descent

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#### Locality and descent
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[[!include descent and locality - contents]]
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#Contents#
* table of contents
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## Idea
From the point of view of infinity category theory (nPOV), _descent_ is the study of generalizations of the [[sheaf]] condition on [[presheaf|presheaves]] to presheaves with values in [[higher category theory|higher categories]]. Those higher presheaves that satisfy descent are called [[infinity-stack]]s.

More generally,
**descent theory** studies existence and (non)uniqueness of an object $u$ in a (possibly higher) category $C_X$ provided some "inverse image" functor $f^*:C_X\to C_Y$ which applied to $u$ produces an object in some (possibly higher) category $C_Y$, or a collection of inverse image functors $\{f^*_\alpha:C_X\to C_{Y_\alpha}\}_{\alpha\in I}$ is given. Labels $Y_\alpha$ are considered as labels of local regions, over which objects in $C_{Y_\alpha}$ live and the inverse image functor is considered as some sort of restriction along geometric morphism of spaces from $Y_\alpha\to X$. In favourable cases, the nonuniqueness is parametrized by equipping the object $f^*(u)$ with additional "gluing" data $\xi$. The pair $(f^*(u),\xi)$ is called a __descent datum__, the existence of a reconstruction procedure of $u$ from $(f^*(u),\xi)$ is also called a descent, and it describes the property that the (higher) category of descent data in $C_Y$ is equivalent to the category $C_X$, or at least that it embeds via a canonical fully faithful functor. Descent theory in 1-categorical context has been first formulated by Grothendieck in FGA using pseudofunctors and in [[SGA1]] using fibered categories.

The most important case is when there is a descent (in the sense of [[equivalence of higher categories]]) along an [[inverse image]] functor along every cover of a [[Grothendieck topology]] or its higher analogue; though many cases (for example [[descent in noncommutative algebraic geometry]]) do not fit into this framework. These cases of descent along all covers is also called (higher) stack theory and may be phrased in modern viewpoint as a characterization of $(\infty,1)$-[[(infinity,1)-sheaf|sheaves]] (i.e. $\infty$-[[infinity-stack|stacks]]) among all $(\infty,1)$-[[(infinity,1)-presheaf|presheaves]] as those $(\infty,1)$-presheaves which are [[local objects]] with respect to certain morphisms $Y \to X$ which are to be regarded as [[covers]] or [[hypercover]] of the $(\infty,1)$-presheaf $X$: the idea is that an $(\infty,1)$-sheaf "descends from the cover $Y$ down to $X$".

More concretely

* every [[(∞,1)-category of (∞,1)-sheaves]] is characterized as being a sub-[[(∞,1)-topos]] $Sh(S) \hookrightarrow PSh(S)$ of the $(\infty,1)$-topos of [[(∞,1)-presheaves]] on some (small) [[(∞,1)-category]] $S$;

* every such $(\infty,1)$-topos is a [[reflective (∞,1)-subcategory]] of $PSh(S)$, hence a [[localization of an (∞,1)-category]] at a collection $W = \{Y \to X\}$ of morphisms which are sent to equivalences by the left adjoint of the inclusion;

* and the sheaves in $Sh(S) \hookrightarrow PSh(S)$ are precisely the [[local objects]] with respect to this collection $W$ of morphisms, i.e. precisely those objects $A \in PSh(S)$ such that $PSh(S)(X,A) \to PSh(S)(Y,A)$ is an isomorphism in the [[homotopy category of an (infinity,1)-category|homotopy category]], which we shall write $PSh(S)(X,A) \stackrel{\simeq}{\to} PSh(S)(Y,A)$ in the following paragraphs.

* This condition is essentially the _descent conditon_.

In concrete models for the [[(∞,1)-category of (∞,1)-sheaves]] -- notably in terms of the [[model structure on simplicial presheaves]] -- the morphisms $Y \to X$ in $W$ usually come from [[hypercovers]]  $Y \to X$;

in this case the above condition becomes $PSh(S)(X,A) \stackrel{\simeq}{\to} PSh(S)(colim^\Delta Y_\bullet, A)$ which is equivalent to $ PSh(S)(X,A) \stackrel{\simeq}{\to} lim^\Delta PSh(S)(Y_\bullet, A) $. This in turn is usually equivalently written

$$
  A(X)
  \stackrel{\simeq}{\to}
   Desc(Y_\bullet \to X, A)
   := lim^\Delta A(Y_\bullet)
  \,.
$$

And this is the form of the [[local object]]-condition which is usually called **descent condition**.

## Descent for ordinary sheaves

Descent is best understood as a direct generalization of the  situation for 0-stacks, i.e. ordinary sheaves, which we briefly  recall in a language suitable for the following generalization.

For $S$ any small [[category]] and [[Set]] the category of small sets, write $\mathrm{PSh}(S) = [S^{op}, Set]$ for the category of [[presheaf|presheaves]] on $S$. Categories of this form enjoy various nice properties which are familiar from $Set$ itself,  and which are summarized by saying that $\mathrm{PSh}(S)$ is a [[topos]]. The relevance of
this for the present purpose is that there is a natural notion of morphisms of topoi, which are [[functors]] respecting this structure in some sense: these are called [[geometric morphisms]].

A [[category of sheaves]] on $S$ is a sub-topos of $PSh(S)$ in that it is a [[full and faithful functor]] $Sh(S)\hookrightarrow PSh(S)$ which is a [[geometric morphism]].

One finds that the [[reflective subcategory]] $Sh(S) \hookrightarrow PSh(S)$ of sheaves inside presheaves is the [[localization]] of $PSh(S)$ at morphisms $f : Y \to X$ called [[local isomorphisms]], which are determined by and determine the choice of topos-inclusion. A [[presheaf]] $A$ is a [[sheaf]] precisely if it is a [[local object]] with respect to these [[local isomorphisms]], that is precisely if

$$
  Hom_{PSh(S)}(X,A) \stackrel{Hom_{PSh(S)}(f,A)}{\to} Hom_{PSh(S)}(Y,A)
$$

is an [[isomorphism]] for all [[local isomorphisms]] $f$.

This locality condition is in fact the _descent_ condition: the sheaf has to descend from $Y$ down to $X$. More concretely, this condition is called a _descent condition_ when evaluated on morphisms $f : Y \to X$ which are [[hypercovers]]:

namely if $\pi : Y^1 \to X$ is a [[local epimorphism]] with respect to the [[coverage]] that corresponds to the [[localization]] and if $\pi_2 : Y^2 \to Y^1 \times_X Y^1$ is a [[local epimorphism]], then with

$$
  Y^\bullet := (Y^2 \rightrightarrows Y^1)
$$

being the two canonical morphisms out of $Y^2$, it follows that the canonical morphism

$$
  colim Y^\bullet \to X
$$

is a [[local isomorphism]].

(This is exercise 16.6 in [[Categories and Sheaves]]).

Therefore for a  [[presheaf]] $A$ to be a [[sheaf]], it is necessary that

$$
  Hom_{PSh(S)}(X,A) \stackrel{\simeq}{\to} Hom_{PSh(S)}(colim Y^\bullet, A)
$$

is an [[isomorphism]]. The [[colimit]] may be taken out of the [[hom-functor]] to make this equivalently

$$
  Hom_{PSh(S)}(X,A)
   \stackrel{\simeq}{\to}
  lim Hom_{PSh(S)}(Y^\bullet, A)
  \,.
$$

It is convenient, suggestive and common to write $A(X) := Hom_{PSh(S)}(X,A)$, $A(Y^\bullet) := Hom_{PSh(S)}(Y^\bullet,A)$, following the spirit of the [[Yoneda lemma]] whether or not $X$ and/or $Y^\bullet$ are [[representable functor|representable]]. In that notation the above finally becomes

$$
  A(X)
  \stackrel{\simeq}{\to}
  lim A(Y^\bullet)
  \,.
$$

This is the form of the condition that is most commonly called the _descent condition_.

## Descent for simplicial presheaves

For more references and background on the following see [[descent for simplicial presheaves]].

A well-studied class of models/presentations for an [[(∞,1)-category of (∞,1)-sheaves]] is obtained using the [[model structure on simplicial presheaves]] on an  ordinary (1-categorical) [[site]] $S$, as follows.

Let $[S^{op}, SSet]$ be the [[SimpSet|SSet]]-[[enriched category]] of [[simplicial presheaf|simplicial presheaves]] on $S$.

Recall from [[model structure on simplicial presheaves]] that there is the _global_ and the _local_ injective simplicial model structure on $[S^{op}, SSet]$ which makes it a [[simplicial model category]] and that the local model structure is a (Bousfield-)localization of the global model structure.

So in terms of simplicial presheaves the [[localization of an (∞,1)-category]] that we want to describe, namely [[∞-stackification]], is modeled as the [[localization of a simplicial model category]].

Recall that the [[(∞,1)-category]] modeled/presented by a [[simplicial model category]] is the full [[SimpSet|SSet]]-subcategory on fibrant-cofibrant objects.
According to [section 6.5.2](http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf#page=528)  of [[Higher Topos Theory|HTT]] we have:

* the full simplicial subcategory on fibrant-cofibrant objects of $[S^{op}, SSet]$ with respect to the _global_ injective model structure is (the [[SimpSet|SSet]]-[[enriched category]] realization of) the $(\infty,1)$-category  $PSh_{(\infty,1)}(S)$ of [[(∞,1)-presheaves]] on $S$.

* the full simplicial subcategory on fibrant-cofibrant objects of $[S^{op}, SSet]$ with respect to the _local_ injective model structure is (the [[SimpSet|SSet]]-[[enriched category]] realization of) the $(\infty,1)$-category  $\bar{Sh}_{(\infty,1)}(S)$ which is the [[hypercompletion]] of the $(\infty,1)$-category $Sh_{(\infty,1)}(S)$ of [[(∞,1)-sheaves]] on $S$.

Since with respect to the local or global injective model structure all objects are automatically cofibrant, this means that $\bar Sh_{(\infty,1)}(S)$ is the full sub-$(\infty,1)$-category of $PSh_{(\infty,1)}(S)$ on simplicial presheaves which are fibrant with respect to the local injective model structure: these are the [[∞-stacks]] in this model.

By the general properties of [[localization of an (∞,1)-category]] there should be a class of morphisms $f : Y \to X$ in $PSh_{(\infty,1)}(S)$ -- hence between injective-fibrant objects in $[S^{op}, PSh(S)]$ -- such that the simplicial presheaves representing $\infty$-stacks are precisely the [[local objects]] with respect to these morphisms.

The general idea of descent in this simplicial context is the precise analog of the situation for ordinary sheaves, but with ordinary [[limit|(co)limits]] replaced everywhere with the [[limit in quasi-categories|(∞,1)-categorical (co)limits]], which in terms of the [[presentable (infinity,1)-category|presentation]] by the [[model structure on simplicial presheaves]] amounts to the [[homotopy limit|homotopy (co)limit]].

So for $Y \to X$ a morphism of simplicial presheaves, the condition that a simplicial presheaf $A$ is [[local object|local]] with respect to it, hence satisfies descent with respect to it, is generally that

$$
  \begin{aligned}
    RHom(X,A)  \stackrel{}{\to} & RHom(Y,A)
    \\
    & \simeq RHom(hocolim_n Y_n, A)
    \\
    & \simeq holim_n RHom(Y_n, A)
    \\
    & =: holim_n A(Y_n)
  \end{aligned}
$$

is a weak equivalence, where $RHom$ denotes the corresponding $(\infty,1)$-categorical hom, i.e. the derived hom with respect to the [[model structure on simplicial presheaves]] -- for instance the ordinary simplicial hom if both $Y$ and $A$ are fibrant with respect to the given model structure.

The details on which morphisms $Y \to X$ one needs to check against here have been worked out in

* D. Dugger, S. Hollander, D. Isaksen, _Hypercovers and simplicial presheaves_ ([pdf](http://hopf.math.purdue.edu//Dugger-Hollander-Isaksen/hypspre.pdf))

We now describe central results of that article.

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

For $X \in S$ an object in the [[site]] regarded as a simplicial presheaf and $Y \in [S^{op}, SSet]$ a simplicial presheaf on $S$, a morphism $Y \to X$ is a **[[hypercover]]** if it is a _local acyclic fibration_, i.e. of for all $V \in S$ and all diagrams
$$
  \array{
    \Lambda^k[n]\otimes V &\to & Y
    \\
    \downarrow && \downarrow
    \\
    \Delta^n\otimes V &\to&
    X
  }
  \;\; respectively \;\,
  \array{
    \partial \Delta^n\otimes V &\to & Y
    \\
    \downarrow && \downarrow
    \\
    \Delta^n\otimes V &\to&
    X
  }
$$
there exists a covering  [[sieve]] $\{U_i \to V\}$ of $V$ with respect to the given [[Grothendieck topology]] on $S$ such that for every $U_i \to V$ in that [[sieve]] the pullback of the above diagram to $U$ has a lift
$$
  \array{
    \Lambda^k[n]\otimes U_i &\to & Y
    \\
    \downarrow &\nearrow & \downarrow
    \\
    \Delta^n\otimes U_i &\to&
    X
  }
  \;\; respectively \;\,
  \array{
    \partial \Delta^n\otimes U_i &\to & Y
    \\
    \downarrow &\nearrow& \downarrow
    \\
    \Delta^n\otimes U_i &\to&
    X
  }
  \,.
$$
=--

If $S$ is a [[Verdier site]] then every such hypercover $Y \to X$ has a refinement by a hypercover which is cofibrant with respect to the projective global [[model structure on simplicial presheaves]]. We shall from now on make the assumption that the hypercovers $Y \to X$ we discuss are cofibrant in this sense. These are called _split hypercovers_. (This works in many cases that arise in practice, see the discussion after [DHI, def. 9.1](http://hopf.math.purdue.edu//Dugger-Hollander-Isaksen/hypspre.pdf#page=29).)

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

The objects of $Sh_{(\infty,1)}(S)$ -- i.e. the fibrant objects with respect to the projective model structure on $[S^{op}, SSet]$ -- are precisely those objects $A$ of $PSh_{(\infty,1)}(S)$ -- i.e. [[Kan complex]]-valued simplicial presheaves -- which
**satisfy descent for all split hypercovers**, i.e. those for which for all split hypercover $f : Y \to X$ in $[S^{op}, SSet]$ we have that
$$
  [S^{op}, SSet](X,A) \stackrel{\simeq}{\to}
  [S^{op}, SSet](Y,A)
$$
is a [[model structure on simplicial sets|weak equivalence of simplicial sets]].
=--

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

This is [DHI, thm 1.3](http://hopf.math.purdue.edu//Dugger-Hollander-Isaksen/hypspre.pdf#page=3) formulated in the light of [DHI, lemma 4.4 (ii)](http://hopf.math.purdue.edu//Dugger-Hollander-Isaksen/hypspre.pdf#page=9).
=--

Notice that by the [[co-Yoneda lemma]] every simplicial presheaf $F : S^{op} \to SSet$, which we may regard as a presheaf $F : \Delta^{op}\times S^{op} \to Set$, is isomorphic to the [[weighted limit|weighted colimit]]

$$
    F \simeq colim^\Delta F_\bullet
$$

which is equivalently the [[end|coend]]

$$
  F \simeq \int^{[n] \in \Delta} \Delta^n \cdot F_n
  \,,
$$

where $F_n$ is the Set-valued presheaf of $n$-cells of $F$ regarded as an $SSet$-valued presheaf under the inclusion $Set \hookrightarrow SSet$, and
where the [[SimpSet|SSet]]-weight is the canonical cosimplicial simplicial set $\Delta$, i.e. for all $X \in S$

$$
  F : X \mapsto \int^{[n] \in \Delta} \Delta^n \times F(X)_n
  \,.
$$

In particular therefore for $A$ a [[Kan complex]]-valued presheaf the descent condition reads

$$
  [S^{op}, SSet](X,A) \stackrel{\simeq}{\to}
  [S^{op}, SSet](colim^\Delta Y_\bullet,A)
  \simeq
  lim^\Delta [S^{op}, SSet](Y_\bullet,A)
  \,.
$$

With the shorthand notation introduced above the **descent condition** finally reads, for all global-injective fibrant simplicial presheaves $A$ and hypercovers $U \to X$:

$$
  A(X) \stackrel{\simeq}{\to} lim^\Delta A(Y_\bullet)
  \,.
$$

The right hand here is often denoted $Desc(Y_\bullet \to X, A)$, in which case this reads

$$
  A(X) \stackrel{\simeq}{\to} Desc(Y_\bullet \to X, A)
  \,.
$$

## Descent for strict $\omega$-groupoid valued presheaves

While simplicial sets are a very convenient model for general reasoning about higher weak categories and [[∞-groupoids]], often concrete computations in particular with $(\infty)$-groupoids are more convenient in the context of more strictified models.

Notably, by the generalized [[Dold-Kan correspondence]] the [[oriental|$\omega$-nerve]] injects [[crossed complex]]es -- nonabelian generalizations of [[chain complex]]es of abelian groups which are equivalent to [[strict ∞-groupoids]] -- to simplicial sets

$$
  CrsCmplx \stackrel{\simeq}{\to}
  Str \omega Grpd \stackrel{N}{\to}
  SSet
  \,.
$$

Since for instance something as simple as an abelian group $A$ regarded as a complex of groups in degree $n$ (hence as an $n$-group) already bcomes a somewhat involved object to understand under the nervet operation,

_it is desirable to have a means to control descent for simplicial presheaves which happen to factor through the $\omega$-nerve directly in the context of $Str \omega Cat$._

In his work on descent

* Ross Street, _Categorical and combinatorial aspects of descent theory_, [arXiv:math.CT/0303175](http://arxiv.org/abs/math.CT/0303175)

Ross Street considered presheaves with values in [[strict ∞-categories]]

$$
  A : S^{op} \to Str \omega Cat
$$

and declared the descent $\omega$-category with respect to a simplicial object $Y_\bullet : \Delta^{op} \to S$ to be the [[weighted limit]] in $Str\omega-Cat$-[[enriched category theory]]

$$
  lim^{F \Delta} A(U^\bullet)
  \,,
$$

where $O := F \Delta : \Delta \to Str \omega Cat$ are the [[orientals]], i.e. the free $\omega$-categories on the simplicial [[simplex|simplices]]

$$
  O = F \circ \Delta
  \,,
$$

where $F : SSet \to Str\omega Cat$ is the [[right adjoint]] to the [[oriental|∞-nerve]] $N : Str \omega Cat \to SSet$.

The two precscriptions

$$
 \array{
  lim^\Delta N A(U^\bullet) & in SSet
  \\
  \\
  lim^{F \Delta} A(U^\bullet) & in Str \omega Grpd
 }
$$

have a very similar appearance. The following theorem asserts if and when they are actually equivalent.

+-- {: .un_theorem }
###### Theorem (Dominic Verity)

There exists a canonical comparison map

$$
  N(Desc_{Street}(U^\bullet, A))
  :=
  N(lim^{F \Delta} A(U^\bullet))
  \;\;\;\stackrel{\;\;\;\;\;\;\;\;\;}{\hookrightarrow}\;\;\;
  Desc_{simp}(U^\bullet, N \circ A)
  :=
  holim_\bullet N(A(U^\bullet))
  \,.
$$

This is a [[model structure on simplicial sets|weak equivalence]] of [[Kan complex]]es if the cosimplicial simplicial set $N(A(U^\bullet))$ is [[Reedy category|Reedy fibrant]].

=--

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

The full proof is given at [[Verity on descent for strict omega-groupoid valued presheave|Verity on descent for strict omega-groupoid valued presheaves]].

=--

## Descent in terms of gluing conditions {#AsGluing}

We unwrap the expression

$$
  lim^\Delta A(Y_\bullet)
$$

for the descent data for a presheaf $A$ with respect to a (hyper)cover $Y \to X$

This [[weighted limit]] (whether taken in $SSet$- or in $Str \omega Cat$-[[enriched category theory]]) is given by the [[end|coend]]

$$
  lim^W\Delta A(Y_\bullet)
  \simeq
  \int^{[n] \in \Delta} [\Delta^n, A(Y_n)]
  \,.
$$

Unwrapping what this means one finds that an object/vertex of this is a choice of $n$-simplex in each $A(Y_n)$, subject to conditions which say that the boundary of this $n$-simplex must be obtained from pullback of $A$ along the maps $Y_n \to Y_{[n-1]}$ of the $(n-1)$-simplex in $A(Y_{n-1})$

Namely an object in

$
  lim^W\Delta A(Y_\bullet)
  \simeq
  \int^{[n] \in \Delta} [\Delta^n, A(Y_n)]
$

is a commuting diagram

$$
  \array{
    \uparrow &&&& \uparrow && \uparrow
    \\
    [2]
    &&&&
    \Delta^2 &\stackrel{f}{\to}&
    A(Y_2)
    \\
    \uparrow &&&& \uparrow && \uparrow
    \\
    [1]
    &&&&
    \Delta^1 &\stackrel{g}{\to}&
    A(Y_1)
    \\
    \uparrow &&&& \uparrow && \uparrow
    \\
    [0]
    &&&&
    \Delta^0 &\stackrel{a}{\to}&
    A(Y_0)
  }
$$

where the vertical arrows indicate all the simplicial maps of the cosimplicial objects $\Delta$ and $A(Y_\bullet)$.

So this is

* on $Y_0$ an object $a \in A(Y_0)$;

* on "double intersections" (might be a cover of double intersections) $Y_1$ a gluing isomorphism $g : \pi_1^* a \to \pi_2^* a$ which identifies the two copies of $a$ obtained by pullback along the two projection maps $\pi_1, \pi_2 : U \times_X U \to U$.

* on "triple intersections" $Y_2$ a gluing 2-isomorphism
$
 \array{
   && \pi_2^* a
   \\
   &
   {}^{\pi_{12}^* g}\nearrow
   &\Downarrow^f&
   \searrow^{\pi_{23}^* g}
   \\
   \pi_1^* a
   &&
     \stackrel{\pi_{13}^* g}{\to}
   &&
   \pi_3^* a
 }
$
which identifies the different gluing 1-isomorphisms.

And so on.

### Gluing for ordinary stacks

The article

* Sharon Hollander, _A Homotopy Theory for Stacks_ ([arXiv](http://arxiv.org/abs/math.AT/0110247))

spells out how the familiar formulation of the descent condition for ordinary [[stacks]] is equivalent to the corresponding descent condition for simplicial presheaves, discussed above.

## Codescent

Sometimes one wishes to compute the descent objects for presheaves of the form

$$
  [B(-), A] : S^{op} \to SSet
  \,,
$$

where $B : S \to [S^{op}, SSet]$ is a given presheaf-valued co-presheaf. For instance in the context of [[schreiber:Differential Nonabelian Cohomology|differential nonabelian cohomology]] one is interested in the co-presheaf that assigns [[fundamental ∞-groupoids]]

$$
  B := \Pi : U \mapsto (V \mapsto S(V \times \Delta^\bullet, U))
$$

in which case the presheaf

$$
  [\Pi(-),A]
$$

would assign to $U \in S$ the pre-$\infty$-stack of "trivial $A$-principial bundles with flat connection".

For $Y \to X$ a given (hyper)cover, the descent object for $[B(-), A]$ can be expressed as

$$
  \begin{aligned}
    Desc(Y_\bullet \to X, [B(-), A])
    & := lim^\Delta [B(Y_\bullet),A]
    \\
    & \simeq \int^{[n]\in \Delta} [ \Delta^n, [B(Y_n),A] ]
    \\
    & \simeq \int^{[n]\in \Delta}
      [ \Delta^n \otimes B(Y_n), A]
    \\
    & \simeq [\int_{[n] \in \Delta} \Delta^n \otimes B(Y_n)\;,\; A ]
     \\
      & \simeq [colim^\Delta B(Y_\bullet), A]
  \end{aligned}
  \,.
$$

This way the descent for $[B(-),A]$ on the object $U = colim^\Delta U_\bullet$ is reexpressed as descent for $A$ of the $B$-modified object $colim^\Delta B(Y_\bullet)$. Following Street, this we may call the **[[codescent]]** object, as it co-represents descent.  See also [[pseudo-extranatural transformation]].

## Monadic descent

In some context the descent condion may algebraically be encoded in an [[adjunction]]. This leads to the notion of [[monadic descent]]. See there for more details.

## Related concepts

* **descent**

  * [[cover]]

  * [[cohomological descent]]

  * [[fibered category]], [[descent morphism]]
  * [[Bénabou-Roubaud theorem]]
  * [[monadic descent]]:

    * [[Sweedler coring]], [[Amitsur complex]],
    * [[higher monadic descent]], [[descent in noncommutative algebraic geometry]]

  * [[van Kampen colimit]]
  * [[descent along a torsor]], [[Schneider's descent theorem]]

* [[sheaf]], [[(2,1)-sheaf]], [[2-sheaf]] [[(∞,1)-sheaf]], [[stack]]

## References

The original references:

* {#Grothendieck60} [[Alexander Grothendieck]], *Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats*, Séminaire [[N. Bourbaki]] exp. no190 (1960) 299-327 [[numdam:SB_1958-1960__5__299_0](http://www.numdam.org/item/?id=SB_1958-1960__5__299_0)]

* [[A. Grothendieck]], M. Raynaud et al. _Revêtements étales et groupe fondamental_ ([[SGA1]]), Lecture Notes in Mathematics __224__, Springer 1971 (retyped as [math.AG/0206203](http://arxiv.org/abs/math/0206203); published  version Documents Mathématiques __3__, Société Mathématique de France, Paris 2003)

* [[John Duskin]], *An outline of a theory of higher dimensional descent*, Bull. de la Soc. Math. de  Belgique **41** (1989) 249-277

*  [[Angelo Vistoli]], _Grothendieck topologies, fibered categories and descent theory_ [MR2223406](http://www.ams.org/mathscinet-getitem?mr=2223406); [math.AG/0412512](http://arxiv.org/abs/math/0412512) pp. 1--104 in Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, _Fundamental algebraic geometry. Grothendieck's [[FGA explained]]_, Mathematical Surveys and Monographs __123__, Amer. Math. Soc. 2005. x+339 pp. [MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001)

* [[Jacob Lurie]], _[[Descent Theorems]]_

* Daniel Schäppi, _Descent via Tannaka duality_, [arxiv/1505.05681](http://arxiv.org/abs/1505.05681)

The connection between the concept of monadicity and descent is proved in:

* [[Jean Bénabou]], [[Jacques Roubaud]], _Monades et descente_, C. R. Acad. Sc. Paris, t. 270 (12 Janvier 1970), Serie A, 96--98, ([link](http://gallica.bnf.fr/ark:/12148/bpt6k480298g/f100), Bibliothèque nationale de France)

Making good use of this connection, we have:

* {#JanelidzeTholen04}[[George Janelidze]], [[Walter Tholen]], _Facets of Descent III_, Applied Categorical Structures __12__ (2004) 461–-477

A survey, with lots of new results, of the categorical perspective on descent theory can be found in:

* [[George Janelidze]], Manuela Sobral, [[Walter Tholen]], _Beyond Barr exactness: effective descent morphisms_, Ch. 8 of Categ. Foundations, (eds. Maria Cristina Pedicchio, Walter Tholen) Enc. Math. Appl. 2003

Further developments with categorical perspectives and generalizations can be found in:

* {#JanelidzeTholen97}[[George Janelidze]], [[Walter Tholen]], _Facets of Descent II_, Applied Categorical Structures __5__:3 (1997) 229-248

* [[Ross Street]], _Categorical and combinatorial aspects of descent theory_, [arXiv:math.CT/0303175](http://arxiv.org/abs/math.CT/0303175)

* [[Fernando Lucatelli Nunes]], *Pseudo-Kan extensions and descent theory*, 2018, TAC [33-15](http://www.tac.mta.ca/tac/volumes/33/15/33-15abs.html)

* [[Renato Betti]]. _Adjointness in descent theory_, Journal of Pure and Applied Algebra 116.1-3 (1997): 41-47.

[[!redirects descent and codescent]]
[[!redirects codescent]]
[[!redirects descent theory]]
[[!redirects ∞-descent]]


descent algebra

If $W$ is a finite [[Coxeter group]] then the associated __Solomon's descent algebra__ is certain subalgebra $\Sigma[W]$ of the integral [[group algebra]] $\mathbf{Z}[W]$ discovered/introduced in

* L. Solomon, _A Mackey formula in the group ring of a Coxeter group_, J. Algebra __41__, (1976), 255-268, [MR0444756](http://www.ams.org/mathscinet-getitem?mr=444756), doi

Other literature

* Mike Atkinson, _A new proof of a theorem of Solomon_, Bull. London Math. Soc. __18__ (1986), 351-354; _Solomon's descent algebra revisited_, Bull. London Math. Soc. 24 (1992), 545-551.
* Goetz Pfeiffer, _A quiver presentation for Solomon's descent algebra_, [arxiv/0709.3914](http://arxiv.org/abs/0709.3914)
* [[Israel Gelfand]], D. Krob, Alain Lascoux, B. Leclerc, [[Vladimir Retakh|V. S. Retakh]], J.-Y. Thibon, _Noncommutative symmetric functions_, [hep-th/9407124](http://arxiv.org/abs/hep-th/9407124)

[[!redirects Solomon's descent algebra]]


descent along a torsor

Let $C$ be a category equipped with [[Grothendieck topology]] $\tau$, and $\pi:\mathcal{F}\to C$ a [[fibered category]] which is in fact a [[stack]] over the Grothendieck [[site]] $(C,\tau)$. Let $G$ be an [[group object]] in $C$ and $P$ a $G$-[[torsor]] in $C$ over $X\cong P/G$.
Then the $G$-equivariant fiber over $P$ (i.e. the category $F^G(P)$ of $G$-[[equivariant object]]s over $P$ in the fibered category $\mathcal{F}$ over $C$)
is equivalent to the usual fiber $\mathcal{F}(U)= \pi^{-1}(id_U)$ over $X$; this statement is referred to as the descent along the $G$-torsor $P$.

In particular, the assignment of the category of quasicoherent sheaves to every scheme gives a pseudofunctor over the categroy of schemes with the faithfully flat topology. In the faithfully flat topology, for an algebraic group scheme $G$ and a $G$-torsor $P\to X$ in faithfully flat topology, the category of equivariant quasicoherent sheaves over $P$ is equivalent to the category of quasicoherent sheaves over $X$.

Sometimes, the theorem on the descent along torsors is quite imprecisely referred to as the equivariant descent theorem.

An analogue in affine noncommutative algebraic geometry is the [[Schneider's descent theorem]] relating relative [[Hopf module]]s over the Hopf-Galois extension with the usual modules over the base. It has generalizations for other distributive laws, e.g. when the Hopf modules are replaced by [[entwined modules]]; also the generalization when
the Hopf algebras are replaced by weak Hopf algebras, Hopf algebroids and alike. There
are also globalized (non-affine) versions using coaction [[compatible localization]]s.

*  [[Angelo Vistoli]], _Grothendieck topologies, fibered categories and descent theory_ [MR2223406](http://www.ams.org/mathscinet-getitem?mr=2223406); [math.AG/0412512](http://arxiv.org/abs/math/0412512) pp. 1--104 in Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, _Fundamental algebraic geometry. Grothendieck's [[FGA explained]]_, Mathematical Surveys and Monographs __123__, Amer. Math. Soc. 2005. x+339 pp. [MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001)
* [[Tomasz Brzeziński]], _On synthetic interpretation of quantum principal bundles_, AJSE D - Mathematics 35(1D): 13-27, 2010 [arXiv:0912.0213](http://arxiv.org/abs/0912.0213)
* [[Z. Škoda]], _Some equivariant constructions in noncommutative algebraic geometry_ Georgian Mathematical Journal __16__ (2009), No. 1, 183–202, [arXiv:0811.4770](http://arxiv.org/abs/0811.4770) [MR2011b:14004](http://www.ams.org/mathscinet-getitem?mr=2527623)

[[!redirects descent along torsors]]


descent and codescent > history

< [[descent and codescent]]
[[!redirects descent and codescent -- history]]


descent and locality - contents

**[[localization]]**

* [[local object]], [[local morphism]]

* [[reflective localization]]

  * [[category of sheaves]], [[(∞,1)-category of (∞,1)-sheaves]]

* [[Bousfield localization of model categories|Bousfield localization]], [[simplicial localization]]

**[[descent]]**

* [[cover]]

* [[descent object]], [[descent morphism]]

  * [[matching family]]

* [[sheaf]], [[(2,1)-sheaf]]/[[stack]], [[2-sheaf]],[[(∞,1)-sheaf]]/[[∞-stack]]

* [[cohomological descent]]

* [[monadic descent]], [[higher monadic descent]]

  * [[Sweedler coring]], [[descent in noncommutative algebraic geometry]]

[Edit this sidebar](/nlab/edit/descent+and+locality+-+contents)


descent for L-infinity algebras

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--

> under construction

> so far these are notes taken in talks by [[Ezra Getzler]]

#Contents#
* table of contents
{:toc}

## $L_\infty$-algebras

The notion of [[L-infinity algebra]] is something that naturally arises in [[deformation theory]]
and in [[descent]] problems.

A [[dg-Lie algebra]] is a [[chain complex]] of [[vector space]]s

$$
  \cdots
  \to
  V^{-1}
  \stackrel{d}{\to}
  V^0
  \stackrel{d}{\to}
  V^1
  \stackrel{d}{\to}
  V^2
  \to
  \cdots
$$

and is equipped with a bracket operation

$$
  [-,-] : V^i \otimes V^j \to V^{i + j}
$$

which is

* bilinear

* graded antisymmetric: $[x,y] = - (-1)^{deg(x) deg(y)} [y,x]$

* satisfies the graded Jacobi identity $[x,[y,z]] = [[x,y],z] + (-1)^{deg(x) deg(y)} [y,[x,z]]$.

* is graded Leibnitz: $d[x,y] = [d x,y] + (-1)^{deg(x)} [x, d y]$.
(aka a graded derivation)

**Note** If $deg(x)$ is odd, then $[x,x]$ need not vanish. (see also [[super Lie algebra]]).

Let $L$ be a [[dg-Lie algebra]], degreewise finite dimensional ("of finite type" in the language of [[rational homotopy theory]]).

We can form its [[Chevalley-Eilenberg algebra]] (see there for details) of cochains

$$
  CE(L) = (\wedge^\bullet L[1]^*, d)
$$

a [[quasi-free dga]].

(N.B. In full generality, read this as

$$
  CE(L) = ((\wedge^\bullet L[1])^*, \delta)
$$
and regard
$$
 \wedge^\bullet L[1]
$$
as a graded commutative COalgebra.)

The underlying graded algebra we may dually think of as functions on some
space, a (so-called) **formal graded manifold**.

The total differential

$$
  \delta = \delta_1 + \delta_2
$$

where the first is $d$ and the second is the dual of the bracket: $[-,-]^*$ extended as a
graded derivation.

Being a [[derivation]], dually we may think of this as a [[vector field]]
on our formal smooth manifold. This is sometimes called an [[NQ-supermanifold]].

**Definition**

An [[L-infinity algebra]] (see there)
of finite type  is the evident generalization of this
inroduced by [[Jim Stasheff]] and [[Tom Lada]]

what kind of link is needed here??

 in the early 1990s:

An [[L-infinity algebra]]  is equivalent to  ( in the degreewise finite dimensional case) a
free graded-commutative algebra equipped with a differential of degree +1.

Now the differential corresponds to a sequence of $n$-ary brackets. For $n = 1$ this is
the differential on the complex, for $n = 2$ this is the binary bracket from
above, and then there are higher brackets.

### Morphisms of $L_\infty$-algebras

One can consider two notions of morphisms: strict ones and general ones.

A strict one would be a linear map of the underlying vector spaces that
strictly preserves all the brackets.

A general definitin of morphisms is: in terms of the dual [[dg-algebra]]s
just a morphism of these, going the opposite way. In the dual formulation this
is due to Lada and Stasheff.

We may also think of this as a morphism of [[NQ-supermanifold]]s.

All this arose in this form probably most vivedly in the [[BFV-BRST formalism]] or in the [[BV-BRST formalism]].

So in components such a morphism $f : L \to K$ of $L_\infty$-algebras
consists of $n$-ary maps

$$
 f_k : L^{i_1} \otimes \cdots L^{i_k} \to K^{i_1 + \cdots + 1 - k}
$$

(where the shift in the indices is due to the numbering convention here only).

### The homotopical category of $L_\infty$-algebra

We will now describe on the [[category]] of $L_\infty$-algebras the structure of
a [[category of fibrant objects]].

The issue is that the category of $L_\infty$-alghebras as defined above has not
all products and coproducts.

But we can turn it into a [[category of fibrant objects]].

#### A notion of category of fibrant objects {#GetzlerCatFibOb}

This is analogous to (in fact an example of the same general fact)
how [[Kan complex]]es inside all [[simplicial set]]s
are the fibrant objects of the [[model structure on simplicial sets]] but do
not form among themselves a [[model category]] but a
[[category of fibrant objects]].

See [[Kan complex]] for more...

We now look at the axioms for our category of fibrant objects. It is a slight
variant of those in [[BrownAHT]] described at
[[category of fibrant objects]] and draws bit from work of Dwyer and Kan.

Let $C$ be a [[category]]. The axioms used here are the following.

1. There is a [[subcategory]] $W \subset C$
   whose morphisms are called **weak equivalences**,
   such that this makes $C$ into a
   [[category with weak equivalences]].

1. There is another [[subcategory]] $F \subset C$, whose morphisms
   are called  **fibrations** (and those that are also
   in $W$ are called **acyclic fibrations**) , such that

   * it contains all [[isomorphism]]s;

   * the [[pullback]] of a fibration is again a fibration.

   * the pullback of an acyclic fibrations is an acyclic fibration.

1. $C$ has all [[product]]s and in particular a [[terminal object]] $*$.

#### Filtered $L_\infty$-algebras as a Getzler-category of fibrant objects {#Filtered}

Write $\mathbb{L}$ for the category of _filtered_ [[L-infinity algebra]]s

Let $L^\bullet$ be a [[graded vector space]]

a decreasing filtrration on it is

$$
  L = F^0 L \supset F^1 L \supset \cdots
$$

such that $L$ is the [[limit]] over this

$$
  L \simeq \lim_{\leftarrow} L/F^1 K
$$

i.e. if $(x_i \in F^i L)$ then $\sum_{i=0}^\infty x_i$ exists

something missing here

$$
  [-,- , \dots ]_k
$$

has filtration degree 0 if $k \gt 0$ or filtration degree 1 if $k = 0$.

The differential $d x = [x]_1$ has the property

namely?

Then $gr(d)$ is a  true differential $gr(L)$

Morphisms:

$$
  f_k : L^{\otimes k} \to N
$$

where we take $f_k$ to have filtration degree 0 for $k \gt 0$ and filtration degree 1
for $k = 0$.

So $gr(f_1)$ is a morphism of complexes from $gr(L)$ to $gr(N)$.

**Definition** A morphism $f$ is a **weak equivalence** if $gr(f)$ is a
[[quasi-isomorphism]] of complexess.

It is a **fibration** if $gr(f_1)$ is surjective.

**Theorem** This defines the structure of
a (Getzler-version of a) [[category of fibrant objects]]
as defined above.

Given $C$ be a Getzler-[[category of fibrant objects]].

Define a new Getzler-category of fibrant objects $s C$ as follows:

* The [[object]]s  of $s C$ are [[simplicial object]]s in $C$, subject to
  a condition stated in the following item.

* As in the [[Reedy model structure]], for $X_\bullet$ a simplicial objects
  let $M_k X_\bullet$ be the corresponding _matching object_ , defined by the
  [[pullback]] diagram

  $$
    \array{
      M_k X_\bullet &\to& (X_{k-1})^{k+1}
      \\
      \downarrow && \downarrow
      \\
      (X_{k-2})^{\left(k+1 \atop 2\right)}&\to& (M_{k+1} X_\bullet)^{k+1}
    }
    \,.
  $$

  Here the right vertical morphism is assumed to be a fibration, hence so
  is the left vertical morphism.

  So $M_k X_\bullet$ comes with a map $X_k \to M_k X_\bullet$. We assume that this
  is a fibration. This allows us to define $M_{k+1} X_\bullet$ and to continue
  the induction.

  So this defines a _Reedy fibrant object_ .

So the objects of $s C$ are Reedy fibrant objects $X_\bullet$ and morphisms are morphisms
of simplicial objects.

The weak equivalences in $s C$ are taken to be the levelwise weak equivalences.

The fibrations are taken to be the Reedy fibrations, as in the [[Reedy model structure]],
i.e. those morphisms $X_\bullet \to Y_\bullet$ such that
$X_k \to Y_k \times_{M_k Y_\bullet} M_k X_\bullet$ is a fibration for all $k$.

So this is just the [[full subcategory]] of the [[Reedy model structure]] on
$[\Delta^{}op], C$ on the fibrant objects.

There is still a fourth axiom for Getzler-cats of fibrant objects to be stated, which
is the existence of [[path space object]]s. We take this to be gven by a
**path space functor**

$$
  P : C \to s C
$$

which is such that

1. for all $X$ the face maps of $(P X)_\bullet$ are weak
equivalences;

1. $P$ preserves fibrations and acyclic fibrations;

1. $(P X)_0$ is [[natural isomorphism|naturally isomorphic]] to $X$.

**Examples**

If $C$ is the category of [[Kan complex]]es, then $P_k X = sSet(\Delta[k],X)$.

**Proposition**

For our category $\mathcal{L}$ of filtered $L_\infty$-algebras we may set

$$
  P_k L = L \otimes_{compl} \Omega^\bullet(\Delta^k)
   =
   \lim_{\leftarrow} L \otimes \Omega / F^i L \otimes \Omega
   \,,
$$
where $\otimes_{compl}$ denotes the completed tensor product, more commonly denoted $\hat \otimes$.

**Remark**

We may also speak of cofibrant objects in a (Getzler-) [[category of fibrant objects]]:

those objects $X$ such that for all acyclic fibrations $f : A \to B$ the induced
map $C(X,A) \to C(X,B)$ is surjective (i.e. those with [[left lifting property]]
again acyclic fibrations).

#### Maurer-Cartan elements

All the above is designed to make the following come out right.

Generally, $C(*,X)$ is the set of points ([[global element]]s) of $X$.

A morphism from the terminal object into an $L_\infty$-algebra is a [[Maurer-Cartan element]]
in the $L_\infty$-algebra.

Such a point is just an element of degree 1 and filtration degree 1 that satisfies the equation

$$
  \sum_{k= 0}^{\infty} \frac{1}{k!}
   [\omega, \cdots, \omega]_k = 0
   \,.
$$

In the case of [[dg-Lie algebra]]s, this is just the familiar Maurer-Cartan equation
$$
  d \omega + \frac{1}{2}[\omega, \omega] = 0
  \,.
$$

We have that $C(*, P_k X)$ is a functor from $C$ to the category of [[Kan complex]]es.

For the category of Kan complexes, it is the identity functor.

For filtered $L_\infty$-algebras it gives

$$
  L \mapsto MC_\bullet(L) = MC(L \otimes \Omega^\bullet(\Delta^\bullet))
$$

This functor $MC_\bullet$ takes fibrations to fibrations and acyclic fibrations to
acyclic fibrations and weak equivalences to weak equivalences.

Other applications to sheaves of $L_\infty$-algebras

Evaluate on a [[Cech-nerve]] to get a cosimplicial $L_\infty$-algebra

$L^\bullet = (L^0 , L^1, \cdots)$

$L^k = \prod_{i_0, \cdots, i_k} L(U_{i_0} \cap \cdots \cap U_{i_k})$

$$
  Tot(L^\bullet) = \int_{k \in \Delta} L^k \otimes \Omega^\bullet(\Delta^k)
  \,.
$$

If $L$ is a [[dg-Lie algebra]], then

$$
  MC_1(L)
  =
  \left\{
     \omega_0 + \omega_1 d t
     |\quad
     \omega_1 \in F^1 L^1 [t],\quad \omega_1 \in F^1 L^0 [t]
     , \quad
     d_L \omega_0 + [\omega_1, \omega_1] = 0,
     \quad d_{dR} \omega_0 + [\omega_1, \omega_0] = 0
  \right\}
$$

Now define the **Deligne groupoid** as in Getzler' integration article.

We find inside the large Kan complex of MC-elements a smaller one that is still
equivalent.

$$
  \gamma_1(L)
  \left\{
    \omega \in MC_1(L) | \omega_1 is constant
  \right\}
$$

To get this impose a gauge condition known from homological perturbation theory.

A _context_ is

$$
  L \stackrel{\overset{f}{\to}}{\underset{g}{\leftarrow}} M
$$

$$
  g \circ f = Id_L
$$

$$
  f \circ g = Id - (d_M h + h d_M)
$$

$$
  g \circ h = 0, \; h \circ f = 0, \; h \circ h = 0
$$

$$
  MC(L) \simeq \{\omega \in MC(M) | h \omega = 0\}
$$

See Kuranishi's article in Annals to see where the motivation for all this comes from.

+-- {: .query}

[[Jim Stasheff]]: citation please and how much does **all** refer to??

=--

**Example** Consider the space of Schouten Lie algebras

$$
  L^k = \Gamma(X, \wedge^{k+1} T X)
$$

Then $MC_(L)$ is the set of [[Poisson manifold|Poisson brackets]] $\mathcal{O}(\hbar)$.

For let $P \in MC(L)$. Then $\pi_1(MC_\bullet(L), P)$ is the
locally Hamiltonian diffeomorphisms / Hamiltonian diffeos.

$\pi_2(MC_\bullet(L), P)$ is the set of Casimir operators of $P$.

for $k \gt 2$ $\pi_k(MC_\bullet(L), P)$.

## Descent for $L_\infty$-algebra valued sheaves {#Descent}

Associated to an [[L-infinity algebra]] $L$ is a [[Kan complex]] whose set of $k$-cells is the set of [[Maurer-Cartan element]]s on the $n$-[[simplex]]

$$
  MC_k(L) = MC( L \otimes \Omega^\bullet(\Delta^k) )
  \,.
$$

Now assume that we have a [[sheaf L-∞ algebras]] over a [[topological space]] $X$. Let $\{U_\alpha \to X\}$ be an [[open cover]] of $X$.

On $k$-fold intersections we form

$$
  L^k = \oplus_{\alpha_0,\cdots, \alpha_k}
  L(U_{\alpha_0, \cdots, \alpha_k})
  \,.
$$

The problem of [[descent]] is to glue all this to a single $L_\infty$ algebra given by the [[totalization]]  [[end]]

$$
  Tot(L^\bullet) = \int_k L^k \otimes \Omega^\bullet(\Delta^k)
$$

and check if that is equivalent to the one assigned to $X$.

We now want to compare the $\infty$-stack of $L_\infty$-algebras and that of the "integration" to the Kan complexes of Maurer-Cartan elements, so compare

$$
  MC_\bullet(Tot(L^\bullet))
$$

with

$$
  Tot(MC_\bullet(L^\bullet))
$$

Notice that we have an evident map

$$
  MC(\int_l L^l \otimes \Omega^\bullet(\Delta^l) \otimes \Omega^\bullet(\Delta^k) )
  \to
  MC(\int_l L^l \otimes \Omega^\bullet(\Delta^l \times \Delta^k))
$$

Hinich shows in a special case that this is a [[homotopy equivalence]].

It is easy to prove it for abelian $L_\infty$-algebras.

**Theorem** (Getzler)

This is indeed a homotopy equivalence.

**Proof** By E.G.'s own  account he has "a terrible proof" but thinks a nicer one using induction should be possible.

### Gauge fixing

Recall the notion of "context" from above, which is a collection of maps

$$
  L \stackrel{\overset{f}{\to}}{\underset{g}{\leftarrow}}
  M
  \stackrel{h}{\to} M
$$

between filtered complex-like things,

+-- {: .query}

meaning?? more general or complex with additional structure??

=--

satisfying some conditions.

We can arrange this such that $h$ imposes a certain gauge condition on $L$, or something

> I missed some details here...

**Theorem**

With

$$
  MC(M,h) :=
  \left\{
    \omega \in MC(M) |
    h \omega = 0
  \right\}
$$

we have

$$
  g : MC(M,h) \stackrel{\simeq}{\to} MC(L)
$$

$$
  MC_\bullet(M,h) \stackrel{g \simeq}{\to} MC_\bullet(L)
  \stackrel{MC_\bullet(f) \simeq}{\to}
  MC_\bullet(M)
$$

**Proof** Along the lines of Kuranishi's construction:

$$
 h
  \left(
    [-]_0 + d_M \omega + \sum_{k = 1}^\infty
    \frac{1}{k!} [\omega, \cdots , \omega]^h
  \right)
  = 0
  \,.
$$

So the big $\infty$-groupoid that drops out of the integration procedure is equivalent to the smaller one which is obtained from it by applying that gauge fixing condition.

It would be nice if in the definition of the MC complex we could replace differential forms on the $n$-simplex with just [[cochain on a simplicial set|simplicial cochains]] on $\Delta[n]$.

$$
  MC(L \otimes C^\bullet(\Delta^\bullet))
  \,.
$$

This would make the construction even smaller.

+-- {: .query}

What's the problem?

=--

If $L$ is abelian, then this is the [[Eilenberg-MacLane space]] which features in the [[Dold-Kan correspondence]].

This is true if one takes care of some things. This is part of the above "terrible proof".

Because one can prove that using explicit Eilenberg-MacLane's homotopies that proove the [[Eilenberg-Zilber theorem]] in terms of [[cochain on a simplicial set|simplicial cochains]] we have an equivalence

$$
  MC(
    L \otimes C^\bullet(\Delta[k] \otimes \Delta[l])
  )
  \to
  MC( L \otimes C^\bullet(\Delta[k] \times \Delta[l]))
$$

## References

The discussion of the Deligne groupoid (the $\infty$-groupoid "integrating" an $L_\infty$-algebra) and the gauge condition on the Maurer-Cartan elements is

* [[Ezra Getzler]], _Lie theory for nilpotent L-infinity algebras_ ([arXiv:0404003](http://arxiv.org/abs/math/0404003))

> A reference for the theorem above seems not to be available yet, but I'll check.

[[!redirects descent for L-∞ algebras]]


descent for simplicial presheaves

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[simplicial presheaf|Simplicial presheaves]] equipped with the [[model structure on simplicial presheaves]] are one model/presentation for the [[(∞,1)-category of (∞,1)-sheaves]] on a given [[site]].

The fibrant object $\bar X$ that a [[simplicial presheaf]] $X : S^{op} \to SSet$ is weakly equivalent to with respect to this model structure is the [[∞-stackification]] of $X$. One expects that [[∞-stacks]]/[[(∞,1)-sheaves]] are precisely those [[(∞,1)-presheaves]] which satisfy a kind of [[descent]] condition.

Precsisely what this condition is like for the particular model constituted by simplicial presheaves with the given Jardine [[model structure on simplicial presheaves]] was worked out in

* Daniel Dugger, Sharon Hollander, Daniel C. Isaksen, _Hypercovers and simplicial presheaves_ ([web](http://www.math.uiuc.edu/K-theory/0563/))

and

* Toën-Vezzosi, _Segal topoi and stacks over Segal categories_ ([pdf](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1046.508&rep=rep1&type=pdf))

recalled as corollary 6.5.3.13 in

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

The following is a summary of these results.

The main point is that the fibrant objects are essentially those simplicial presheaves, which satisfy [[descent]] with respect not just to [[covers]], but to [[hypercovers]].

Localizations of [[(∞,1)-presheaves]] at [[hypercovers]] are called [[hypercompletions]] in [section 6.5.3](http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf#page=534) of [[Higher Topos Theory]]. Notice that in [section 6.5.4](http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf#page=539) of [[Higher Topos Theory]] it is argued that it may be more natural _not_ to localize at [[hypercovers]], but just at [[covers]] after all.

## Details

A well-studied class of models/presentations for an [[(∞,1)-category of (∞,1)-sheaves]] is obtained using the [[model structure on simplicial presheaves]] on an  ordinary (1-categorical) [[site]] $S$, as follows.

Let $[S^{op}, SSet]$ be the [[SimpSet|SSet]]-[[enriched category]] of [[simplicial presheaf|simplicial presheaves]] on $S$.

Recall from [[model structure on simplicial presheaves]] that there is the _global_ and the _local_ injective simplicial model structure on $[S^{op}, SSet]$, and that the local model structure is a (Bousfield-)localization of the global model structure.

According to [section 6.5.2](http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf#page=528)  of [[Higher Topos Theory|HTT]] we have:

* the full simplicial subcategory on fibrant-cofibrant objects of $[S^{op}, SSet]$ with respect to the _global_ injective model structure is (the [[SimpSet|SSet]]-[[enriched category]] realization of) the $(\infty,1)$-category  $PSh_{(\infty,1)}(S)$ of [[(∞,1)-presheaves]] on $S$.

* the full simplicial subcategory on fibrant-cofibrant objects of $[S^{op}, SSet]$ with respect to the _local_ injective model structure is (the [[SimpSet|SSet]]-[[enriched category]] realization of) the $(\infty,1)$-category  $\bar{Sh}_{(\infty,1)}(S)$ which is the [[hypercompletion]] of the $(\infty,1)$-category $Sh_{(\infty,1)}(S)$ of [[(∞,1)-sheaves]] on $S$.

Since with respect to the local or global injective model structure all objects are automatically cofibrant, this means that $\bar Sh_{(\infty,1)}(S)$ is the full sub-$(\infty,1)$-category of $PSh_{(\infty,1)}(S)$ on simplicial presheaves which are fibrant with respect to the local injective model structure: these are the [[∞-stacks]] in this model.

By the general properties of [[localization of an (∞,1)-category]] there should be a class of morphisms $f : Y \to X$ in $PSh_{(\infty,1)}(S)$ -- hence between injective-fibrant objects in $[S^{op}, PSh(S)]$ -- such that the simplicial presheaves representing $\infty$-stacks are precisely the [[local objects]] with respect to these morphisms.

This was worked out in

* D. Dugger, S. Hollander, D. Isaksen, _Hypercovers and simplicial presheaves_ ([pdf](http://hopf.math.purdue.edu//Dugger-Hollander-Isaksen/hypspre.pdf))

We now describe central results of that article.

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

For $X \in S$ an object in the [[site]] regarded as a simplicial presheaf and $Y \in [S^{op}, SSet]$ a simplicial presheaf on $S$, a morphism $Y \to X$ is a **[[hypercover]]** if it is a _local acyclic fibration_, i.e. of for all $V \in S$ and all diagrams
$$
  \array{
    \Lambda^k[n]\otimes V &\to & Y
    \\
    \downarrow && \downarrow
    \\
    \Delta^n\otimes V &\to&
    X
  }
  \;\; respectively \;\,
  \array{
    \partial \Delta^n\otimes V &\to & Y
    \\
    \downarrow && \downarrow
    \\
    \Delta^n\otimes V &\to&
    X
  }
$$
there exists a covering  [[sieve]] $\{U_i \to V\}$ of $V$ with respect to the given [[Grothendieck topology]] on $S$ such that for every $U_i \to V$ in that [[sieve]] the pullback of the abve diagram to $U$ has a lift
$$
  \array{
    \Lambda^k[n]\otimes U_i &\to & Y
    \\
    \downarrow &\nearrow & \downarrow
    \\
    \Delta^n\otimes U_i &\to&
    X
  }
  \;\; respectively \;\,
  \array{
    \partial \Delta^n\otimes U_i &\to & Y
    \\
    \downarrow &\nearrow& \downarrow
    \\
    \Delta^n\otimes U_i &\to&
    X
  }
  \,.
$$
=--

If $S$ is a [[Verdier site]] then every such hypercover $Y \to X$ has a refinement by a hypercover which is cofibrant with respect to the projective global [[model structure on simplicial presheaves]]. We shall from now on make the assumption that the hypercovers $Y \to X$ we discuss are cofibrant in this sense. These are called _split hypercovers_. (This works in many cases that arise in practice, see the discussion after [DHI, def. 9.1](http://hopf.math.purdue.edu//Dugger-Hollander-Isaksen/hypspre.pdf#page=29).)

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

The objects of $Sh_{(\infty,1)}(S)$ -- i.e. the fibrant objects with respect to the projective model structure on $[S^{op}, SSet]$ -- are precisely those objects $A$ of $PSh_{(\infty,1)}(S)$ -- i.e. [[Kan complex]]-valued simplicial presheaves -- which
**satisfy descent for all split hypercovers**, i.e. those for which for all split hypercover $f : Y \to X$ in $[S^{op}, SSet]$ we have that
$$
  [S^{op}, SSet](X,A) \stackrel{\simeq}{\to}
  [S^{op}, SSet](Y,A)
$$
is a [[model structure on simplicial sets|weak equivalence of simplicial sets]].
=--

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

This is [DHI, thm 1.3](http://hopf.math.purdue.edu//Dugger-Hollander-Isaksen/hypspre.pdf#page=3) formulated in the light of [DHI, lemma 4.4 ii)](http://hopf.math.purdue.edu//Dugger-Hollander-Isaksen/hypspre.pdf#page=9).
=--

Notice that by the [[co-Yoneda lemma]] every simplicial presheaf $F : S^{op} \to SSet$, which we may regard as a presheaf $F : \Delta^{op}\times S^{op} \to Set$, is isomorphic to the [[weighted limit|weighted colimit]]
$$
    F \simeq colim^\Delta F_\bullet
$$
which is equivalently the [[end|coend]]
$$
  F \simeq \int^{[n] \in \Delta} \Delta^n \cdot F_n
  \,,
$$
where $F_n$ is the Set-valued presheaf of $n$-cells of $F$ regarded as an $SSet$-valued presheaf under the inclusion $Set \hookrightarrow SSet$, and
where the [[SimpSet|SSet]]-weight is the canonical cosimplicial simplicial set $\Delta$, i.e. for all $X \in S$
$$
  F : X \mapsto \int^{[n] \in \Delta} \Delta^n \times F(X)_n
  \,.
$$
In particular therefore for $A$ a [[Kan complex]]-valued presheaf the descent condition reads
$$
  [S^{op}, SSet](X,A) \stackrel{\simeq}{\to}
  [S^{op}, SSet](colim^\Delta Y_\bullet,A)
  \simeq
  lim^\Delta [S^{op}, SSet](Y_\bullet,A)
  \,.
$$

With the shorthand notation introduced above the **descent condition** finally reads, for all global-injective fibrant simplicial presheaves $A$ and hypercovers $U \to X$:
$$
  A(X) \stackrel{\simeq}{\to} lim^\Delta A(Y_\bullet)
  \,.
$$
The right hand here is often denoted $Desc(Y_\bullet \to X, A)$, in which case this reads
$$
  A(X) \stackrel{\simeq}{\to} Desc(Y_\bullet \to X, A)
  \,.
$$

## In terms of homotopy limit

(expanded version of remark 2.1 in [DHI](http://hopf.math.purdue.edu//Dugger-Hollander-Isaksen/hypspre.pdf))

Using the [[Bousfield-Kan map]] every simplicial presheaf $F$ is also weakly equivalent to the weighted limit over $F_\bullet$ with weight given by $N(\Delta/(-)) : \Delta \to SSet$.

$$
  lim^{N(\Delta/(-))} F_\bullet \stackrel{\simeq}{\to}
  lim^\Delta F_\bullet
  \,.
$$

But by the discussion at [[weighted limit]], the left hand computes the [[homotopy limit]] of $F_\bullet$ (since $F_\bullet$ is objectwise fibrant, since $F_n$ factors through $Set \hookrightarrow SSet$), hence we have a weak equivalence

$$
  holim F_\bullet \stackrel{\simeq}{\to}
  F
  \,.
$$

Often the descent condition is therefore formulated with the cover $U$ replaced by its homotopy limit, whence it reads

$$
  [S^{op}, SSet](X,A) \stackrel{\simeq}{\to}
  [S^{op}, SSet](hocolim U_\bullet,A)
  \,.
$$

With $A$ global-injective fibrant this is equivalent to

$$
  [S^{op}, SSet](X,A) \stackrel{\simeq}{\to}
  holim [S^{op}, SSet](U_\bullet,A)
  \,.
$$

Using the notation introduced above this becomes finally

$$
  A(X) \stackrel{\simeq}{\to} holim A(U_\bullet)
  \,.
$$

## Related concepts

* [[model structure on simplicial presheaves]]

* [[(infinity,1)-sheaf|$(\infty,1)$-sheaf]]

[[!redirects homotopy descent]]


descent in noncommutative algebraic geometry

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Gluing of quasicoherent sheaves via localizations

The [[quasicoherent sheaves]] have natural analogues in some formalisms of [[noncommutative algebraic geometry]]. The Zariski open sets generalize to some nice class of flat localizations, and then one can consider faithfully flat descent. However the analogues of a cover and sheaf condition are more subtle.

Let $A$ be an [[abelian category]] and $\{B_\lambda\}_{\lambda\in\Lambda}$ a finite family of abelian categories; and $Q^*_\lambda : A\to B_\lambda$ exact localization functors having fully faithful right adjoints $Q_{\lambda *}:B_\lambda\to A$. We say that the family ${Q_{\lambda *}}_{\lambda\in\Lambda}$ is conservative if a morphism in $A$ is invertible iff $Q_{\lambda *}(f)$ is invertible for each $\lambda$.

There is a standard way to construct a [[comonad]] $G$ on the product category $\prod_\lambda B_\lambda$. Under the conditions of a comonadic version of Barr's monadicity theorem one can reconstruct $A$ as the category equivalent to the category of $G$-comodules.

For Gabriel localizations of categories of modules over rings, such a gluing theorem, called "globalization theorem" is in A. Rosenberg's 1988 Stockholm book and obtained independently in Willaert-van Oystaeyen 1995 article, in both cases proved by nontrivial localization theory. Applying [[comonad|comonadicity]] theorems is much more straightforward.

On the other hand the family of all cocovers (conservative families) by flat localizations and the family of all nice localizations do not have as good properties as the families of open sets and covers in topology. For example, the intersection operation on open sets is commutative, while the consecutive application of two localization functors depends on the order among the two in general. For example, the [[Ore localization]] of a ring $R$ on a left Ore subset $S\subset S^{-1}R$ is a ring; but the localization of that localized ring $S^{-1}R$ by applying the localization functor for another Ore subset $T\subset R$ is $T^{-1}R\otimes_R S^{-1} R$ which is not necessarily a ring in a natural way. In fact it has a canonical structure of a ring exactly in the case when the two Ore localization functors commute as functors on the original category of left modules over $R$. The considerations of this compatibility of localization functors has been studied in the language of torsion theories by Verschoeren and van Oystaeyen in 1970s. In fact, this situation is akin to semiseparatedness condition on schemes introduced by Thomason-Trobaugh: a scheme is semiseparated if it has an affine cover where the intersection of any two affines is again affine. Every separated scheme is semiseparated. Examples of [[noncommutative scheme|noncommutative schemes]] are rarely semiseparated in the sense of cocovers by flat localizations: for example most quantum flag varieties are not semiseparated.

Another problem is that the family of all cocovers by flat  localizations (even for the categories of modules) is typically not a Grothendieck cotopology: in the dual language of topologies the stability axiom fails, even if one modifies the notion of the pullback form categorical product to the dual of the operation of meet for torsion theories. In terms of torsion theories, this stems from the fact that the lattice of torsion theories is not a distributive lattice.

* Z. Škoda, _Noncommutative localization in noncommutative geometry_,  London Math. Society Lecture Note Series 330 ([pdf](http://www.maths.ed.ac.uk/~aar/books/nlat.pdf)), ed.  A. Ranicki; pp. 220--313, [math.QA/0403276](http://arxiv.org/abs/math.QA/0403276).

## Flat descent for modules over rings

For the purposes of the descent for modules over rings, comonads can be replaced by [[corings]]. Alternatively the descent data can be represented by connections for the Amitsur complex. Symmetries are important in that analysis.

* P. Nuss, _Noncommutative descent and non-abelian cohomology_,  $K$-Theory __12__  (1997),  no. 1, 23--74.

* T. Brzeziński, R. Wisbauer, _Corings and comodules_, London Math. Soc. Lec. Note Series 309, Cambridge 2003.

* C. Menini, D. Ştefan, _Descent theory and Amitsur cohomology of triples_,
J. Algebra __266__ (2003), no. 1, 261--304.

## (Co)monadic descent and noncommutative schemes

The monadicity and comonadicity properties are at heart of the definition of a relative [[noncommutative scheme]]

* [[A. L. Rosenberg]], _Noncommutative schemes_, Compos. Math. __112__ (1998) 93–125, [doi](http://dx.doi.org/10.1023/A:1000479824211)

## Galois descent

In noncommutative case, the faithfully flat descent along torsor (equivalence of the $G$-equivariant quasicoherent sheaves on the $G$-scheme $X$ and the quasicoherent sheaves on $X/G$) is well understood when the group is replaced by (the opposite) to a [[Hopf algebra]] and when the base and total [[noncommutative scheme]]s are affine. This is the case of [[Hopf-Galois extension|Hopf-Galois descent]] and generalizations for entwinings. One can combine this vertical descent with the horizontal descent on localizations, provided the localizations are compatible with the coactions of the Hopf algebra.

Coring language is also very natural in generalizing the Galois theory to noncommutative rings.

## Nonflat descent

Unlike the descent for the categories of quasicoherent sheaves, where apart from semiseparatedness, the descent problem has a natural solution (say in comonadic language) the descent for affine schemes fails. In that case, Grothendieck essentially used the symmetry in commutative case. Another problem is that localizations are hard to find, especially very noncommutative rings (roughly those of infinite Gelfand-Kirillov dimensions) do not have nice covers by flat localizations. Cohn localizations sometimes work but they are not flat; therefore the descent has to be replaced by some sort of nonflat descent, for example the descent at the [[derived algebraic geometry|derived]] level.
That kind of analogues are not really noncommutative schemes as they are not locally affine in Zariski-like (=faithfully flat by localizations) topology.

## Gluing for sheaves of sets on $NAff$

Yet another question is how to generalize the functor of points approach to [[schemes]] and locally affine spaces in some Grothendieck topology on $Aff$ (or $Aff/S$) to the noncommutative setup. Rosenberg for that reason introduced considerations of sheaves on/in [[Q-categories]] which can be used to define noncommutative algebraic spaces of various kinds in functor of points approach. In some cases, like noncommutative smooth topology (Kontsevich-Rosenberg) usual Grothendieck topologies on $NAff$ suffice.

## Related entries

* [[descent]]: [[cover]], [[monadic descent]], [[Sweedler coring]], [[higher monadic descent]], [[Eilenberg-Moore category]], [[Hopf-Galois extension]]
* [[localization]]
* [[sheaf]], [[(2,1)-sheaf]], [[2-sheaf]] [[(∞,1)-sheaf]]

[[!redirects gluing in noncommutative algebraic geometry]]
[[!redirects descent in noncommutative geometry]]


descent morphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Descent and locality
+-- {: .hide}
[[!include descent and locality - contents]]
=--
=--
=--

# Descent morphisms
* table of contents
{: toc}

## Idea

Given a [[fibered category]], a morphism along which the induced [[comparison functor]] between the [[category of descent data]] and the [[codomain fibration]] is [[fully faithful functor|fully faithful]] (resp. an [[equivalence of categories]]) is said to be a *descent morphism* (resp. *effective descent morphism*).

## The case of codomain fibration

### Definition
 {#Definition}

Let $C$ be a [[category]] with [[pullbacks]] and [[coequalizers]].  For any [[morphism]] $p\colon A\to B$, we have an [[internal category]] $ker(p)$ defined by $A\times_B A \rightrightarrows A$ (the [[kernel pair]] of $p$).  The [[category of descent data]] for $p$ is the category $C^{ker(p)}$ (the "[[descent object]]") of internal diagrams on this internal category.  Explicitly, an object of $C^{ker(p)}$ is a morphism $C\to A$ together with an action $A\times_B C \to C$ satisfying suitable axioms.

The evident internal functor $ker(p) \to B$ (viewing $B$ as a [[discrete category|discrete]] internal category) induces a *comparison functor* $C^B \to C^{ker(p)}$.  We say that $p$ is:

* a **descent morphism** if this [[comparison functor]] is [[fully faithful functor|fully faithful]], and
* an **effective descent morphism** if this comparison functor is an [[equivalence of categories]].

It is a little unfortunate that the more important notion of *effective descent* has the longer name, but it seems unwise to try to change it (although the [[Elephant]] uses "pre-descent" and "descent").

### Properties

Let $C$ be a category with pullbacks.

+-- {: .un_theorem}
###### Theorem
$p\colon A\to B$ is a descent morphism if and only if $p$ is a [[pullback-stability|stable]] [[regular epimorphism]].
=--

In particular, descent morphisms are closed under [[pullback]] and [[composition]].  Moreover, in a [[regular category]], the descent morphisms are precisely the regular epimorphisms.

Perhaps more surprising is:

+-- {: .un_theorem}
###### Theorem
Effective descent morphisms are closed under pullback and composition.
=--

See ([JT](#JT), 2.4), ([ST](#ST)) and ([RST](#RST)) for proofs.

## General case

In general, [[descent]] is about higher [[sheaf]] conditions (i.e. [[stack]] conditions). More precisely, being an $n$-stack means that all
covers in the base are effective $n$-categorical descent morphism. Hence the morphism being of effective descent is a building block, the single morphism case of a stack condition.

Thus, being an effective descent morphism says that the corresponding fibered category is a 1-stack ("2-sheaf") for the singleton covering family $p$.  Similarly, $p$ is a descent morphism iff the codomain fibration is a pre-stack (that is, a 2-separated 2-presheaf) for $p$.

More generally, we may use the terms "descent morphism" and "effective descent morphism" relativized to any [[fibration]] or [[indexed category]] rather than the codomain fibration.

We can also, of course, generalize to higher categories: an [[n-category]] with pullbacks has an analogue of a "codomain fibration", and we can ask for stack conditions on it.  This is most common in the case of [[(infinity,1)-categories]]; see the page [[descent]] for more information and links.

Descent can sometimes (for this we need to have also the direct image functor) be rephrased in terms of the [[monadicity theorem]]; see [[monadic descent]].

## Examples

If $C$ is [[exact category|exact]], or has stable [[reflexive coequalizers]], then every
[[regular epimorphism]] is an effective descent morphism.  (See, for instance, section B1.5 of the [[Elephant]].)  In particular, this is the case for any [[topos]].

However, there are also important effective descent morphisms in non-exact categories.

* In [[Top]], there are characterizations of effective descent morphisms, see [CJ20](#EDM).

* In the category [[Loc]] of [[locales]], every [[triquotient map]] is an effective descent morphism.  These includes [[open map|open]] surjections and also [[proper map|proper]] surjections.

...

Of course, there are also many effective descent morphisms relative to fibrations other than the codomain fibration.  If $A$ is a [[stack]] for a particular [[Grothendieck topology]], then every singleton cover in that topology will be, by definition, an effective descent morphism relative to $A$.  A few important examples are:

* Every [[fpqc site|fpqc cover]] of [[schemes]] is an effective descent morphism relative to the indexed category $QCoh(-)$ of [[quasicoherent sheaves]].

...

## Related concepts

* [[descent]]

  * [[cover]]

  * [[cohomological descent]]

  * **descent morphism**

  * [[monadic descent]],

    * [[Sweedler coring]]

    * [[higher monadic descent]]

    * [[descent in noncommutative algebraic geometry]]

* [[sheaf]], [[(2,1)-sheaf]], [[2-sheaf]] [[(∞,1)-sheaf]]

## References

Original articles:

* {#Grothendieck60} [[Alexander Grothendieck]], Def. 1.7 in: *Technique de descente et théorèmes d’existence engéométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats*, Séminaire N. Bourbaki exp. no190 (1960) 299-327 [[numdam:SB_1958-1960__5__299_0](http://www.numdam.org/item/?id=SB_1958-1960__5__299_0)]

See also:

* {#JT} [[George Janelidze|G. Janelidze]], and [[Walter Tholen|W. Tholen]]. _How algebraic is the change-of-base functor?._ Category Theory: Proceedings of the International Conference held in Como, Italy, July 22–28, 1990. Springer Berlin Heidelberg, 1991.

* {#ST} M. Sobral, [[Walter Tholen|W. Tholen]], _Effective descent morphisms and effective equivalence relations_, Category Theory 1991, CMS Conference Proceedings __13__ (1992), 421--433

* {#RST} J. Reiterman, M. Sobral, [[Walter Tholen|W. Tholen]], _Composites of effective descent maps_, Cahiers __34__ (1993), 193--207, [numdam](http://www.numdam.org/item?id=CTGDC_1993__34_3_193_0)

* {#EDM} [[Maria Manuel Clementino]], [[George Janelidze]], _Another note on effective descent morphisms of topological spaces and relational algebras_. Topology Appl. 273 (2020), 106961. [DOI](https://doi.org/10.1016/j.topol.2019.106961).

A criterion for categories (including [[quasi-abelian categories]]) under which effective descent morphisms and descent morphisms coincide is established in

* [[Marino Gran]], Olivette Ngaha Ngaha, _Effective descent morphisms in star-regular categories_, Homology, Homotopy and Applications 15:2 (2013), 127-144.  [doi](http://dx.doi.org/10.4310/hha.2013.v15.n2.a7).

[[!redirects descent morphism]]
[[!redirects descent morphisms]]
[[!redirects effective descent morphism]]
[[!redirects effective descent morphisms]]

[[!redirects effective descent]]

[[!redirects descent map]]
[[!redirects descent maps]]
[[!redirects effective descent map]]
[[!redirects effective descent maps]]


descent object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Quite generally, one says that an [[object]] $A$ in a [[category]] or [[higher category theory|higher category]] $\mathcal{C}$ satisfies _[[descent]]_ along a given [[morphism]] $p : \hat X \to X$ in $\mathcal{C}$ if it is a _$p$-[[local object]]_, hence if the induced map -- the _[[descent morphism]]_

$$
  \mathcal{C}(X, A) \to \mathcal{C}(\hat X, A)
$$

is an [[equivalence]]. We may read this as saying that every collection of $A$-data on $\hat X$ "[[descent|descends]]" down along $p$ to $X$.

In the context the [[hom object]] $\mathcal{C}(\hat X, A)$ is also called the **descent object**.

While roughly synonyms, typically one speaks of "descent" instead of [[local objects|locality]] when $\mathcal{C}$ is a category of [[presheaves]] or higher presheaves ([[(2,1)-presheaves]], [[(∞,1)-presheaves]], [[(∞,n)-presheaves]]).

In this case, in turn, the objects $X$ above are typically [[representables]] of a given [[site]] (or higher site) and $\hat X$ is either the [[Cech nerve]]  of a [[covering]] family with respect to a chosen [[coverage]]/[[Grothendieck topology]], or is the [[colimit]] of this Čech nerve: the corresponding [[sieve]] (the _[[codescent object]]_).

The [[descent]] condition then says that the presheaf $X$ satisfies the [[sheaf]]-condition ([[stack]]-condition, [[(∞,1)-sheaf]]/[[∞-stack]]-condition, etc.) for this given covering family.

Whether one takes $\hat X$ to be the [[Cech nerve]] or the corresponding [[sieve]] depends on [[homotopy theory|homotopical]] details of the setup. If $\mathcal{C}$ is taken to be an [[(∞,1)-category]], then it typically does not matter. But if $\mathcal{C}$ is instead just a [[homotopical category]] presenting the desired higher category, then $\hat X$ needs to satisfy some extra conditions (such as [[cofibrant object|cofibrancy]]) to ensure that $\mathcal{C}(\hat X, A)$ is indeed the correct descent object, and not too small.

For instance when working with the [[model structure on functors|injective]] [[model structure on simplicial presheaves]], every object is cofibrant and we can take $\hat X$ to be the [[sieve]]. But when working with the projective model structure then (as discussed there) $\hat X$ needs to be [[split hypercover|split]], which means that we need to use the [[Cech nerve]] and even ensure that the corresponding [[covering family]] behaves like a [[good cover]] (or, more generally, form a [[split hypercover]]).

## Details

### For ordinary presheaves
 {#ForPresheaves}

For ordinary [[presheaves]], a descent object is a set of _[[matching families]]_

More in detail, let $C$ be a [[site]], let $X \in C$ be an object, $\{U_i \to X\}$ a [[covering]] family and $S(\{U_i\}) \hookrightarrow X$ the corresponding [[sieve]].

Then for $A : C^{op} \to Set$ any [[presheaf]] on $C$, the descent object with respect to this covering is the [[hom set]]

$$
  Desc(\{U_i\}, A) = PSh(S(\{U_i\}), A)
  \,.
$$

This is discussed in detail at [[sheaf]], so just briefly:

the [[sieve]] may be realized as the [[coequalizer]]

$$
  \coprod_{i, j} U_i \cap U_j \stackrel{\to}{\to}
  \coprod_i U_i \to S(\{U_i\})
  \,.
$$

Accordingly the hom out of this realizes the descent object as the [[equalizer]]

$$
  Desc(\{U_i\}, A) \to
  \prod_{i} A(U_i)
  \stackrel{\to}{\to}
  \prod_{i, j} A(U_i \cap U_j)
  \,.
$$

Writing this out in components shows that this is the set of [[matching families]].

If the [[descent morphism]]

$$
  [C^{op}, Set](X, A) \to Desc(\{U_i\}, A)
$$

is an [[isomorphism]] one says that $A$ satisfies the [[sheaf]]-condition with respect to the [[cover]] $\{U_i \to X\}$. If this morphism is only a [[monomorphism]] one says that $A$ satisfies the [[separated presheaf]]-condition.

### For groupoid valued presheaves / pseudofunctors
 {#ForGroupoidValuedPresheaves}

For $A : C^{op} \to $ [[Grpd]] a [[2-functor]] (hence a "[[pseudofunctor]]" if $C$ is an ordinary [[category]] regarded as a [[2-category]]) and for $\hat X \to X$ a [[covering]] morphism in $C$, the descent object now is a [[groupoid]]

$$
  Desc(\hat X, A) := [C^{op}, Grpd](\hat X, A) \in Grpd
  \,.
$$

If the [[descent morphism]]

$$
  [C^{op}, Grpd](X, A) \to Desc(\hat X, A)
$$

is an [[equivalence of groupoids]], one says that $A$ satisfies the _[[(2,1)-sheaf]]_- or _[[stack]]_-condition with respect to the [[cover]] $\hat X \to X$. If it is just a [[full and faithful functor]], one says (sometimes) that $A$ satisfies the condition for a _[[separated prestack]]_ with respect to this cover.

Similar statements hold for the case of 2-functors with values in [[Cat]]. Here one also often talks about a _[[stack]]-condition_, though less ambiguous would be to speak of _[[2-sheaf]]-conditions_.

By the [[Grothendieck construction]] one may identify pseudofunctors $C^{op} \to Cat$ equivalently with [[fibered categories]] (or just [[categories fibered in groupoids]] for $C^{op} \to Grpd$) over $C$, and all of the above has analogs in this dual description.

### For simplicial presheaves

See _[[descent]]_.

### For strict $\omega$-category-valued presheaves

In ([Street](#Street)) a proposal for a definition of descent objects for presehaves with values in [[strict ∞-categories]] was proposed. Additional homotopical conditions to ensure that this gives the right answer were discussed in ([Verity](#Verity)).

+-- {: .num_defn}
###### Definition
Let $C$ be a category, let $E_1\stackrel{\stackrel{d_1}{\to}}{\stackrel{d_0}{\to}}E_0\xrightarrow{p}B$ be morphisms where the parallel arrows $\mathcal{E}:=\{d_0,d_1:E_1\to E_0\}$ are seen as _a_ diagram, let $X\in C_0$ be an object.

Applying the functor $C(-,X)$ to this sequence gives

$$C(E_1,X)\xleftarrow{C(d_0,X),C(d_1,X)}C(E_0,X)\xleftarrow{C(p,X)}C(B,X)$$

If this diagram is for all $X\in C_0$ an equalizer diagram $B$ is called **codescent object for** the diagram $\mathcal{E}$.

=--

+-- {: .num_defn}
###### Definition
Let $E_0\to\E_1\to E_2$ be a diagram where $E_0\xrightarrow{\partial_0}E_1\xrightarrow{\partial_0}E_2$, $E_0\xleftarrow{\iota_0}E_1\xrightarrow{\partial_1}E_2$, $E_0\xrightarrow{\partial_1}E_1\xrightarrow{\partial_2}E_2$ satisfying $\partial_s\partial_r=\partial_r\partial_{s-1}$ for $r\lt s$ and $\iota_0\partial_0=\iota_0\partial_1$ (these are the identities characterizing a truncated cosimplicial category).

Then the **descent category** $\Desc E$ of $E$ has as objects pairs $(F,f)$ where $F\in E_0$, $f:\partial_1 F\to \partial_0 F$ such that $\iota_0 f=\id_F$ and $\partial_0 f=\partial_2( f)\circ \partial_0 (f)$ and a morphism $(F,f)\to (G,g)$ consists of a morphism $(u:F\to G)\in E_1$ such that $\partial_0 u\circ f=g\circ \partial_1 u$.
=--

+-- {: .num_instance}
###### Instance
Let $A$, $X$ be categories.

Then $\Desc [N(A),X]\cong[A,X]$
=--

## Related concepts

See also _[[descent]]_ and [[category of descent data]].

Discussion related to the computation of descent objects is also at _[[model structure on cosimplicial simplicial sets]]_.

## References

See also the references at [[descent]].

A definition of descent objects for presheaves with values in strict $\omega$-categories was proposed in

* [[Ross Street]], [Categorical and combinatorial aspects of descent theory](http://arxiv.org/pdf/math/0303175)
  {#Street}

A discussion of a missing condition on this definition is in

* [[Dominic Verity]], _Relating descent notions_ ([pdf](http://ncatlab.org/nlab/files/VerityDescent.pdf) [[Verity on descent for strict omega-groupoid valued presheaves|nLab]])
  {#Verity}

[[!redirects descent objects]]

[[!redirects descent data]]


descent of affine schemes

In [[algebraic geometry]] the most basic and often considered case of [[descent theory]] is the descent of [[quasicoherent sheaves]]. However, [[SGA I]] also considers the descent of affine schemes over a base scheme.

* [[SGA I]].8.2: Descente des préschémas affines sur un autre

The basic theorem is that the fibered category $\mathcal{F}$ of [[affine morphism]]s $f: X\to S$ over the site of base schemes $S$ with fpqc topology is a Grothendieck 1-stack (i.e. every fpqc morphism is of effective $\mathcal{F}$-descent).

This theorem uses more specific reasoning than the theory of descent for quasicoherent sheaves and unlike the former, it does not generalize to the noncommutative case.

[[!redirects descent for affine schemes]]


descent spectral sequence

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[morphism]] of [[discrete]] [[commutative rings]], $\phi:A\to B$, one is often interested in computing the [[Amitsur cohomology]] with respect to $\phi$. This is the dualization of [[Cech cohomology]] for a morphism of spaces, except it takes place before applying a "Spec" functor to the rings. Often, the Amitsur cohomology of a morphism of rings, especially with coefficients in some $B$-module, contains useful information regarding [[descent]] along $\phi$. More generally, when we pass to a morphism of, say, [[ring spectra]] $\phi:A\to B$, we obtain a [[cosimplicial]] ring spectrum, and in this homotopical context, the generalization of [[cohomology]] groups are the [[homotopy groups]] of its [[totalization]]. Indeed, if we restrict to [[Eilenberg-Mac Lane spectra]] whose homotopy is concentrated in degree zero, we regain the usual notion of discrete Amitsur cohomology. Thus, the descent spectral sequence is the [[spectral sequence]] associated to cosimplicial Amitsur complex of a morphism of [[algebra]] objects in some [[category]]. In nice cases, the descent spectral sequence for a morphism $\phi:A\to B$ converges to the homotopy groups of $B$.
## Examples

* The (unstable) [[Adams spectral sequence]] is an example for descent along $Spec(E) \to Spec(R)$. (...) ([Hess 10, section 5.3](#Hess10))

## References

For [[higher monadic descent]]: section 5.3 of

* [[Kathryn Hess]], _A general framework for homotopic descent and codescent_, [arXiv/1001.1556](http://arxiv.org/abs/1001.1556)
 {#Hess10}

* Menini, Claudia; Ştefan, Dragoş, _Descent theory and Amitsur cohomology of triples_, J. Algebra 266 (2003), no. 1, 261--304.

[[!redirects descent spectral sequences]]


Descent Theorems

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--

This page collects links related to the article

* [[Jacob Lurie]],

  _Descent Theorems_

 ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XI.pdf))

on [[descent]] of [[quasicoherent sheaves]] in the [[Nisnevich topology]] and [[etale topology]] in [[E-∞ geometry]].

category: reference


descriptive complexity

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[Complexity theory]] is an area of research in [[computer science]] that aims to determine the amount of resources ([[time]] and [[space]] usually) that an [[algorithm]] needs to decide a query. **Descriptive complexity** is a branch of complexity theory that uses sentences in finite [[model theory]] to describe queries. It turns out that the [[languages]] of these sentences directly correspond to known [[complexity classes]].

## Related concepts

* [[complexity theory]]

## References

* Neil Immerman, _Descriptive Complexity_, Graduate texts in computer science (Springer-Verlag New York Inc.) (1999)

* Ronald Fagin, _Generalized first-order spectra and polynomial-time recognizable sets_, Complexity of Computation, ed. R. Karp, SIAM-AMS Proceedings, Vol. Vol. 7 (1974):43-73.


descriptive set theory

#Contents#
* table of contents
{:toc}

## Idea

Descriptive set theory is the study of the structures and hierarchies of [[subsets]] of [[real numbers]] (or more generally of subsets of [[Polish spaces]]) that are definable by [[formulas]] with real parameters in [[second-order arithmetic]].

Such subsets include [[Borel sets]] and more generally [[projective sets]] that are defined by alternating between taking [[images]] under [[projection]] maps of previously defined sets and taking [[complements]] of previously defined sets. Once the domain of topologists of the Polish schools and Russian analysts of the early 20th century, descriptive set theory is now considered a central area of logic in which [[set theory]] and [[computability theory]] ([[recursion theory]]) meet and interact.

## Related concepts

* [[set theory]]

* [[Mikhail Yakovlevich Suslin]]

[[!redirects descriptive set theories]]


determinant

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

# Contents
* table of contents
{: toc}

## Idea

The determinant is the (essentially unique) universal alternating [[multilinear map]].

## Definition

### Preliminaries on exterior algebra

Let [[Vect]]${}_k$ be the [[category]] of [[vector space|vector spaces]] over a [[field]] $k$, and assume for the moment that the [[characteristic]] $char(k) \neq 2$. For each $j \geq 0$, let

$$sgn_j \colon S_j \to \hom(k, k)$$

be the 1-dimensional [[sign representation]] on the [[symmetric group]] $S_j$, taking each transposition $(i j)$ to $-1 \in k^\times$. We may linearly extend the sign action of $S_j$, so that $sgn$ names a (right) $k S_j$-[[module]] with underlying [[vector space]] $k$. At the same time, $S_j$ acts on the $j^{th}$ [[tensor product]] of a vector space $V$ by permuting tensor factors, giving a left $k S_j$-module structure on $V^{\otimes j}$. We define the [[Schur functor]]

$$\Lambda^j \colon Vect_k \to Vect_k$$

by the formula

$$\Lambda^j(V) = sgn_j \otimes_{k S_j} V^{\otimes j}.$$

It is called the $j^{th}$ **alternating power** (of $V$).

Another point of view on the alternating power is via [[superalgebra]]. For any [[cosmos]] $\mathbf{V}$ let $CMon(\mathbf{V})$ be the category of [[commutative monoid]] objects in $\mathbf{V}$. The forgetful functor $CMon(\mathbf{V}) \to \mathbf{V}$ has a [[left adjoint]]

$$\exp(V) = \sum_{n \geq 0} V^{\otimes n}/S_n$$

whose values are naturally regarded as graded by degree $n$.

This applies in particular to $\mathbf{V}$ the category of [[supervector spaces]]; if $V$ is a supervector space concentrated in odd degree, say with component $V_{odd}$, then each symmetry $\sigma: V \otimes V \to V \otimes V$ maps $v \otimes w \mapsto -w \otimes v$ for elements $v, w \in V_{odd}$. It follows that the graded component $\exp(V)_n$ is concentrated in $parity(n)$ degree, with component $\Lambda^n(V_{odd})$.

+-- {: .num_prop}
###### Proposition
There is a canonical natural isomorphism $\Lambda^n(V \oplus W) \cong \sum_{j + k = n} \Lambda^j(V) \otimes \Lambda^k(W)$.
=--

+-- {: .proof}
###### Proof
Again take $\mathbf{V}$ to be the category of supervector spaces. Since the left adjoint $\exp: \mathbf{V} \to CMon(\mathbf{V})$ preserves [[coproducts]] and since the tensor product $\otimes$ of $\mathbf{V}$ provides the coproduct for commutative monoid objects, we have a natural isomorphism

$$\exp(V \oplus W) \cong \exp(V) \otimes \exp(W).$$

Examining the grade $n$ component $\exp(V \oplus W)_n$, this leads to an identification

$$\exp(V \oplus W)_n = \sum_{j + k = n} \exp(V)_j \otimes \exp(W)_k.$$

and now the result follows by considering the case where $V, W$ are concentrated in odd degree.
=--

+-- {: .num_cor}
###### Corollary
If $V$ is $n$-[[dimension|dimensional]], then $\Lambda^j(V)$ has dimension $\binom{n}{j}$. In particular, $\Lambda^n(V)$ is 1-dimensional.
=--

+-- {: .proof}
###### Proof
By induction on dimension. If $\dim(V) = 1$, we have that $\Lambda^0(V)$ and $\Lambda^1(V)$ are $1$-dimensional, and clearly $\Lambda^n(V) = 0$ for $n \geq 2$, at least when $char(k) \neq 2$.

We then infer

$$\array{
\Lambda^j(V \oplus k) & \cong & \sum_{p + q = j} \Lambda^p(V) \otimes \Lambda^q(k) \\
 & \cong & \Lambda^j(V) \oplus \Lambda^{j-1}(V)
}$$

where the dimensions satisfy the same recurrence relation as for [[binomial coefficients]]: $\binom{n+1}{j} = \binom{n}{j} + \binom{n}{j-1}$.
=--

More concretely: if $e_1, \ldots, e_n$ is a basis for $V$, then expressions of the form $e_{n_1} \otimes \ldots \otimes e_{n_j}$ form a [[basis of a vector space|basis]] for $V^{\otimes j}$. Let $e_{n_1} \wedge \ldots \wedge e_{n_j}$ denote the [[image]] of this element under the [[quotient]] map $V^{\otimes j} \to \Lambda^j(V)$. We have

$$e_{n_1} \wedge \ldots \wedge e_{n_i} \wedge e_{n_{i+1}} \wedge \ldots \wedge e_{n_j} = -e_{n_1} \wedge \ldots \wedge e_{n_{i+1}} \wedge e_{n_i} \wedge \ldots \wedge e_{n_j}$$

(consider the transposition in $S_j$ which swaps $i$ and $i+1$) and so we may take only such expressions on the left where $n_1 \lt \ldots \lt n_j$ as forming a spanning set for $\Lambda^j(V)$, and indeed these form a basis. The number of such expressions is $\binom{n}{j}$.

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

In the case where $char(k) = 2$, the same development may be carried out by simply decreeing that $e_{n_1} \wedge \ldots \wedge e_{n_j} = 0$ whenever $n_i = n_{i'}$ for some pair of distinct indices $i$, $i'$.

=--

Now let $V$ be an $n$-dimensional space, and let $f \colon V \to V$ be a [[linear map]]. By the corollary, the map

$$\Lambda^n(f) \colon \Lambda^n(V) \to \Lambda^n(V),$$

being an [[endomorphism]] on a 1-dimensional space, is given by multiplying by a scalar $D(f) \in k$. It is manifestly [[functor|functorial]] since $\Lambda^n$ is, i.e., $D(f g) = D(f) D(g)$. The quantity $D(f)$ is called the **determinant** of $f$.

### Determinant of a matrix

We see then that if $V$ is of dimension $n$,

$$\det \colon End(V) \to k$$

is a [[homomorphism]] of multiplicative [[monoids]]; by commutativity of multiplication in $k$, we infer that

$$\det(U A U^{-1}) = \det(A)$$

for each [[inverse|invertible]] [[linear map]] $U \in GL(V)$.

If we choose a [[basis of a vector space|basis]] of $V$ so that we have an identification $End(V) \cong Mat_n(k)$, then the determinant gives a [[function]]

$$\det \colon Mat_n(k) \to k$$

or by restriction to invertible matrices with invertible determinants a function

$$\det\colon GL_n(k) \to k^*$$

that takes products of $n \times n$ [[matrices]] to products in $k$. The determinant however is of course independent of choice of basis, since any two choices are related by a change-of-basis matrix $U$, where $A$ and its transform $U A U^{-1}$ have the same determinant. The above map is furthermore natural in $k$, hence is a [[natural transformation]] $det\colon GL_n\rightarrow -^*$ from the [[general linear group]] to the [[group of units]] of a [[field]] (or more generally a [[ring]]), which are both functors from [[Field]] (or more generally [[Ring]]) to [[Grp]].

By following the definitions above, we can give an explicit formula:

$$
  \det(A)
    \;=\;
  \sum_{\sigma \in S_n} sgn(\sigma) \prod_{i = 1}^n a_{i \sigma(i)}
  \;
$$

This may equivalently be written using the [[Levi-Civita symbol]] $\epsilon$ and the [[Einstein summation convention]] as

\[
  \label{DeterminantInTermsOfLCSymbol}
  \det(A)
    \;=\;
    a_{1 j_1}
    a_{2 j_2}
    \cdots
    a_{n j_n}
    \,
    \epsilon^{j_1 j_2 \cdots j_n}
\]

which in turn may be re-written more symmetrically as

\[
  \label{DeterminantInTermsOfLCSymbols}
  \det(A)
    \;=\;
    \frac{1}{n!}
    \epsilon^{ i_1 i_2 \cdots i_n }
    \,
    a_{i_1 j_1}
    a_{i_2 j_2}
    \cdots
    a_{i_n j_n}
    \,
    \epsilon^{j_1 j_2 \cdots j_n}
\]

## Properties

We work over [[fields]] of arbitrary [[characteristic]]. The determinant satisfies the following properties, which taken together uniquely characterize the determinant. Write a square [[matrix]] $A$ as a row of column [[vectors]] $(v_1, \ldots, v_n)$.

1. $\det$ is separately linear in each column vector:
$$\det(v_1, \ldots, a v + b w, \ldots, v_n) = a\det(v_1, \ldots, v, \ldots, v_n) + b\det(v_1, \ldots, w, \ldots, v_n)$$

1. $\det(v_1, \ldots, v_n) = 0$ whenever $v_i = v_j$ for distinct $i, j$.

1. $\det(I) = 1$, where $I$ is the identity matrix.

Other properties may be worked out, starting from the explicit formula or otherwise:

* If $A$ is a diagonal matrix, then $\det(A)$ is the product of its diagonal entries.

* More generally, if $A$ is an upper (or lower) triangular matrix, then $\det(A)$ is the product of the diagonal entries.

* If $E/k$ is a [[field extension]] and $f$ is a $k$-linear map $V \to V$, then $\det(f) = \det(E \otimes_k f)$. Using the preceding properties and the [[Jordan normal form]] of a matrix, this means that $\det(f)$ is the product of its [[eigenvalues]] (counted with multiplicity), as computed in the [[algebraic closure]] of $k$.

* If $A^t$ is the transpose of $A$, then $\det(A^t) = \det(A)$.

### Cramer's rule

A simple observation which flows from these basic properties is

+-- {: .num_prop}
###### Proposition
**(Cramer's Rule)**

Let $v_1, \ldots, v_n$ be column vectors of dimension $n$, and suppose
$$w = \sum_j a_j v_j.$$
Then for each $i$ we have
$$a_j \det(v_1, \ldots, v_i, \ldots, v_n) = \det(v_1, \ldots, w, \ldots, v_n)$$
where $w$ occurs as the $i^{th}$ column vector on the right.
=--

+-- {: .proof}
###### Proof
This follows straightforwardly from properties 1 and 2 above.
=--

For instance, given a square matrix $A$ such that $\det(A) \neq 0$, and writing $A = (v_1, \ldots, v_n)$, this allows us to solve for a vector $a$ in an equation

$$A \cdot a = w$$

and we easily conclude that $A$ is invertible if $\det(A) \neq 0$.

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

This holds true even if we replace the field $k$ by an arbitrary commutative [[ring]] $R$, and we replace the condition $\det(A) \neq 0$ by the condition that $\det(A)$ is a unit. (The entire development given above goes through, _mutatis mutandis_.)

=--

### Characteristic polynomial and Cayley-Hamilton theorem

Given a linear endomorphism $f: M\to M$ of a finite rank free unital module over a commutative unital ring, one can consider the zeros of the [[characteristic polynomial]] $\det(t \cdot 1_V - f)$. The coefficients of the polynomial are the concern of the [[Cayley-Hamilton theorem]].

### Over the real numbers: volume and orientation

A useful intuition to have for determinants of [[real number|real]] matrices is that they measure _change of volume_. That is, an $n \times n$ matrix with real entries will map a standard unit cube in $\mathbb{R}^n$ to a parallelpiped in $\mathbb{R}^n$ (quashed to lie in a hyperplane if the matrix is singular), and the determinant is, up to sign, the volume of the parallelpiped. It is easy to convince oneself of this in the planar case by a simple dissection of a parallelogram, rearranging the dissected pieces in the style of Euclid to form a rectangle. In algebraic terms, the dissection and rearrangement amount to applying shearing or [[elementary column operations]] to the matrix which, by the properties discussed earlier, leave the determinant unchanged. These operations transform the matrix into a diagonal matrix whose determinant is the area of the corresponding rectangle. This procedure easily generalizes to $n$ dimensions.

The sign itself is a matter of interest. An invertible transformation $f \colon V \to V$ is said to be **[[orientation]]-preserving** if $\det(f)$ is positive, and **orientation-reversing** if $\det(f)$ is negative. Orientations play an important role throughout geometry and algebraic topology, for example in the study of orientable manifolds (where the tangent bundle as $GL(n)$-bundle can be lifted to a $GL_+(n)$-bundle structure, $GL_+(n) \hookrightarrow GL(n)$ being the subgroup of matrices of positive determinant). See also [[KO-theory]].

Finally, we include one more property of determinants which pertains to matrices with real coefficients (which works slightly more generally for matrices with coefficients in a [[local field]]):

### As a polynomial in traces of powers
 {#AsAPolynomialInTracesofPowers}

On the [[relation between determinant and trace]]:

If $A$ is an $n \times n$ [[matrix]], the determinant of its [[exponential]] equals the [[exponential]] of its [[trace]]

$$
  \det(\exp(A)) = \exp(tr(A))
  \,.
$$

More generally, the determinant of $A$ is a [[polynomial]] in the [[traces]] of the [[powers]] of $A$:

For $2 \times 2$-matrices:

$$
  det(A)
  \;=\;
  \tfrac{1}{2}\left( tr(A)^2 - tr(A^2) \right)
$$

For $3 \times 3$-matrices:

$$
  det(A)
  \;=\;
  \tfrac{1}{6}
  \left(
    (tr(A))^3
    -
    3 tr(A^2) tr(A)
    +
    tr(A^3)
  \right)
$$

For $4 \times 4$-matrices:

$$
  det(A)
  \;=\;
  \tfrac{1}{24}
  \left(
    (tr(A))^4
    -
    6 tr(A^2)(tr(A))^2
    +
    3 (tr(A^2))^2
    +
    8 tr(A^3) tr(A)
    -
    6 tr(A^4)
  \right)
$$

Generally for $n \times n$-matrices ([Kondratyuk-Krivoruchenko 92, appendix B](#KondratyukKrivoruchenko92)):

\[
  \label{DeterminantAsPolynomialInTracesOfPowers}
  det(A)
  \;=\;
  \underset{
     { k_1,\cdots, k_n \in \mathbb{N} }
     \atop
     { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n }
  }{\sum}
  \underoverset{ l = 1 }{ n }{\prod}
  \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l !  }
  \left(tr(A^l)\right)^{k_l}
\]

+-- {: .proof}
###### Proof of (eq:DeterminantAsPolynomialInTracesOfPowers)

It is enough to prove this for semisimple matrices $A$ (matrices that are [[diagonalizable matrix|diagonalizable]] upon passing to the [[algebraic closure]] of the ground field) because this [[subset]] of matrices is [[Zariski topology|Zariski]] [[dense subset|dense]] (using for example the nonvanishing of the [[discriminant]] of the [[characteristic polynomial]]) and the set of $A$ for which the equation holds is [[Zariski topology|Zariski]] [[closed subset|closed]].

Thus, without loss of generality we may suppose that $A$ is [[diagonal matrix|diagonal]] with $n$ [[eigenvalues]] $\lambda_1, \ldots, \lambda_n$ along the diagonal, where the statement can be rewritten as follows. Letting $p_k = tr(A^k) = \lambda_1^k + \ldots + \lambda_n^k$, the following identity holds:

\[
\prod_{i=1}^n \lambda_i = \underset{
     { k_1,\cdots, k_n \in \mathbb{N} }
     \atop
     { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n }
  }{\sum}
  \underoverset{ l = 1 }{ n }{\prod}
  \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l !  }
  p_l^{k_l}
\]

This of course is just a [[polynomial]] identity, one closely related to various of the [[Newton identities]] that concern symmetric polynomials in indeterminates $x_1, \ldots, x_n$. Thus we again let $p_k = x_1^k + \ldots + x_n^k$, and define the [[elementary symmetric polynomials]] $\sigma_k = \sigma_k(x_1, \ldots, x_n)$ via the generating function identity

\[
\sum_{k \geq 0} \sigma_k t^k = \prod_{i=1}^n (1 + x_i t).
\]

Then we compute

\[\array{
\sum_{k \geq 0} \sigma_k t^k & = & \prod_{i=1}^n (1 + x_i t) \\
 & = & \exp\left(\sum_{i=1}^n \log(1 + x_i t)\right) \\
 & = & \exp\left(\sum_{i=1}^n \sum_{k \geq 1} (-1)^{k+1} \frac{x_i^k}{k} t^k \right)\\
 & = & \exp\left( \sum_{k \geq 1} (-1)^{k+1} \frac{p_k}{k} t^k\right)
}\]

and simply match coefficients of $t^n$ in the initial and final series expansions, where we easily compute

\[
x_1 x_2 \ldots x_n = \sum_{n = k_1 + 2k_2 + \ldots + n k_n} \prod_{l=1}^n \frac1{(k_l)!} \left(\frac{p_l}{l}\right)^{k_l} (-1)^{k_l+1}
\]

This completes the proof.
=--

## In terms of Berezinian integrals

see [[Pfaffian]] for the moment

## Related entries

* [[matrix]], [[linear algebra]], [[exterior algebra]], [[characteristic polynomial]]

* [[principal submatrix]]

* [[quasideterminant]], [[Berezinian]],[[Jacobian]], [[Pfaffian]], [[hafnian]],  [[Wronskian]]

* [[Dieudonné determinant]]

* [[resultant]], [[discriminant]], [[hyperdeterminant]]

* [[functional determinant]]

* [[determinant line]], [[determinant line bundle]], [[Pfaffian line bundle]]

* [[density bundle]]

* [[pseudoscalar]]

* [[Slater determinant]]

* [[permanent]]

## References

See also

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

One derivation of the formula (eq:DeterminantAsPolynomialInTracesOfPowers) for the determinant as a polynomial in traces of powers is spelled out in appendix B of

* {#KondratyukKrivoruchenko92} L. A. Kondratyuk, I. Krivoruchenko, _Superconducting quark matter in $SU(2)$ colour group_, Zeitschrift für Physik A Hadrons and Nuclei March 1992, Volume 344, Issue 1, pp 99–115 ([doi:10.1007/BF01291027](https://doi.org/10.1007/BF01291027))

[[!redirects determinants]]

[[!redirects Cramer's rule]]
[[!redirects Cramer's rules]]


determinant line

#Contents#
* table of contents
{:toc}

## Definition

For $k$ a [[field]] and $V$ a $k$-[[vector space]], its _determinant line_ is the 1-[[dimension]]al vector space

$$
  det V = \wedge^{dim V} V
  \,.
$$

## Related concepts

* [[determinant]],

* [[determinant line bundle]]