXiv:0902.2499](http://arxiv.org/abs/0902.2499))

More discussion in the generality of [[E-infinity arithmetic geometry]] is in

* {#Lurie} [[Jacob Lurie]], _[[Rational and p-adic Homotopy Theory]]_

Discussion for $K(1)$-local $E_\infty$-rings is in

* {#Hopkins} [[Michael Hopkins]], _$K(1)$-local $E_\infty$-Ring spectra_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/knlocal.pdf))

and discussion of power operations in [[Morava E-theory]] is in

* [[Matthew Ando]], _Isogenies of formal group laws and power operations in the cohomology theories $E_n$_, Duke Math. J. Volume 79, Number 2 (1995), 423-485 ([Euclid](http://projecteuclid.org/euclid.dmj/1077285158))

* {#Strickland98} [[Neil Strickland]], _Morava E-theory of symmetric groups_ ([arXiv:math/9801125](http://arxiv.org/abs/math/9801125))

Comments on the analogy between power operations in homotopy theory and [[Lambda ring]] structure in [[Borger's absolute geometry]] are in

* {#Guillot06} [[Pierre Guillot]], _Adams operations in cohomotopy_ ([arXiv:0612327](http://arxiv.org/abs/math/0612327))

* {#MoravaSanthanam12} [[Jack Morava]], Rakha Santhanam, _Power operations and Absolute geometry_, 2012 ([pdf](http://www.lemiller.net/media/slidesconf/AbsolutePower.pdf))

[[!redirects power operations]]


power series

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

\tableofcontents

## Definition

A power series in a [[variable]] $X$ and with [[coefficients]] in a [[ring]] $R$ is a [[series]] of the form

$$
  \sum_{n = 0}^\infty a_n X^n
$$

where $a_n$ is in $R$ for each $n\ge 0$. Given that there are no additional convergence conditions, a power series is also termed emphatically as a __formal power series__. If $R$ is commutative, then the collection of formal power series in a variable $X$ with coefficients in $R$ forms a commutative ring denoted by $R [ [ X ] ]$.

More generally, a power series in $k$ commuting variables $X_1,\ldots, X_k$ with coefficients in a ring $R$ has the form $\sum_{n_1=0,n_2=0,\ldots, n_k = 0}^\infty a_{n_1\ldots n_k} X_1^{n_1} X_2^{n_2}\cdots X_k^{n_k}$. If $R$ is commutative, then the collection of formal power series in $k$ commuting variables $X_1,\ldots, X_k$ form a formal power series ring denoted by $R [ [ X_1,\ldots, X_k
] ]$.

More generally, we can consider noncommutative (associative unital) ring $R$ and words in noncommutative
variables $X_1,\ldots, X_k$ of the form
$$
w = X_{i_1}\cdots X_{i_m}
$$
(where $m$ has nothing to do with $k$) and with coefficient $a_w \in R$ (here $w$ is a word of any length, not a multiindex in the previous sense). Thus the power sum is of the form $\sum_w a_w X_w$ and they form a formal power series ring in variables $X_1,\ldots, X_k$ denoted by $R\langle \langle X_1,\ldots, X_k \rangle\rangle$. Furthermore, $R$ can be even a noncommutative [[semiring]] in which case the words belong to the free monoid on the set $S = \{ X_1,\ldots, X_k\}$, the partial sums are then belong to a monoid semiring $R\langle S\rangle$. The formal power series then also form a semiring, by the multiplication rule
$$
\sum_{r} a_r X_r \cdot \sum b_s X_s = \sum_w \sum_{u,v; w = u v} a_u b_v X_w
$$
Of course, this implies that in a specialization, $b$-s commute with variables $X_{i_k}$; what is usually generalized to take some endomorphisms into an account (like at noncommutative polynomial level of partial sums where we get skew-polynomial rings, i.e. iterated Ore extensions).

### As adic completions

Let $R$ be a [[commutative ring]], and let $R[X]$ be the [[polynomial ring]] on one indeterminant $X$. Then $(X)$ is a [[maximal ideal]] in $R[X]$, and results in an [[adic topology]] on $R[X]$. The ring of formal power series in $R$ is the [[adic completion]] of the [[limit]] of the quotient of $R[X]$ by powers of $(X)$:

$$R[[X]] \coloneqq \underset{\leftarrow}\lim R/(X)^n$$

The ring of formal power series for multiple indeterminants $X_i$ is constructed iteratively: because $R[[X_1, X_2, \ldots X_n]]$ is a commutative ring, one could construct the [[polynomial ring]] $R[[X_1, X_2, \ldots X_n]][X_{n+1}]$ on the indeterminant $X_{n+1}$. As above, $(X_{n+1})$ is a [[maximal ideal]] in $R[[X_1, X_2, \ldots X_n]][X_{n+1}]$ with a corresponding [[adic topology]], and one can then take the [[adic completion]]
$$R[[X_1, X_2, \ldots X_n]][[X_{n+1}]] \coloneqq \underset{\leftarrow}\lim R[[X_1, X_2, \ldots X_n]]/(X_{n+1})^m$$
The resulting commutative ring is usually just written as $R[[X_1, X_2, \ldots X_n, X_{n+1}]]$.

## Examples

### Polynomials

For a natural number $k$, a power series $\sum_{n=0}^\infty a_n X^n$ such that $a_n = 0$ for all $n \gt k$ is a [[polynomial]] of degree at most $k$.

### Taylor series

* [[Taylor series]]

### MacLaurin series

For $f \in C^\infty(\mathbb{R})$ a [[smooth function]] on the [[real line]], and for $f^{(n)} \in C^\infty(\mathbb{R})$ denoting its $n$th [[derivative]] its [[MacLaurin series]] (its [[Taylor series]] at $0$) is the power series

$$
  \sum_{n = 0}^\infty
   \frac{1}{n!} f^{(n)}(0) x^n
  \,.
$$

If this power series [[convergence|converges]] to $f$, then we say that $f$ is _[[analytic function|analytic]]_.

### Laurent series

* [[Laurent series]]

### Puiseux series

* [[Puiseux series]]

## Properties

* An element $a = a_0 + a_1 x + a_2 x^2 + \ldots$ in $R[ [x] ]$ is (multiplicatively) invertible iff $a_0$ is invertible.

This follows easily from the observation that we can invert $1 + x b$ for any power series $b$ by forming $1 - x b + x^2 b^2 - \ldots$ and collecting only finitely many terms in each degree. As a simple corollary,

* If $R$ is a [[local ring]], then the power series ring $R[ [X] ]$ is also a local ring.

### Functional substitution and inversion

+-- {: .num_prop}
###### Proposition
$R[ [x_1, \ldots, x_n] ]$ equipped with the ideal $(x_1, \ldots, x_n)$ is the free [[formal group law|adic]] $R$-algebra on $n$ generators, in the sense that it is the value of the left adjoint $Pow$ to the forgetful functor

$$Ideal: AdicRAlg \to Set: (A, I) \mapsto I$$

as applied to the set $\{x_1, \ldots, x_n\}$.
=--

+-- {: .proof}
###### Proof
The idea is that for each adic $R$-algebra $(S, I)$ and element $(s_1, \ldots s_n) \in I^n$, there is a unique adic algebra map $R[ [x_1, \ldots, x_n] ] \to S$ that sends $x_i$ to $s_i$; this adic algebra map sends a power series $\sum a_{k_1, \ldots, k_n} x_1^{k_1} x_n^{k_n}$ to the sequence of truncations

$$\left(\sum_{k_1 + \ldots + k_n \lt k} a_{k_1, \ldots, k_n} s_1^{k_1} \ldots s_n^{k_n} \mod I^k\right)_k$$

belonging to $\underset{\longleftarrow}{\lim}_k S/I^k \cong S$.
=--

It follows that we may define a clone or cartesian operad as follows: the $n^{th}$ component is the set $I_n = (x_1, \ldots, x_n) \subset R[ [x_1, \ldots, x_n] ]$ which is the monad value $Ideal Pow(\{x_1, \ldots, x_n\})$. Letting $M$ denote the monad $Ideal \circ Pow$, with monad multiplication $\mu$, and $[n]$ the set $\{x_1, \ldots, x_n\}$, the clone multiplication

$$I_n \times I_k^n \to I_k$$

is the composition of the maps

$$M(n) \times M(k)^n \cong M(n) \times \hom([n], M(k)) \stackrel{1 \times func}{\to} M(n) \times \hom(M(n), M M(k)) \stackrel{eval}{\to} M M(k) \stackrel{\mu(k)}{\to} M(k)$$

The clone multiplication thus defined is called *substitution of power series*; it takes a tuple consisting of $p(x_1, \ldots, x_n) \in I_n, q_1(x_1, \ldots x_k) \in I_k, \ldots q_n(x_1, \ldots, x_k) \in I_k)$ to a power series denoted as

$$p(q_1(x_1, \ldots, x_k), \ldots q_n(x_1, \ldots, x_k)).$$

The resulting clone or operad yields, in the particular case $k = n = 1$, an associative substitution operation

$$x R[ [x] ] \times x R[ [x] ] \stackrel{sub}{\to} x R[ [x] ]$$

with $sub(p, q) = p \circ q$ the power series $p(q(x))$.

+-- {: .num_prop #InvertibleElements}
###### Proposition
The group of invertible elements in the substitution monoid $x R[ [x] ]$ consists of power series of the form $a_1 x + a_2 x^2 + \ldots$ where $a_1$ is multiplicatively invertible in the ring $R$.
=--

In other words, we can functionally invert a power series provided that the linear coefficient $a_1$ is invertible in $R$.

+-- {: .proof}
###### Proof
Given power series $a = a_1 x + a_2 x^2 + \ldots$ and $b = b_1 x + b_2 x^2 + \ldots$, we may read off coefficients of the composite $a \circ b$ as

$$(a \circ b)_k = \sum_{n \geq 1} a_n \sum_{k = k_1 + \ldots + k_n} b_{k_1} b_{k_2} \ldots b_{k_n}$$

where in particular $(a \circ b)_1 = a_1 b_1$. Now $a$ is the left functional inverse of $b$, or $b$ is the right inverse of $a$, if $(a \circ b)(x) = x$, i.e., if $(a \circ b)_k = 1$ if $k = 1$ and $0$ otherwise. The first equation says simply $(a \circ b)_1 = a_1 b_1 = 1$ which implies $a_1$ is invertible. Conversely, if $a_1$ is multiplicatively invertible and $b_1 = a_1^{-1}$, then the equations

$$\array{
\sum_{n \geq 1} a_n \sum_{k = k_1 + \ldots + k_n} b_{k_1} b_{k_2} \ldots b_{k_n} & = 1\; if\; k = 1 \\
 & = 0\; if\; k \neq 1
}$$

may be uniquely solved for the remaining $a_i$\'s given the $b_j$\'s, and uniquely solved for the remaining $b_j$\'s given the $a_i$\'s, by an inductive procedure: for $k \neq 1$ we have

$$a_1 b_k + a_k b_1^k + \; terms\; a_n b_{k_1} \ldots b_{k_n} = 0$$

and this allows us to solve for $b_k$,

$$b_k = -a_1^{-1}(a_k b_1^k + \; terms\; a_n b_{k_1} \ldots b_{k_n})$$

given the values $a_1, \ldots, a_k$ and earlier $b$-values $b_{k_j}$ for $k_j \lt k$ given by inductive hypothesis. Similarly we can solve for $a_k$ in terms of given coefficients $b_1, \ldots, b_k$ and earlier $a$-values $a_n$, $n \lt k$. Thus every power series $a$ has a right inverse if $a_1^{-1}$ exists, and $b$ has a left inverse if $b_1^{-1}$ exists, and this completes the proof.
=--

### Formal differentiation

One way to define the formal differentiation operator, as a function $\frac{\partial}{\partial X}:R[[X]] \to R[[X]]$, is via the usual formula

$$
  \frac{\partial}{\partial X}\left(\sum_{n = 0}^\infty a_n X^n\right) \coloneqq \sum_{n = 0}^\infty a_{n + 1} (n + 1) X^n.
$$

Then $\frac{\partial}{\partial X}$ is an $R$-[[linear function]] on $R[[X]]$ which satisfies the [[Leibniz rule]], meaning that it is a [[derivation]] and $R[[X]]$ is a [[differential algebra]].

Here is a conceptual story underlying the formalism. Let $D = R[\varepsilon]/(\varepsilon^2)$ be the [[representing object]] for [[derivations]] (the "[[ring of dual numbers]]"). Let $\delta: R[ [X] ] \to R[ [X] ] \otimes_R D \cong R[ [X] ][\varepsilon]/(\varepsilon^2)$  be the unique topological $R$-algebra map (under the $(X)$-[[adic topologies]] described above) that sends $X$ to $X + \varepsilon$.  (If it helps, think $\delta(q) = q(X + \varepsilon)$.)

\begin{definition}
For $p \in R[ [X] ]$, the *derivative* $p'$ is the unique element of $R[ [X] ]$ satisfying

$$\delta(p) = p(X) + p'(X)\varepsilon.$$
\end{definition}

We leave as an exercise the proof that the two definitions of derivative match:

$$p'(X) = \frac{\partial}{\partial X} p(X).$$

(Hint: the restriction of $p \mapsto p'$ to $R[X]$ is by construction a derivation such that $X' = 1$, and $(X^k)' = k X^{k-1}$ by [[induction]]. This induces derivations on quotient algebras $R[X]/(X^n)$, satisfying the same formula. Then pass to the inverse limit.)

\begin{proposition}
**(Chain rule)** For $p \in R[ [X] ]$ and $q \in x R[ [X ] ]$,

$$(p \circ q)' = (p' \circ q) \cdot q'.$$
\end{proposition}

See [[chain rule#FormalAlgebra|here]] for a conceptual proof, using the universal property of adic completion.

Relatedly but in a slightly different direction, we can consider differentiation in coalgebraic terms. Suppose the commutative ring $R$ is a [[commutative algebra]] over $\mathbb{Q}$ (thus permitting division by nonzero [[integers]]). Then the set $R[ [X]]$ may be identified with the [[terminal coalgebra of an endofunctor|terminal coalgebra]] $R^\mathbb{N}$ of the [[endofunctor]] $R \times - \colon Set \to Set$ via the map

$$R[ [X]] \to R^\mathbb{N}\; : \; \sum_{n \geq 0} \frac{a_n X^n}{n!} \mapsto (a_0, a_1, a_2, \ldots)$$

whereby the coalgebra structure on $R^\mathbb{N}$,

$$R^\mathbb{N} \to R \times R^\mathbb{N}\; \colon \; (a_n)_{n \geq 0} \mapsto \langle a_0, (a_{n+1})_{n \geq 0} \rangle,$$

corresponds to

$$R[ [X]] \to R \times R[ [X]]\; \colon\; f(X) \mapsto \langle f(0), f'(X) \rangle.$$

One may then apply [[coinduction|coinductive]] techniques to prove various facts. One illustration is given [[coinduction#Examples|here]], where coinduction on power series is used to prove the general binomial theorem

$$(1 + x)^r \coloneqq \exp(r \log(1 + x)) = \sum_{k \geq 0} \frac{r^\underline{k} x^k}{k!}$$

where, remarkably, $r$ is an arbitrary element of $R$.

## Ring of power series as rings with infinitesimals

| [[ring]] with [[infinitesimals]] | [[function]] |
---------------------|-----------------|
| [[dual numbers]] | [[differentiable function]] |
| [[Weil ring]] | [[smooth function]] |
| [[power series ring]] | [[analytic function]] |

### Purely real and purely infinitesimal elements

Suppose that $K$ is a [[Archimedean ordered field]] and $K[[\epsilon]]$ is the ring of power series in $K$. Since $K[[\epsilon]]$ is a [[local ring]], the quotient of $K[[\epsilon]]$ by its ideal of non-invertible elements $\epsilon K[[\epsilon]]$ is the [[residue field]] $K$ itself, and the canonical function used in defining the quotient is the function $\Re:K[[\epsilon]] \to K$ which takes a number $a \in K[[\epsilon]]$ to its purely real component $\Re(a) \in K$ and takes $\Re(\epsilon) = 0$. Since $K[[\epsilon]]$ is an ordered $K$-algebra, there is a [[strictly monotone]] [[ring homomorphism]] $h:K \to K[[\epsilon]]$. An element $a \in K[[\epsilon]]$ is **purely real** if $h(\Re(a)) = a$, and an element $a \in K[[\epsilon]]$ is **purely [[infinitesimal]]** if it is in the [[fiber]] of $\Re$ at $0 \in K$. Zero is the only element in $K[[\epsilon]]$ which is both purely real and purely infinitesimal.

### Analytic functions

Suppose that $K$ is a [[sequentially Cauchy complete]] [[Archimedean ordered field]] with lattice structure, and $K[[\epsilon]]$ is the ring of power series of $K$. Then [[analytic functions]] are each definable on $K$ using the algebraic, order, metric, and convergence structure on $K$.

The ring homomorphism $h:K \to K[[\epsilon]]$ preserves [[analytic functions]]: given a natural number $n \in \mathbb{N}$ and a purely infinitesimal element $\eta \in \epsilon K[[\epsilon]]$, then for every [[analytic function]] $f \in C^\infty(K)$, there is a function $f_{K[[\epsilon]]}:K[[\epsilon]] \to K[[\epsilon]]$ such that for all elements $x \in K$, $f_{K[[\epsilon]]}(h(x)) = h(f(x))$ and
$$f_{K[[\epsilon]]}(h(x) + \eta) = \sum_{i = 0}^{\infty} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}(x)\right) \eta^i$$

## Related concepts

* [[convergence radius]]

* [[asymptotic series]], [[transseries]]

* [[Tate algebra]]

* [[characteristic series]]

* [power series and their formal spectra](formal+scheme#FormalPowerSeries)

## References

* Wikipedia, _[Formal power series](https://en.wikipedia.org/wiki/Formal_power_series)_

A formalization in [[homotopy type theory]] and there in [[Coq]] is discussed in section 4 of

* Álvaro Pelayo, [[Vladimir Voevodsky]], [[Michael Warren]], _A preliminary univalent formalization of the p-adic numbers_ ([arXiv:1302.1207](http://arxiv.org/abs/1302.1207))

The discussion of the differentiation of a converging power series term by term is at

* Tim Gowers's blog: [differentiating-power-series](https://gowers.wordpress.com/2014/02/22/differentiating-power-series)

category: analysis, algebra

[[!redirects power series]]
[[!redirects formal power series]]

[[!redirects power series ring]]
[[!redirects power series rings]]

[[!redirects formal power series ring]]
[[!redirects formal power series rings]]

[[!redirects formal power series algebra]]
[[!redirects formal power series algebras]]

[[!redirects power series algebra]]
[[!redirects power series algebras]]


power set

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Set theory
+-- {: .hide}
[[!include set theory - contents]]
=--
=--
=--

* table of contents
{: toc}

## Definition

Given a [[set]] $S$, the __power set__ of $S$ is the set $\mathcal{P}S$ of all [[subsets]] of $S$.  Equivalently, it is

* the set $\TV^S$ of all [[functions]] from $S$ to the set $\TV$ of [[truth values]].  This is often written $2^S$, since there are (at least in [[classical logic]]) exactly $2$ truth values;

* the collection of [[subobjects]] of $X$ in the [[topos]] [[Set]].

* the [[slice category]] $Inj/S$, where [[Inj]] is the [[wide subcategory]] of [[Set]] with morphisms restricted to [[injection|injections]]. This is similar to the subobject definition but is more unpacked. $Inj/S$ has objects that are injections to $S$ and morphisms that are commuting triangles of injections.

## Foundational status

One generally needs a specific axiom in the [[foundations of mathematics]] to ensure the existence of power sets.  In [[material set theory]], this can be phrased as follows:

* If $S$ is a set, then there exists a set $\mathcal{P}$ such that $A \in \mathcal{P}$ if $A \subseteq S$.

One can then use the [[axiom of separation]] ([[bounded separation]] is enough) to prove that $\mathcal{P}$ may be chosen so that the subsets of $A$ are the *only* members of $\mathcal{P}$; the [[axiom of extensionality]] proves that this $\mathcal{P}$ is unique.

Alternatively, one could include a [[powerset structure]], a primitive unary operator $\mathcal{P}(S)$ such that for all sets $S$, if for all sets $A$ and sets $B$, $B \in A$ implies that $B \in S$, then $A \in \mathcal{P}(S)$.

In [[structural set theory]], we state rather that there exists a set $\mathcal{P}$ which indexes the subsets of $A$ and prove uniqueness [[generalised the|up to unique isomorphism]].

In [[predicative mathematics]], the existence of power sets (along with other "impredicative" axioms) is not accepted.  However we can still speak of a power set as a [[proper class]], sometimes called a __power class__.

One can use power sets to construct [[function sets]]; the converse also works using [[excluded middle]] (or anything else that will guarantee the existence of the set of truth values).  In particular, power sets exist in any theory containing excluded middle and function sets; thus predicative theories which include function sets must also be [[constructive mathematics|constructive]].

### In dependent type theory

In [[dependent type theory]], if one has a [[univalent]] [[type of all propositions]] $(\mathrm{Prop}, \mathrm{El})$, then given a type $S$, the power set of $S$ is the [[function type]] $\mathcal{P}S \coloneqq S \to \mathrm{Prop}$. The power set of a type is always a set, because $\mathrm{Prop}$ is always a set by [[univalence]]; and if the [[codomain]] of a function type is a set, then the function type itself is a set.

An element of a power set $P:\mathcal{P}S$ is a [[predicate]]. The type

$$\sum_{x:S} \mathrm{El}(P(x))$$

is the corresponding [[subtype]] of $S$, with canonical [[embedding]] given by the first projection function defined in the [[elimination rules]] of the [[negative type|negative]] [[dependent sum type]].

$$\pi_1:\left(\sum_{x:S} \mathrm{El}(P(x))\right) \to S$$

There is also a local membership relation $(-)\in_S(-):\mathcal{P}(S \times \mathcal{P}S)$ defined by $a \in_S B \coloneqq B(a)$ for all $a:S$ and $B:\mathcal{P}S$, where $B(a)$ is defined in the [[elimination rules]] for [[function types]].

## Properties

### General

* The power set $\mathcal{P}S$ canonically carries a [[partial order]] by containment, making it a [[poset]]: $A$ precedes $B$ means that $A$ is a [[subset]] of $B$ ($A \subseteq B$).

* [[Cantor's theorem]] states that there exists no [[surjection]] from $S$ to $\mathcal{P}S$; as there does exist such an [[injection]], one concludes that
$$ {|S|} \lt {|\mathcal{P}S|} $$
in the usual arithmetic of [[cardinal numbers]].

* Power sets live in the category [[Set]].  Given an object $S$ of any [[category]], one can similarly form a [[poset]] of [[subobjects]] of $S$; the category is called [[well-powered category|well-powered]] when this poset is [[small category|small]].  One also has an internal notion of power set (a [[power object]]) in a [[topos]].

* The power set construction constitutes an [[equivalence of categories]] between the [[opposite category]] [[Set]]$^{op}$ and that of [[complete atomic Boolean algebras]]. See at _[Set -- Properties -- Opposite category and Boolean algebras](Set#OppositeCategory)_. Restricted to [[finite sets]], the power set construction constitutes an [[equivalence of categories]] between the [[opposite category]] of [[FinSet]] and that of finite [[Boolean algebras]]. See at _[FinSet -- Opposite category](FinSet#OppositeCategory)_.

### Power set functor

The power set construction gives rise to two functors, the contravariant power set functor $Set^op \to Set$ and the covariant power set functor $Set \to Set$. The first sends a function $f\colon S\to T$ to the _preimage_ function $f^*\colon P(T) \to P(S)$, whereas the second sends $f$ to the _image_ function $f_*\colon P(S) \to P(T)$.

## Related concepts

* A [[closure operator]] on a power set is a _[[Moore closure]]_.

* [[power object]]

* [[poly-morphism]]

category: foundational axiom

[[!redirects power set]]
[[!redirects power sets]]
[[!redirects power-set]]
[[!redirects power-sets]]
[[!redirects powerset]]
[[!redirects powersets]]

[[!redirects power class]]
[[!redirects power classes]]
[[!redirects power type]]
[[!redirects power types]]

[[!redirects axiom of power set]]
[[!redirects axiom of power sets]]
[[!redirects power set axiom]]
[[!redirects powerset axiom]]

[[!redirects power set functor]]
[[!redirects power set functors]]
[[!redirects covariant power set functor]]
[[!redirects covariant power set functors]]
[[!redirects contravariant power set functor]]
[[!redirects contravariant power set functors]]

[[!redirects powerset functor]]
[[!redirects powerset functors]]
[[!redirects covariant powerset functor]]
[[!redirects covariant powerset functors]]
[[!redirects contravariant powerset functor]]
[[!redirects contravariant powerset functors]]


power-associative algebra

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

# Power-associative algebras
* table of contents
{: toc}

## Idea

In an [[associative algebra]], or more generally in a [[semigroup]], there is no difficulty in defining $x^n$ for $x$ any element and $n$ any positive [[natural number]] (as well as $x^0$ when the algebra has an [[identity element]]).  In a [[nonassociative algebra]], or more generally a [[magma]], we can have many definitions, starting with $x^3$ (which could be $(x x) x$ but could be $x (x x)$ instead).

However, it *may* be that all possible ways of parenthesizing the expression for $x^n$ are equal.  In this case, we call the algebra _power-associative_.

## Definition

Let $\mathcal{C}$ be a [[multicategory]], and let $A$ be a [[magma object]] in $\mathcal{C}$, that is an object $A$ equipped with a bimorphism (binary multimorphism) $m\colon A, A \to A$.  Consider the [[subcategory|submulticategory]] of $\mathcal{C}$ generated by $A$ and $m$; for each [[natural number]] $n$, it has a [[family]] of $n$-ary multimorphisms whose size is the [[Catalan number]] $C(n)$ of $n$.

The magma object $A$ is __power-associative__ if, for each $n$, all of these $n$-ary multimorphisms are equal.  Traditionally, we make no requirement for $n = 0$, since $C(0) = 0$, but arguably we should require a unique $0$-ary multimorphism in the subalgebra too; then we get a power-associative magma with an [[identity element]].

If $\mathcal{C}$ is [[Set]], then a magma object is simply a [[magma]], and we have a __power-associative magma__.  If $\mathcal{C}$ is $K$-[[Mod]] for $K$ a [[commutative ring]], then a magma object is a [[nonassociative algebra]] over $K$, and we have a __power-associative algebra__.

## Properties

The submagma of every power-associative [[invertible magma]] $M$ [[generator|generated]] by an element $a \in M$ is a [[cyclic group]]. This means in particular there is a $\mathbb{Z}$-[[action]] on $M$ $(-)^{(-)}:M\times\mathbb{Z}\to M$ called the __[[power]]__.

## Examples

Every [[associative algebra]] or [[semigroup]] is of course power-associative.  More generally, every [[alternative algebra]] is also power-associative.  Every [[Jordan algebra]], although not necessarily alternative, is power-associative.  Every [[Cayley–Dickson algebra]], even beyond the [[octonions]] (the last alternative one), is power-associative.

## See also

* [[polynomial function]]

[[!redirects power function]]
[[!redirects power functions]]

[[!redirects power-associative algebra]]
[[!redirects power-associative algebras]]
[[!redirects power associative algebra]]
[[!redirects power associative algebras]]
[[!redirects power-associative magma]]
[[!redirects power-associative magmas]]
[[!redirects power associative magma]]
[[!redirects power associative magmas]]
[[!redirects power-associativity]]
[[!redirects power associativity]]

[[!redirects power-associative]]
[[!redirects power associative]]


powered and copowered category

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

#Contents#
* table of contents
{:toc}

## Definition

Throughout, let $\mathcal{V}$ be a [[Bénabou cosmos]] for [[enriched category theory]].

+-- {: .num_defn #TensoringAndCotensoring}
###### Definition
**([[tensoring]] and [[cotensoring]])**

Let $\mathcal{C}$ be a $\mathcal{V}$-[[enriched category]].

1. A _[[powering]]_ (or _[[cotensoring]]_) of $\mathcal{C}$ over $\mathcal{V}$ is

   1. a [[functor]]

      $$
        [-,-]
        \;\colon\;
        \mathcal{V}^{op} \times \mathcal{C} \longrightarrow \mathcal{C}
      $$

   1. for each $v \in \mathcal{V}$ a [[natural isomorphism]] of the form

      \[
        \label{NaturalIsomorphismForCotensoring}
        \mathcal{V}(v, \mathcal{C}(c_1,c_2)  )
        \;\simeq\;
        \mathcal{C}(c_1,[v,c_2])
      \]

1. A _[[copowering]]_ (or _[[tensoring]]_) of $\mathcal{C}$ over $\mathcal{V}$ is

   1. a [[functor]]

      $$
        (-)\otimes(-)
        \;\colon\;
        \mathcal{V} \times \mathcal{C} \longrightarrow \mathcal{C}
      $$

   1. for each $v \in \mathcal{V}$ a [[natural isomorphism]] of the form

      \[
        \label{NaturalIsomorphismForTensoring}
        \mathcal{C}(v \otimes c_1, c_2)
        \;\simeq\;
        \mathcal{V}(v, \mathcal{C}(c_1,c_2)  )
      \]

If $\mathcal{C}$ is equipped with a (co-)powering it is called _(co-)powered_
over $\mathcal{V}$.

=--

## Properties

+-- {: .num_prop #TensoringLeftAdjointToCotensoring}
###### Proposition
**([[tensoring]] [[left adjoint]] to [[cotensoring]])**

If $\mathcal{C}$ is both [[tensored and cotensored category|tensored and cotensored]] over $\mathcal{V}$ (Def. \ref{TensoringAndCotensoring}), then for fixed $v \in \mathcal{V}$ the operations of [[tensoring]] with $v$ and of [[cotensoring]] with $\mathcal{V}$ form a pair of [[adjoint functors]]

$$
  \mathcal{C}
    \underoverset
      {\underset{ [v,-] }{\longrightarrow}}
      {\overset{ v \otimes (-) }{\longleftarrow}}
      {\phantom{AA}\bot\phantom{AA}}
  \mathcal{C}
$$

=--

+-- {: .proof}
###### Proof

The hom-isomorphism characterizing the pair of [[adjoint functors]] is provided by the [[composition]] of
the [[natural isomorphisms]] (eq:NaturalIsomorphismForCotensoring) and (eq:NaturalIsomorphismForTensoring):

$$
  \array{
    \mathcal{C}(v \otimes c_1, c_2)
    \;\simeq\;
    \mathcal{V}(v, \mathcal{C}(c_1,c_2)  )
    \;\simeq\;
    \mathcal{C}(c_1,[v,c_2])
  }
$$

=--

It follows that:

+-- {: .num_prop #InTensoredCotensoredCategoryInitialObjectIsEnrichedInitial}
###### Proposition
**(in [[tensored and cotensored categories]] [[initial object|initial]]/[[terminal objects]] are enriched initial/terminal)**

If $\mathcal{C}$ is both [[tensored and cotensored category|tensored and cotensored]] over $\mathcal{V}$ then

1. an [[initial object]] $\emptyset$ of the underlying [[category]] of $\mathcal{C}$ is also enriched initial, in that the [[hom-object]] out of it is the [[terminal object]] $\ast$ of $\mathcal{V}$

   $$
     \mathcal{C}(\emptyset, c) \;\simeq\; \ast
   $$

1. a [[terminal object]] $\ast$ of the underlying category of $\mathcal{C}$ is also enriched terminal, in that the [[hom-object]] into it is the [[terminal object]] of $\mathcal{V}$:

   $$
     \mathcal{C}(c, \ast) \;\simeq\; \ast
   $$

=--

+-- {: .proof}
###### Proof

We discuss the first claim, the second is [[formal duality|formally dual]].

By prop. \ref{TensoringLeftAdjointToCotensoring}, tensoring is a [[left adjoint]]. Since [[left adjoints preserve colimits]], and since an [[initial object]] is the [[colimit]] over the [[initial category|empty diagram]], it follows that

$$
  v \otimes \emptyset \;\simeq\; \emptyset
$$

for all $v \in \mathcal{V}$, in particular for $\emptyset \in \mathcal{V}$. Therefore the natural isomorphism (eq:NaturalIsomorphismForTensoring) implies for all $v \in \mathcal{V}$ that

$$
  \mathcal{C}(\emptyset, c)
  \;\simeq\;
  \mathcal{C}( \emptyset \otimes \emptyset, c )
  \;\simeq\;
  \mathcal{V}( \emptyset, \mathcal{C}(\emptyset, c) )
  \;\simeq\;
  \ast
$$

where in the last step we used that the [[internal hom]] $\mathcal{V}(-,-) = [-,-]$ in $\mathcal{V}$ sends [[colimits]] in its first argument to [[limits]] ([this prop.](hom-functor+preserves+limits#InternalHomPreservesLimits)) and used that a terminal object is [[generalized the|the]] [[limit]] over the [[empty category|empty diagram]].

=--

## Examples

+-- {: .num_example}
###### Example

The [[symmetric monoidal category|symmetric]] [[closed monoidal category]] $\mathcal{V}$ is tensored and cotensored over itself, with tensoring being its [[tensor product]] and [[powering]] being its [[internal hom]].

=--

+-- {: .num_example}
###### Example

For $\mathcal{C}$ a [[small category|small]] $\mathcal{V}$-[[enriched category]], the $\mathcal{V}$-enriched [[category of enriched presheaves]] $[\mathcal{C}^{op}, \mathcal{V}]$ is tensored and cotensored ([this Prop.](geometry+of+physics+--+categories+and+toposes#UniversalPropertyOfTensoringAndPoweringOfFunctorsToTopcg))

=--

## Related concepts

* [[enriched model category]]

## References

* [[G. Max Kelly]], *Adjunction for enriched categories*, in: *Reports of the Midwest Category Seminar III*, Lecture Notes in Mathematics **106**, Springer (1969) [[doi:10.1007/BFb0059145](https://doi.org/10.1007/BFb0059145)]

[[!redirects powered and copowered categories]]

[[!redirects tensored and cotensored category]]
[[!redirects tensored and cotensored categories]]


powering

> This entry is about the [[formal dual]] to [[tensoring]] in the generality of [[category theory]]. For the different concept of _[[cotensor product]]_ of [[comodules]] see there.

***

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

#Contents#
* table of contents
{:toc}

## Idea

In a [[closed monoidal category|closed]] [[symmetric monoidal category]] $V$ the [[internal hom]] $[-,-] : V^{op} \times V \to V$ satisfies the [[natural isomorphism]]

$$
  \big[v_1,[v_2,v_3]\big]
  \;\simeq\;
  \big[v_2,[v_1,v_3]\big]
$$

for all [[objects]] $v_i \in V$ ([prop.](closed+monoidal+category#TensorHomIsoInternalizes)). If we regard $V$ as a $V$-[[enriched category]] we write $V(v_1,v_2) \mathrel{:=} [v_1,v_2]$ and this reads

$$
  V\big(v_1,V(v_2,v_3)\big)
  \;\simeq\; V\big(v_2,V(v_1,v_3)\big)
  \,.
$$

If we now pass more generally to any $V$-[[enriched category]] $C$ then we still have the enriched [[hom object]] functor $C(-,-) : C^{op} \times C \to V$. One says that $C$ is _powered_ over $V$ if it is in addition equipped also with a mixed operation $\pitchfork : V^{op} \times C \to C$ such that $\pitchfork(v,c)$ behaves as if it were a hom of the object $v \in V$ into the object $c \in C$ in that it comes with natural isomorphisms of the form

$$
  C\big(c_1, \pitchfork(v,c_2)\big)
  \;\simeq\;
  V\big(v, C(c_1,c_2)\big)
  \,.
$$

## Definition

+-- {: .un_defn}
###### Definition

Let $V$ be a [[closed monoidal category|closed]] [[monoidal category]].  In a $V$-[[enriched category]] $C$, the **power** of an object $y\in C$ by an object $v\in V$ is an object $\pitchfork(v,y) \in C$ with a [[natural isomorphism]]

$$
  C(x, \pitchfork(v,y)) \cong V(v, C(x,y))
$$

where $C(-,-)$ is the $V$-valued hom of $C$ and $V(-,-)$ is the [[internal hom]] of $V$.

We say that $C$ is **powered** or **cotensored** over $V$ if all such power objects exist.

=--

+-- {: .un_remark}
###### Remark

Powers are frequently called _cotensors_ and a $V$-category having all powers is called _cotensored_, while the word "power" is reserved for the case $V=$ [[Set]].  However, there seems to be no good reason for making this distinction.  Moreover, the word "tensor" is fairly overused, and unfortunate since a tensor (= a [[copower]]) is a [[colimit]], while a cotensor (= power) is a [[limit]].

=--

## Properties

* Powers are a special sort of [[weighted limit]]: in particular, where the domain is the unit $V$-category. Conversely, all weighted limits can be constructed from powers together with [[conical limit]]s.  The dual colimit notion of a power is a [[copower]].

## Examples

### In 1-category theory

* $V$ itself is always powered over itself, with $\pitchfork(v_1,v_2) \mathrel{:=} [v_1,v_2]$.

* Every [[locally small category]] $C$ ($V = (Set,\times)$ ) with all [[product]]s  is powered over [[Set]]: the powering operation

  $$
    \pitchfork(S,c) \mathrel{:=} \prod_{s\in S} c
  $$

  of an object $c$ by a set $S$ forms the $|S|$-fold [[cartesian product]] of $c$ with itself, where $|S|$ is the [[cardinality]] of $S$.

  The defining natural isomorphism

  $$
    Hom_C(c_1,\pitchfork(S,c_2))\simeq Hom_{Set}(S,Hom_C(c_1,c_2))
  $$

  is effectively the definition of the product (see [[limit]]).

* In a [[2-category]] $\mathcal{K}$ (seen as a $\mathbf{Cat}$-enriched category), powers by the walking arrow $\downarrow$ are ways to internalize 'generalized arrows' of a given object $A:\mathcal{K}$.
Specifically, ${A^\downarrow} := {\downarrow \pitchfork A}$, called the **object of arrows** of $A$ is, when it exists, an object such that:
$$
\mathcal{K}(X, A^\downarrow) \simeq \mathbf{Cat}(\downarrow, \mathcal{K}(X,A)) = \mathcal{K}(X,A)^\downarrow.
$$
Thus [[generalized elements]] of $A^\downarrow$ correspond to 2-cells between generalized elements of $A$, explaining why $A^\downarrow$ can be considered a 'view from the inside' of the internal structure of $A$.

[[!include powering of ∞-toposes over ∞-groupoids -- section]]

## Related concepts

* [[tensored and cotensored category]]

* [[copower]], [[(∞,1)-copower]]

* [[pullback-power]]

## References

Textbook accounts:

* [[Max Kelly]], section 3.7 of: _Basic concepts of enriched category theory_ ([tac](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html) ,[pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf))

* {#Borceux94} [[Francis Borceux]], Vol 2, Section 6.5 of _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994)

* [[Emily Riehl]], §3.7 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

[[!redirects powerings]]

[[!redirects power]]
[[!redirects powers]]

[[!redirects cotensor]]
[[!redirects cotensoring]]

[[!redirects cotensors]]
[[!redirects cotensored category]]


powering of ∞-toposes over ∞-groupoids -- section

### Powering of $\infty$-toposes over $\infty$-groupoids
  {#PoweringOfInfinityToposesOverInfinityGroupoids}

We discuss how the [[powering]] of [[(infinity,1)-toposes|$\infty$-toposes]] over [[Infinity-Grpd|$Grpd_\infty$]] is given by forming [[mapping stacks]] out of [[locally constant infinity-stacks|locally constant $\infty$-stacks]]. All of the following formulas and their proofs hold verbatim also for [[Grothendieck toposes]], as they just use general abstract properties.

\linebreak

Let $\mathbf{H}$ be an [[(infinity,1)-topos|$\infty$-topos]]

* with terminal [[(infinity,1)-geometric morphism|geometric morphism]] denoted

  \[
    \label{TerminalGeometricMorphismAdjunction}
    \mathbf{H}
    \underoverset
      {\underset{\Gamma}{\longrightarrow}}
      {\overset{LConst}{\longleftarrow}}
      {\;\;\;\;\bot\;\;\;\;}
    Grp_\infty
    \,,
  \]

  where the [[inverse image]] constructs [[locally constant infinity-stacks|locally constant $\infty$-stacks]],

* and with its [[internal hom]] ([[mapping stack]]) [[adjoint (infinity,1)-functor|adjunction]] denoted

  \[
    \label{MappingStackAdjunction}
    \mathbf{H}
      \underoverset
        {\underset{Maps(X,-)}{\longrightarrow}}
        { \overset{ (-) \times X }{\longleftarrow} }
        {\;\;\;\; \bot \;\;\;\;}
    \mathbf{H}
  \]

  for $X \,\in\, \mathbf{H}$.

  Notice that this construction is also [[(infinity,1)-functor|$\infty$-functorial]] in the first argument:
$Maps\big( X \xrightarrow{f} Y ,\, A \big)$ is the morphism which under the [[(infinity,1)-Yoneda lemma|$\infty$-Yoneda lemma]] over $\mathbf{H}$ (which is large but locally small, so that the lemma does apply)  corresponds to

$$
  \mathbf{H}
  \big(
    (-)
    ,\,
    Maps(X,A)
  \big)
  \;\simeq\;
  \mathbf{H}
  \big(
    (-) \times X
    ,\,
    A
  \big)
  \xrightarrow{
    \mathbf{H}
    \big(
      (-) \times f
      ,\,
      A
    \big)
  }
  \mathbf{H}
  \big(
    (-) \times Y
    ,\,
    A
  \big)
  \;\simeq\;
  \mathbf{H}
  \big(
    (-)
    ,\,
    Maps(X,A)
  \big)
  \,.
$$

By definition, for any $S \in Grpd_\infty$ and $X \in \mathbf{H}$ the [[powering]]] is the [[(∞,1)-limit]] over the [[diagram]] constant on $X$

$$
  X^K \,=\, {\lim_\leftarrow}_K X
$$

while the [[tensoring]] is the [[(∞,1)-colimit]] over the diagram constant on $X$

$$
  K \cdot X \,=\, {\lim_{\to}}_K X
  \,.
$$

\begin{remark}
  Under [[Isbell duality]], the powering operations on homotopy types $X$ corresponds to higher order [[Hochschild cohomology]] of suitable algebras of functions on  $X$, as discussed there.
\end{remark}

\begin{proposition}
The *powering* of $\mathbf{H}$ over [[Infinity-Grpd|$Grpd_\infty$]] is given by the [[mapping stack]] out of the [[locally constant infinity-stack|locally constant $\infty$-stacks]]:

$$
  \array{
    Grpd_\infty^{op}
    \times
    \mathbf{H}
    &
    \overset{ LConst^{op} \times \mathrm{id} }{\longrightarrow}
    &
    \mathbf{H}^{op}
    \times
    \mathbf{H}
    &
    \overset{Maps(-,-)}{\longrightarrow}
    &
    \mathbf{H}
  }
$$
in that this operation has the following properties:

1. For all $X,\,A \,\in\, \mathbf{H}$ and $S \,\in\, Grpd_\infty$ we have a [[natural equivalence]]

   \[
     \mathbf{H}
     \Big(
       X
       ,\,
       Maps
       \big(
         LConst(S)
         ,\,
         A
       \big)
     \Big)
     \;\;
     \simeq
     \;\;
     Grpd_\infty
     \Big(
       S
       ,\,
       \mathbf{H}
       \big(
         X
         ,\,
         A
       \big)
     \Big)
   \]

1. In its first argument the operation

   1. sends the [[terminal object]] (the [[point]]) to the identity:

      \[
        \label{MappingStackOutOfLocallyConstantPreservesLimitsInFirstArg}
        Maps
        \big(
          LConst(\ast)
          ,\,
          X
        \big)
        \;\;
        \simeq
        \;\;
         X
      \]

   1. sends [[(infinity,1)-colimits|$\infty$-colimits]] to [[(infinity,1)-limits|$\infty$-limits]]:

      \[
        \label{MappingStackOutOfLConstStacksPreservesColimitInFirstArgument}
        Maps
        \Big(
          \underset{
            \longrightarrow
          }{\lim}
          \,
          LConst\big(S_\bullet\big)
          ,\,
          X
        \Big)
        \;\;
        \simeq
        \;\;
        \underset{
          \longleftarrow
        }{\lim}
        \,
        Maps
        \Big(
          LConst\big(S_\bullet\big)
          ,\,
          X
        \Big)
        \,,
      \]

   where all [[equivalence in an (infinity,1)-category|equivalences]] shown are [[natural equivalence|natural]].

\end{proposition}

\begin{proof}

For the first statement to be proven, consider the following sequence of [[natural equivalences]]:
$$
  \begin{array}{lll}
     \mathbf{H}
     \Big(
       X
       ,\,
       Maps
       \big(
         LConst(S)
         ,\,
         A
       \big)
     \Big)
     &
     \;\simeq\;
     \mathbf{H}
     \big(
       X
       \times
       LConst(S)
       ,\,
       A
     \big)
     &
     \text{(eq:MappingStackAdjunction)}
     \\
     & \;\simeq\;
     \mathbf{H}
     \Big(
       LConst(S)
       ,\,
       Maps
       \big(
         X
         ,\,
         A
       \big)
     \Big)
     &
     \text{(eq:MappingStackAdjunction)}
     \\
     & \;\simeq\;
     Grpd_\infty
     \Big(
       S
       ,\,
       \Gamma
       \,
       Maps
       \big(
         X
         ,\,
         A
       \big)
     \Big)
     &
     \text{ (eq:TerminalGeometricMorphismAdjunction) }
     \\
     & \;\simeq\;
     Grpd_\infty
     \bigg(
       S
       ,\,
       \mathbf{H}
       \Big(
         \ast_{\mathbf{H}}
         ,\,
         Maps
         \big(
           X
           ,\,
           A
         \big)
       \Big)
     \bigg)
     &
     \text{by}\;\text{this Prop.}
     \\
     & \;\simeq\;
     Grpd_\infty
     \Big(
       S
       ,\,
       \mathbf{H}
       \big(
         \ast_{\mathbf{H}}
         \times
         X
         ,\,
         A
       \big)
     \Big)
     &
     \text{(eq:MappingStackAdjunction)}
     \\
     & \;\simeq\;
     Grpd_\infty
     \Big(
       S
       ,\,
       \mathbf{H}
       \big(
         X
         ,\,
         A
       \big)
     \Big)
  \end{array}
$$

For the second statement, recall that
[[hom-functors preserve limits]] in that there are [[natural equivalences|natural]] [[equivalences in an (infinity,1)-category|equivalences]] of the form

\[
  \label{HomFunctorPreservesLimits}
  \mathbf{H}
  \Big(
    \underset{\underset{i}{\longrightarrow}}{\lim}
    \,,
    X_i
    ,\,
    \underset{\underset{j}{\longleftarrow}}{\lim}
    \,,
    A_j
  \Big)
  \;\;
  \simeq
  \;\;
  \underset{\underset{i}{\longleftarrow}}{\lim}
  \,
  \underset{\underset{j}{\longleftarrow}}{\lim}
  \,
  \mathbf{H}
  \Big(
    X_i
    ,\,
    A_j
  \Big)
  \,,
\]

and that $\infty$-toposes have [[universal colimits]], in particular that the [[product]] operation is a [[left adjoint]] (eq:MappingStackAdjunction) and [[left adjoints preserve colimits|hence preserves colimits]]:

\[
  \label{ProductsPreserveColimits}
  (-)
  \,\times\,
  \underset{{\longrightarrow}}{\lim} \, S_\bullet
  \;\;
  \simeq
  \;\;
  \underset{{\longrightarrow}}{\lim} \,
  \big(
    (-)
    \,\times\,
    S_\bullet
  \big)
  \,.
\]

With this, we get the following sequences of [[natural equivalences]]:

$$
  \begin{array}{lll}
    &
    \mathbf{H}
    \bigg(
      (-)
      ,\,
      Maps
      \Big(
        \underset{\longrightarrow}{\lim}
        \,
        LConst(S_\bullet)
        ,\,
        X
      \Big)
    \bigg)
    \\
    & \;\simeq\;
    \mathbf{H}
    \Big(
      (-)
      \times
      \underset{\longrightarrow}{\lim}
      \,
      LConst(S_\bullet)
      ,\,
      X
    \Big)
    &
    \text{ (eq:MappingStackAdjunction) }
    \\
    & \;\simeq\;
    \mathbf{H}
    \Big(
      \underset{\longrightarrow}{\lim}
      \big(
        (-)
        \times
        LConst(S_\bullet)
      \big)
      ,\,
      X
    \Big)
    &
    \text{ (eq:ProductsPreserveColimits) }
    \\
    & \;\simeq\;
    \underset{\longleftarrow}{\lim}
    \,
    \mathbf{H}
    \big(
      (-)
      \times
      LConst(S_\bullet)
      ,\,
      X
    \big)
    &
    \text{ (eq:HomFunctorPreservesLimits) }
    \\
    & \;\simeq\;
    \underset{\longleftarrow}{\lim}
    \,
    \mathbf{H}
    \Big(
      (-)
      ,\,
      Maps
      \big(
        LConst(S_\bullet)
        ,\,
        X
      \big)
    \Big)
    &
    \text{ (eq:MappingStackAdjunction) }
    \\
    & \;\simeq\;
    \mathbf{H}
    \Big(
      (-)
      ,\,
      \underset{\longleftarrow}{\lim}
      \,
      Maps
      \big(
        LConst(S_\bullet)
        ,\,
        X
      \big)
    \Big)
    &
    \text{ (eq:HomFunctorPreservesLimits) }
    \,.
  \end{array}
$$
This implies (eq:MappingStackOutOfLConstStacksPreservesColimitInFirstArgument) by the [[(infinity,1)-Yoneda lemma|$\infty$-Yoneda lemma]] over $\mathbf{H}$ (which is large but locally small, so that the lemma does apply).

Finally (eq:MappingStackOutOfLocallyConstantPreservesLimitsInFirstArg) is immediate from the fact that $LConst$ preserves the terminal object, by definition:
$$
    Maps
    \big(
      LConst(\ast)
      ,\,
      X
    \big)
    \;\simeq\;
    Maps
    \big(
      \ast_{\mathbf{H}}
      ,\,
      X
    \big)
    \;\simeq\;
    X
    \,.
$$
\end{proof}