derived geometry

> This entry is about a certain way of formalizing [[higher geometry]]. For variants and more background, see there.

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###Context###
#### Higher geometry
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[[!include higher geometry - contents]]
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#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
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#Contents#
* table of contents
{:toc}

## Idea and motivation

For $T$ a [[nLab:Lawvere theory]], the _geometry_ modeled on $T$ is encoded in the [[nLab:sheaf topos]] on formal duals of [[nLab:T-algebra]]s. For instance for $T = $ [[nLab:CartSp]] we have that $T$-algebras are [[nLab:smooth algebra]]s and the geometry modeled on them is [[nLab:synthetic differential geometry]].

This statement generalizes to [[nLab:(∞,1)-category theory]]: for $T$ an [[nLab:(∞,1)-algebraic theory]] and $C \subset T Alg^{op}$ a [[nLab:(∞,1)-site]] of formal duals of $\infty$-algebras over $T$, one says that the [[nLab:(∞,1)-topos]] $\mathbf{H} = \infty Sh(C)$ over $C$ encodes **[[nLab:derived geometry]]** modeled on $T$. The objects of $\mathbf{H}$ are also called _[[nLab:derived stack]]s_ on $C$.

The term "derived" here is meant to specifically contrast with [[nLab:∞-stack]]s on just a 1-categorical site. Even if $T$ happens to be just an ordinary Lawvere theory, regarded as a 1-[[nLab:truncated]] $(\infty,1)$-theory, the $\infty$-topos over $T Alg_\infty^{op}$ behaves considerably differently from that over just $T Alg^{op}$.

An object $X$ in derived geometry has both an [[nLab:∞-groupoid]] of internal symmetries as well as an $\infty$-[[nLab:function algebras on ∞-stacks|algebra of functions]]:

1. passing from a sheaf topos over a site to the $\infty$-stacks over that site makes _[[nLab:colimit]]s_ behave well in [[nLab:cohomology]]. For instance a singular [[nLab:quotient]] becomes an [[nLab:orbifold]].

1. passing to derived geometry by making the site a genuine $(\infty,1)$-category makes _[[nLab:limit]]s_ behave well in cohomology. For instance the intersection pairing of non-[[nLab:transversal map|transversal]] [[nLab:smooth manifold]]s comes out correctly when regarding them as [[nLab:derived smooth manifold]]s.

A central class of examples for nontransveral [[nLab:pullback]]s in derived geometry are [[nLab:derived loop space]] objects. For $X \in T Alg^{op}$ an ordinary space, its [[nLab:free loop space object]] computed in the underived $\infty$-topos over $T Alg^{op}$ exists, but simply coincides with $X$, because $X$ is 0-[[nLab:truncated]] in there. But the free loop space object $\mathcal{L}X$ of $X$ computed in the $\infty$-topos over $T Alg_\infty^{op}$ may be very rich: its $\infty$-[[nLab:function algebras on infinity-stacks|function algebra]] is the [[nLab:Hochschild homology]] of $X$. Moreover, the functions on $\mathcal{L}X$ that are invariant under the canonical internal [[nLab:circle]]-[[nLab:action]] are the _closed_ [[nLab:Kähler differential]] forms on $X$.

Examples of derived spaces have appeared long ago as the configuration spaces in [[nLab:gauge theory]]. What is called the [[nLab:BV-BRST complex]] of a gauge theory is the function algebra on the [[nLab:infinitesimal object|infinitesimal]] approximation to a derived orbifold whose internal symmetries are the [[nLab:gauge transformation]]s and whose function complex provides a [[nLab:resolution]] of the locus of solutions to the physical equations of motion.

More recently, much of the motivation for derived geometry came from the observation that the [Goerss-Hopkins-Miller theorem](http://arxiv.org/abs/0910.5130) suggests that there is a [[nLab:A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves|derived moduli space of derived elliptic currves]], that it carries a [[nLab:structured (∞,1)-topos|structure ∞-sheaf]] of [[nLab:E-∞-rings]], and the the [[nLab:global section]]s of that yield the [[nLab:ring spectrum]] of the [[nLab:generalized cohomology theory]] called [[nLab:tmf]].

## Plan

The rough plan is that we

1. talk about some relevant leftover topics from the later chapters of [[nLab:Higher Topos Theory]], continuing where we left of [[Seminar on (∞,1)-Categories and ∞-Stacks|last time]];

1. then discuss some of the specifics of $\infty$-sites for derived geometry;

1. and finally look at the specific case of derived _smooth differential_ geometry.

The following is the beginning of a detailed schedule of talks. The first few talks have been fixed by now, the later talks will be fixed as we move along.

### $\infty$-Sites from $\infty$-algebraic theories {#Sites}

* _Oct 15, 2010_

  **[[nLab:algebraic theories]]** / **[[nLab:Lawvere theories]]**

  The [[nLab:theories]] of [[nLab:group]]s,
  [[nLab:ring]]s, $k$-[[nLab:algebra]]s,
  [[nLab:C-∞-ring]]s and so forth are examples of
  [[nLab:algebraic theories]]. [[nLab:Bill Lawvere]] famously
  noticed that these theories are encoded by
  categories $T$ with products, all whose objects are cartesian powers
  of a generating object $x$. One speaks of [[nLab:Lawvere theories]].
  An [[nLab:algebra over a Lawvere theory|algebra]] $A$ of a Lawvere theory is identified with a product-preserving co-presheaf

  $$
    A : T \to Set
    \,.
  $$

  Here the value $A(x)$ is identified with the underlying set of the algebra and for any morphism $f : x^n \to x$ in $T$ the morphism $A(f) : A(x^n) \simeq A(x)^n \to A(x)$ is an $n$-ary operation in the algebra. Functoriality of $A$ encodes the compatibilities of all these operations, such as associativity.

  The identification of algebras with co-presheaves gives rise to [[nLab:Isbell duality]], which naturally identifies the opposite $T Alg^{op}$ of the category of all $T$-algebras with a category of test spaces: the [[nLab:sheaf topos]] $Sh(C)$ on a small full subcategory $C \hookrightarrow T Alg^{op}$ is the context in which geometry over $T$ takes place. For $T = $ [[nLab:CartSp]] the theory of [[nLab:smooth algebra]]s, this is [[nLab:synthetic differential geometry]].

* _Oct 22, 2010_

  **[[nLab:(∞,1)-algebraic theory]]**

  There is an $\infty$-categorical generalization of the notion of [[nLab:Lawvere theory]] and [[nLab:algebra over a Lawvere theory]]. The general abstract discussion of this is hidden in section 5.5.8 of _[[nLab:Higher Topos Theory]]_ .

  The idea is evident: we say

  * an **$(\infty,1)$-algebraic theory** is an [[nLab:(∞,1)-category]] $T$ with [[nLab:(∞,1)-limit|(∞,1)-product]]s.

  * an **$(\infty,1)$-algebra** over $T$ is an [[nLab:(∞,1)-functor]] $A : T \to $ [[nLab:∞Grpd]] that preserves these products;

  * the $(\infty,1)$-category of all $T$-algebras is the full [[nLab:sub-(∞,1)-category]]  $\infty T Alg \hookrightarrow \infty Func(T, \infty Grpd)$ of the [[nLab:(∞,1)-category of (∞,1)-presheaves|(∞,1)-category of (∞,1)-copresheaves]] on those functors that preserve products.

* _Dec 17, 2010_

  **[[nLab:(2,1)-algebraic theory of E-∞ algebras]]

  An important example of a [[nLab:(∞,1)-algebraic theory]] beyond ordinary 1-[[nlab:algebraic theories]] is the [[nLab:(n,r)-category|(2,1)-category]] of [[nLab:span]]s of [[nLab:finite set]]s. Its [[nLab:∞-algebra over an (∞,1)-algebraic theory|algebras]] turn out to be [[nLab:E-∞ algebra]]s.

  This is the result of

  * [[nLab:James Cranch]], _Algebraic Theories and $(\infty,1)$-Categories_ ([arXiv](http://arxiv.org/abs/1011.3243))

Next, in order to handle $\infty T Alg$ and to compare it to other known structures it is useful to [[nLab:presentable (∞,1)-category|present]] it in terms of a [[nLab:model category]]. This is the topic of the next part.

### Models for $\infty$-algebras over ordinary algebraic theories {#Models}

* _Oct 29, 2010_

  **[[nLab:homotopy T-algebra]]s**

  We know from the [[nLab:model structure on simplicial presheaves]] that every [[nLab:(∞,1)-functor]] $A : T \to \infty Grpd$ is modeled by a strict functor $T \to $ [[nLab:sSet]]. A **homotopy $T$-algebra** is such a functor that preserves products up to weak equivalence in that for all $n \in \mathbb{N}$ the canonical morphism

  $$
     \prod_{i = 1}^n A(p_i) : A(x^n) \to (A(x))^n
  $$

  is a [[nLab:model structure on simplicial sets|weak equivalence]].

  There is a model structure for such homotopy $T$-algebras, given by left [[nLab:Bousfield localization of model categories|Bousfield localization]] of the projective [[nLab:model structure on simplicial presheaves]] at the morphisms $\coprod_n F_T(1) \to F_T(n)$, where $F_T(n)$ is the free $T$-algebra on $n$ generators.

  But it turns out that one can further strictify this: a [[nLab:simplicial algebra|simplicial T-algebra]] is a strict functor $T \to sSet$ that also preserves all products up to isomorphism. There is a [[nLab:model structure on simplicial T-algebras]], going back to Quillen, where the weak equivalences and the fibrations are objectwise those of simplicial sets.

  And there is a [[nLab:Quillen equivalence]] between these model structures for homotopy $T$-algebras and for simplicial $T$-algebras.

  This Quillen equivalence is described in

  * [[nLab:Bernard Badzioch]], _Algebraic theories in homotopy theory_ Annals of Mathematics, 155 (2002), 895-913 ([JSTOR](http://www.jstor.org/stable/3062135))

  The result that strict simplicial algebras model all $\infty$-$T$-algebras is also in

  * [[nLab:Julie Bergner]], _Rigidification of algebras over multi-sorted theories_ , Algebraic and Geometric Topoogy 7, 2007.

* _Nov 5, Nov 19, 2010_

  **[[nLab:monoidal Dold-Kan correspondence]]** and **[[nLab:model structure on dg-algebras]]**

  We have seen that the [[nLab:model structure on simplicial T-algebras]]
  models $\infty$-T-algebras. When $T$ is an _abelian_ Lawvere theory,
  so that every $T$-algebra has an underlying [[nLab:abelian group]],
  it is useful to consider the [[nLab:Moore complex|normalized chains complex]] of a simplicial $T$-algebra: equipped with the [[nLab:cup product]] this is a [[nLab:dg-algebra]] (in non-negative degree).

  There is a standard [[nLab:model structure on dg-algebra]]s that prevails
 much of the literature. For instance in its dual form this is the model structure
 traditionally used in [[nLab:rational homotopy theory]].

  The [[nLab:monoidal Dold-Kan correspondence]] asserts that at least for
 $T$ a theory of ordinary commutative algebras, there is a [[nLab:Quillen equivalence]]

  $$
    T Alg_{proj}^{\Delta^{op}} \simeq dgAlg^+_{proj}
  $$

  between the model structure on simplicial algebras and that on
  chain dg-algebras in non-negative degree. This is a monoidal
  refinement of the standard [[nLab:Dold-Kan correspondence]] that
  identifies [[nLab:simplicial abelian group]]s with [[nLab:chain complex]]es.
  The proof is based on general statements about [[nLab:monoidal Quillen adjunction]]s.

  This means that dg-algebras are yet another equivalent model for (certain) $\infty$-algebras. In applications it is convenient to pass back and forth along this equivalence: the simplicial algebras connect more directly to the abstract theory, but the available toolset of theorems and techniqies for dg-algebras is much larger, and they are generally easier to handle.

* _Nov 12, 2010_

  **[[nLab:model structure on algebras over an operad]]**

  Algebraic theories may also be encoded by [[nLab:operad]]s.
  Accordingly $\infty$-algebras may be regarded as
  [[nLab:∞-algebras over an (∞,1)-operad]]. There is again
  a [[nLab:model category]]-structure available to present these:
  the [[nLab:model structure on algebras over an operad]].

  Many classical examples of $\infty$-algebras are captured by this,
  such as [[nLab:A-∞ algebra]]s / [[nLab:A-∞ space]]s,
  [[nLab:E-∞ algebra]]s, [[nLab:L-∞ algebra]]s.

### Derived geometry over $\infty-C^\infty$-rings {#Smooth}

* _Nov 27, Dec 3, 2010_

  **[[nLab:derived loop space]]s**

  Analogous to how a good deal of the phenomenology of [[nLab:stack]]s is exhibited already by the [[nLab:action groupoid|weak quotient]] $*//G$ we have that much of the phenomenology of derived geometry is exhibited already by [[nLab:free loop space object]]s $\mathcal{L}X$ of a derived space $X$: the [[nLab:(∞,1)-pullback]] of the diagonal on $X$ along itself

  $$
    \mathcal{L}X := X \times_{X \times X} X
    \,.
  $$

  When $X$ is an ordinary object in the inclusion $(\infty,1)Sh(T Alg^{op}) \hookrightarrow (\infty,1)Sh(T Alg_\infty^{op})$ one calls $\mathcal{L}X$ for emphasis also the [[nLab:derived loop space]] of $X$.

  Since this pullback is maximally non-transversal, the derived free loop space is quite different from the ordinary free loop space object computed in $(\infty,1)Sh(T Alg^{op})$. Notably when $X$ is 0-[[nLab:truncated]] (just a plain space, no groupoidal morphisms, no derived resoluton) its underived loop space object is just $X$ itself and hence uninteresting, whereas its derived loop space is very rich:

  one finds that the [[nLab:function algebras on ∞-stacks|function algebra on]] $\mathcal{L}X$ is the complex that computes the [[nLab:Hochschild homology]] of the function algebra of $X$:

  $$
    C(\mathcal{L}X) \simeq \mathbf{HH}_\bullet(C(X))
    \,.
  $$

  To some extent this is just the tautological dual reformulation of Hochschild homology as the $(\infty,1)$-categorical ([[nLab:derived functor|derived]]) tensor product

  $$
    \mathbf{HH}_\bullet(A) := A \otimes_{A \otimes A} A
    \,.
  $$

  But there are some noteworthy subtleties. For instance in the traditional literature the derived tensor product is taken in the context of modules, not of algebras.  But one can see that forming a fibrant replacement in the model structure on homotopy $T$-algebras puts an $\infty$-algebra structure back on the Hochschild complex.

  This relation between derived loop spaces and Hochschild homology is very fruitful. It gives a transparent conceptual interpretation to many constructions in Hochschild cohomology and makes all this standard theory applicable to the study of derived geometry.

* _Dec 10 19, 2010_

  **[[nLab:derived smooth manifold]]s**

  The Lawvere theory that encodes standard models of smooth differential geometry ([[nLab:synthetic differential geometry]]) is the category [[nLab:CartSp]] of [[nLab:Cartesian space]]s and [[nLab:smooth function]]s between them. Its algebras are [[nLab:smooth algebra]]s / $C^\infty$-rings. Therefore its $\infty$-algebras are modeled by simplicial $C^\infty$-rings.

  Spaces locally ringed in such smooth $\infty$-algebras are called _derived smooth manifolds_ .

## Examples

* [[derived differential geometry]]

## References

A systematic description of derived geometry using [[nLab:model category]]-theoretic tools was first undertaken in

* [[nLab:Bertrand Toën]], [[nLab:Gabriele Vezzosi]],

  * _Homotopical Algebraic Geometry I: Topos theory_ ([arXiv](http://arxiv.org/abs/math/0207028))

  * _Homotopical Algebraic Geometry II: geometric stacks and applications _ ([arXiv](http://arxiv.org/abs/math/0404373))

This generalizes the Brown-Jardine-Joyal-[[nLab:model structure on simplicial presheaves]] to a [[nLab:model structure on sSet-enriched presheaves]] over an [[nLab:sSet-site]].

A proposal for the precise set of the scene of derived geometry in general abstract [[nLab:(∞,1)-category theory]]-terms is

* [[nLab:Jacob Lurie]], _[[nLab:Structured Spaces]]_

The main point of this is a formalization and identification of special tame objects inside the collection of all [[nLab:derived stack]]s, namely the [[nLab:derived scheme]]s and [[nLab:structured (∞,1)-topos]]es.

The relevance of [[nLab:derived loop space]]s in derived was notably amplified in a series of articles by [[nLab:David Ben-Zvi]] and [[nLab:David Nadler]],

* [[nLab:David Ben-Zvi]], [[nLab:David Nadler]],

  * _Loop Spaces and Langlands Parameters_ ([arXiv:0706.0322](http://arxiv.org/abs/0706.0322))

  * _Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry _ ([arXiv:0805.0157](http://arxiv.org/abs/0805.0157))

  * _Loop Spaces and Connections_ ([arXiv:1002.3636](http://arxiv.org/abs/1002.3636))

    This article uses Toën's theory of [[nLab:function algebras on ∞-stacks]] for showing that the function complex on a derived loop space $\mathcal{L}X$ is under mild conditions the [[nLab:Hochschild homology]] complex of $X$ hence by [[nLab:Hochschild-Kostant-Rosenberg theorem]] the collection of [[nLab:Kähler differential]] forms on $X$, and that the functions on $\mathcal{L}X$ that are invariant under the canonical $S^1$-[[nLab:action]] on $\mathcal{L}X$ are the _closed_ forms. This also gives a geometric interpretation of the old observation by [[nLab:Maxim Kontsevich]] and others, that the differential and grading on the de Rham complex may be understood as induced from automorphisms of the [[nLab:odd line]].

  * _Loop Spaces and Representations_ ([arXiv:1004.5120](http://arxiv.org/abs/1004.5120))

The application of derived geometry to the construction of [[nLab:tmf]] is described in

* [[nLab:Jacob Lurie]], _[[nLab:A Survey of Elliptic Cohomology]]_

The article

* [[nLab:David Spivak]], _Derived smooth manifolds_ ([arXiv:0810.5174](http://arxiv.org/abs/0810.5174))

discussed the $\infty$-version of [[nLab:smooth algebra|C-oo-rings]] -- the algebras over the Lawvere theory [[nLab:CartSp]] -- and the corresponding locally ringed spaces: [[nLab:derived smooth manifold]]s.

Some of the above material is taken from the seminar notes

* [[Urs Schreiber]], _[[schreiber:Seminar on derived differential geometry]]_ (2010)

Survey in the context of [[BV-formalism]]:

* [[Luigi Alfonsi]], §5 in: *Higher geometry in physics*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.07308](https://arxiv.org/abs/2312.07308)]

[[!redirects derived geometries]]


derived global analytic geometry

#Contents#
* table of contents
{:toc}

## Idea

The aim of derived [[global analytic geometry]] is to give a proper setting for the cotangent complex and other derived constructions over a general (ind)-Banach ring, whose specialization to the Banach ring $\mathbb{C}$ of complex numbers gives back [[derived complex analytic spaces]].

## Related subjects

* [[derived analytic geometry]]

* [[derived complex analytic spaces]]

* [[overconvergent global analytic geometry]]

## References

[[Jacob Lurie]] _DAG V: [[Structured Spaces]]_.

[[Jacob Lurie]] _DAG IX: [[Closed Immersions]]_.

[[Frederic Paugam]] _Overconvergent global analytic geometry_ ([arXiv:1410.7971](http://arxiv.org/abs/1410.7971)).


derived group scheme

#derived group scheme#
* automatic table of contents goes here
{:toc}

[To be merged with [[spectral group scheme]].]

## Idea ##

A **general group scheme** is a [[group object]] in [[generalized scheme]]s: it is a generalization to [[higher geometry]] of a [[group scheme]].

## Definition ##

Let $X$ be a $\mathcal{G}$-[[generalized scheme]] for $\mathcal{G}$ the [[geometry (for structured (∞,1)-toposes)]] that defines the desired notion of [[derived scheme]]s.

A **commutative group $\mathcal{G}$-scheme over $X$ is an [[(∞,1)-functor]]

$$
  G : Sch(\mathcal{G})/X \to Ab \infty Grpd
$$

from $\mathcal{G}$-schemes over $X$ to topological abelian groups, such that composition with the forgetful functor $Ab \infty Grpd \to \infty Grpd$ is [[representable functor|representable]] by a [[derived scheme]] flat over $X$.

This is (adapted from) definition 3.1 of

* [[Jacob Lurie]], [[A Survey of Elliptic Cohomology]]

> **warning** careful, this needs a bit more attention. The general idea is obvious, but the details require care. One problem is that in the Elliptic Survey "derived scheme" really refers to [[Spectral Schemes]] and not to the derived schemes discussed in [[Structured Spaces]].

## special cases ##

An important special case is that of a [[derived elliptic curve]].


derived hom-functor

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###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
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=--

#Contents#
* table of contents
{:toc}

## Idea

The (right) [[derived functor]] of an [[enriched hom-functor]].

## Definition

### For enriched model categories
 {#ForEnrichedModelCategories}

We discuss the definition of the derived hom-functor (Def. \ref{DerivedHomFunctorPOnSimplicialModelCategory} below) for [[enriched model categories]] (recalled as Def. \ref{SimplicialModelCategory} below). The key point is to observe (Lemma  \ref{InSimplicialModelCategoryEnrichedHomIsHomotopical} below) that the axioms on [[enriched model categories]] imply that the ordinary [[enriched hom-functor]] is a [[homotopical functor]] when its first argument is [[cofibrant object|cofibrant]] and its second argument is [[fibrant object|fibrant]], so that its [[right derived functor]] exists.

$\,$

+-- {: .num_defn #SimplicialModelCategory}
###### Definition
**([[enriched model category]])**

Let $\mathcal{V}$ be a [[monoidal model category]]. Then a _$\mathcal{V}$-[[enriched model category]]_ is a [[category]] $\mathcal{C}$  equipped with

1. the [[structure]] of $\mathcal{V}$-[[enriched category]], which is also [[tensored and cotensored category|tensored and cotensored]] over $\mathcal{V}$;

1. the [[structure]] of a [[model category]],

such that these two structures are compatible in the following way:

* for every [[cofibration]] $X \overset{f}{\to} Y$ and every [[fibration]] $A \overset{g}{\to} B$ in $\mathcal{C}$, the induced [[pullback powering]]-morphism of [[hom-objects]] in $\mathcal{V}$

  \[
    \label{SimplicialModelCategoryComparisonMorphism}
    \mathcal{C}(Y,A)
      \longrightarrow
     \mathcal{C}(X,A)
       \underset{\mathcal{C}(X,B)}{\times}
     \mathcal{C}(Y,B)
  \]

  is a [[fibration]], and is a [[weak equivalence]] as soon as one of the two morphisms is a [[weak equivalence]] in $\mathcal{C}$.

=--

+-- {: .num_lemma #InSimplicialModelCategoryEnrichedHomIsHomotopical}
###### Lemma
**(in [[enriched model category]] the [[enriched hom-functor]] out of [[cofibrant object|cofibrant]] into [[fibrant object|fibrant]] is [[homotopical functor]])**

Let $\mathcal{C}$ be an [[enriched model category]] (Def. \ref{SimplicialModelCategory}).

If $Y \in \mathcal{C}$ is a [[cofibrant object]], then the [[enriched hom-functor]] out of $Y$

$$
  \mathcal{C}(Y,-)
  \;\colon\;
  \mathcal{C}
    \longrightarrow
  \mathcal{V}
$$

preserves [[fibrations]] and [[acyclic fibrations]].

If $A \in \mathcal{C}$ is a [[fibrant object]], then the [[enriched hom-functor]] into $A$

$$
  \mathcal{C}(-,A)
  \;\colon\;
  \mathcal{C}^{op}
    \longrightarrow
  \mathcal{V}
$$

sends [[cofibrations]] and [[acyclic cofibrations]] in $\mathcal{C}$ to fibrations and acyclic fibrations, respectively, in $\mathcal{V}$.

=--

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

In the first case, consider the comparison morphism (eq:SimplicialModelCategoryComparisonMorphism) for $X =\emptyset$ the [[initial object]], in the second case consider it for $B = \ast$ the [[terminal object]].

Since $\mathcal{C}$ is [[tensored and cotensored category|tensored and cotensored]] over $\mathcal{V}$, it follows (by [this Prop](powered+and+copowered+category#InTensoredCotensoredCategoryInitialObjectIsEnrichedInitial) that

$$
  \mathcal{C}(\emptyset, -)
  \;\simeq\;
  \ast
  \phantom{AA}
  \text{and}
  \phantom{AA}
  \mathcal{C}(-,\ast)
  \;\simeq\;
  \ast\;
  \;\;\;
  \in \mathcal{V}
  \,.
$$

This means that in the first case the comparison morphism
$$
    \mathcal{C}(Y,A)
      \longrightarrow
     \mathcal{C}(X,A)
       \underset{\mathcal{C}(X,B)}{\times}
     \mathcal{C}(Y,B)
$$
(eq:SimplicialModelCategoryComparisonMorphism) becomes equal to the top morphism in the following diagram

$$
  \array{
    \mathcal{C}(Y,A)
       &\overset{\mathcal{C}(Y,g)}{\longrightarrow}&
    \mathcal{C}(Y,B)
    \\
    \Big\downarrow
      &&
    \Big\downarrow
    \\
    \ast &\underset{\phantom{AAA}}{\longrightarrow}& \ast
  }
$$

while in the second case it becomes equal to the left morphism in

$$
  \array{
    \mathcal{C}(Y,A)
       &\overset{\phantom{\mathcal{C}(Y,g)}}{\longrightarrow}&
    \ast
    \\
    {}^{\mathllap{ \mathcal{C}(f,A) }}\Big\downarrow
      &&
    \Big\downarrow
    \\
    \mathcal{C}(X,A)
      &\underset{\phantom{AAA}}{\longrightarrow}&
    \ast
  }
$$

Hence the claim follows by the defining condition on the comparison morphism (eq:SimplicialModelCategoryComparisonMorphism) in an [[enriched model category]].

=--

+-- {: .num_defn #DerivedHomFunctorPOnSimplicialModelCategory}
###### Definition
**([[derived hom-functor]] of an [[enriched model category]])**

Let $\mathcal{C}$ be an [[enriched model category]] (Def. \ref{SimplicialModelCategory}).

By Lemma \ref{InSimplicialModelCategoryEnrichedHomIsHomotopical} and by [[Ken Brown's lemma]], the [[enriched hom-functor]] has a [[right derived functor]] ([this Def.](geometry+of+physics+--+categories+and+toposes#LeftAndRightDerivedFunctorsOnModelCategories)) when its first argument is [[cofibrant object|cofibrant]] and its second argument is [[fibrant object|fibrant]]. This is called the _[[derived hom-functor]]_

$$
  \mathbb{R}hom
  \;\colon\;
  Ho(\mathcal{C})^{op} \times Ho(\mathcal{C})
  \longrightarrow
  Ho(\mathcal{V})
$$

In the presence of [[functorial factorization|functorial]] [[cofibrant resolution]] $Q$ and [[fibrant resolution]] $P$ this is given by the ordinary [[enriched hom-functor]] $\mathcal{C}(-,-)$ as

$$
  \mathbb{R}hom(X,Y)
  \;\simeq\;
  \mathcal{C}(Q X, P Y)
  \,.
$$

=--

## Examples

* For [[simplicial model categories]] see _[[derived hom-space]]_.

* For [[categories of chain complexes]] see _[[Ext]]_.

## Related concepts

* [[hom-set]], [[hom-object]]

* [[hom-functor]]

* [[enriched hom-functor]], [[internal hom-functor]]

[[!redirects derived hom-functors]]


derived infinity-stack > history

< [[derived infinity-stack]]
[[!redirects derived infinity-stack -- history]]


derived intersection

#Contents#
* table of contents
{:toc}

## Idea

A _derived intersection_ or _homotopy intersection_ is an [[intersection]] computed in the [[homotopy theory]] of [[derived geometry]].

## Related concepts

* [[derived critical locus]]

* [[BV-complex]]

[[!redirects derived intersections]]

[[!redirects homotopy intersection]]
[[!redirects homotopy intersections]]


derived inverse image

#Contents#
* table of contents
{:toc}

## Idea

The [[derived functor]] of an [[inverse image]] functor.

A model for the inverse image [[(infinity,1)-functor]].

## Related concepts

* [[abelian sheaf cohomology]]

* [[derived direct image]]

* [[Ext]], [[Tor]]

* [[six operations]]

[[!redirects derived inverse images]]

[[!redirects derived inverse image functor]]
[[!redirects derived inverse image functors]]


derived limit functor

If $A$ is a category and $I$ a small category then we can consider the [[category of functors]] $A^I$, whose objects are the functors from $I$ to $A$. If $A$ admits limits of shape $I$, then the limit $lim = lim_I:A^I\to A$ is a functor which is right adjoint to the constant diagram functor hence it commutes with limits, and in particular it is left exact.

If $A$ has some notion of homotopy theory then usually $A^I$ has the notion as well (e.g. if we work with model category structures or say Abelian categories) hence we can then form right derived functors of $lim$ which are usually denoted $lim^n$ and called the derived limit functors. As usual we can also assemble them into the [[total right derived functor]] $\mathbb{R}lim$.

An important example is where $A$ is a category of modules over a commutative ring, then one has $Ch(A)$ and any $M:I\to A$ can be thought of as $M:I\to Ch(A)$ by thinking of each $M(i)$ as being a chain complex concentrated in dimension 0.

##Related entries

These include [[homotopy limit]], [[lim^1 and Milnor sequences]] and [[cohomology of small categories]], this latter in the case of coefficients in a category of modules. This is a special case of the more general [[Baues-Wirsching cohomology]].

##References

A classic text with links to the theory of modules is

* C. U. Jensen, 1972, _Les foncteurs dérivés de Lim et leurs applications en théorie de modules_, volume 254 of Springer Lecture Notes in Maths.

A proof that derived limit functors give invariants of a corresponding pro-obect can be found in

* [[John Duskin]], _Pro-objects (after Verdier)_, Sém. Heidelberg- Strasbourg1966 -67, Exposé 6, I.R.M.A.Strasbourg.

Some results on the vanishing of 'derived limits' are in

* [[Barbara Osofsky]], _The subscript of $\aleph_n$ projective dimension, and the vanishing of $lim^{(n)}$_, Bull. Amer. Math. Soc. Volume 80, Number 1 (1974), 8 - 26.

[[!redirects derived limit functors]]
[[!redirects lim^n]]


derived loop space

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## Idea

A _derived loop space_ is a [[free loop space object]] in [[derived geometry]].

## Definition

Let $T$ be an [[(∞,1)-algebraic theory]] and $C \subset T Alg_\infty^{op}$ an [[(∞,1)-site]] of [[Isbell duality|formal duals]] of $\infty$-algebras over $T$. Then the [[(∞,1)-topos]] $\mathbf{H} = (\infty,1) Sh(C)$ encodes [[derived geometry]] modeled on $T$.

A **derived loop space** is a [[free loop space object]] in such $\mathbf{H}$.

More specifically, if $T$ is an ordinary [[Lawvere theory]], regarded as a 1-truncated $(\infty,1)$-theory, then $T Alg_{\infty}$ are its simplicial algebras. There is a canonical embedding $T Alg^{op} \hookrightarrow T Alg_\infty^{op}$ of the ordinary algebras into the $\infty$-algebras, so that we may regard $X \in T Alg^{op}$ as an object of $\mathbf{H}$. Then the _derived loop space_ of $X$ is its [[free loop space object]] computed in $\mathbf{H}$.

The point is that the derived loop space of an ordinary $X \in T Alg^{op}$ in general is a significantly richer object than the free loop space object of $X$ as computed just in the underived $(\infty,1)$-topos $(\infty,1)Sh(T Alg^{op})$. In fact, since $X$ is 0-[[truncated]] in $(\infty,1)Sh(T Alg^{op})$, it coincides with its free loop space object there, but the derived loop space does not.

## Function complexes on derived loop spaces: Hochschild homology

The function complex on the derived loop space $\mathcal{L}X$ is the [[Hochschild homology]] complex of $C(X)$. See there for further details. In particular see the section _[Hochschild cohomology -- As function algebra on the derived loop space](Hochschild%20cohomology#HochschildChainComplex)_.

Also see [[free loop space object]] for more information.

## Related concepts

* [[loop space object]], [[free loop space object]],

  * [[delooping]]

  * [[loop space]], [[free loop space]],

  * [[formal loop space]], **derived loop space**

  * [[derived microlocalization]]

  * [[inertia orbifold]]

* [[suspension object]]

  * [[suspension]]

## References

The relevance of derived loop spaces was amplified in a series of articles by [[David Ben-Zvi]] and [[David Nadler]],

* [[David Ben-Zvi]], [[David Nadler]],

  * _Loop Spaces and Langlands Parameters_ ([arXiv:0706.0322](http://arxiv.org/abs/0706.0322))

  * _Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry _ ([arXiv:0805.0157](http://arxiv.org/abs/0805.0157))

  * _Loop Spaces and Connections_ ([arXiv:1002.3636](http://arxiv.org/abs/1002.3636))

    This article uses Toën's theory of [[function algebras on ∞-stacks]] for showing that the function complex on a derived loop space $\mathcal{L}X$ is under mild conditions the [[Hochschild homology]] complex of $X$ hence by [[Hochschild-Kostant-Rosenberg theorem]] the collection of [[Kähler differential]] forms on $X$, and that the functions on $\mathcal{L}X$ that are invariant under the canonical $S^1$-[[action]] on $\mathcal{L}X$ are the _closed_ forms. This also gives a geometric interpretation of the old observation by [[Maxim Kontsevich]] and others, that the differential and grading on the de Rham complex may be understood as induced from automorphisms of the [[odd line]].

  * _Loop Spaces and Representations_ ([arXiv:1004.5120](http://arxiv.org/abs/1004.5120))

[[!redirects derived loop spaces]]


derived microlocalization

#Contents#
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{:toc}

## Idea

[[Microlocalization]] is a tool to study the propagation of singularities of solutions of partial differential systems, in order to study pullbacks of solutions of differential systems (which generalize products of distributions) and prove [[index theorems]].
Derived microlocalization is an adaptation of the theory of microlocalization to the setting of derived [[global analytic geometry]].

## Motivation for a derived notion of microlocalization

In non-smooth situations, the usual normal and conormal bundle used in classical microlocalization is not well behaved, and one needs to take derived versions of them. Moreover, to study [[global analytic index theory]], one needs a formulation of the theory of microlocalization in terms of derived loop stacks, in order to work out a cyclic and Hochshild version of index theorems (similar to the one developed by Kashiwara and Schapira in the [[microlocal formulation of index theory]]) that works on an arbitrary (e.g. integral) basis.

## Loop space and deformation to the normal bundle: method I

We would like to define a family over (say) the disc $D^1$ whose fiber
at $0$ is $LX:=\Hom(S^1,X)$ and whose fiber at $1$ is $[LX/S^1]$.
In the case of a usual scheme, this is done (by Vezzosi) by showing that
$LX$ identifies with $\Hom(BG_a,X)$, and by making $G_a$ act
multiplicatively on the action of $BG_a$ on $LX$, to get the trivial
action at $0$ and the usual action at $1$. This gives the desired family.
Recall that $BG_a$ is the affinization of $B\mathbb{Z}\cong S^1$.

## Loop space and deformation to the normal bundle: method II

[[Microlocalization]] in [[derived geometry]] involves the proper definition of the deformation to the [[normal bundle]] of a closed embedding $Y\subset X$ of global analytic spaces (or even stacks). One needs to consider a [[derived loop space]] approach to this construction because it allows to avoid the use of denominators in the definition of the [[Chern character]] (following Connes-Loday-Toen-Vezzosi) and in the development of more general [[global analytic index theory]] with integral coefficients.

The space of paths on $X$ based on $Y$ is the groupoid acting on $Y$ given by
$$P_Y X:=\{f\in \Hom(\Delta^1,X),\;f(0)\in Y,\;f(1)\in Y\}.$$
More concretely, this is given by the homotopy pullback
$$
\array{
    P_Y X &\to& \Hom(\Delta^1,X)
    \\
    \downarrow && \downarrow^{\mathrlap{ev_0\times ev_1}}
    \\
    Y\times Y &\to& X\times X
}
$$
Notice that the natural projection $P_Y X\to Y\times Y$ makes $P_Y X$ a groupoid (of paths in $X$) acting on $Y$. In the case of the diagonal immersion $Y=M\hookrightarrow M\times M=X$, we get
$$P_Y X\cong LM:=\Hom(S^1,M).$$

There is a natural projection $p:P_Y X\to \Hom(\Delta^1,X)\sim X$ and $P_Y X$ is equiped with the natural structure of a groupoid acting on $Y$ through the projection $P_Y X\to Y\times Y$.
A more explicit description of the loop space (that is obtained by using the homotopy $\Delta^1\sim \Delta^0$) is given by the homotopy pullback
$$P_Y X=Y\times^h_X Y,$$
which clearly has an interesting meaning only in the setting of derived geometry. See also at _[[derived loop space]]_ and at _[[Hochschild cohomology]]_.

To make the above construction work for general Artin stacks, one needs to make it local for the smooth topology. This is done by replacing the loop space groupoid $P_Y X$ acting on $Y$ by its formal completion along the identity morphism. This gives a formal groupoid $\hat{P}_Y X$ that is equivalent modulo the [[HKR]] theorem (in characteristic $0$) to $T_Y[-1]X$ for a general Artin stack. Similarly, we will have $\hat{L}M:=\hat{P}_M(M\times M)\cong T[-1]M$ by [[HKR]].

The deformation to the normal bundle in strict derived global analytic geometry is then simply given by the (**false to be corrected**) formula (with $D^1=\mathbb{M}(R\{X\}^\dagger)$ for $R$ the base ind-Banach ring)
$$
\widetilde{P_Y X}:=\{f\in \Hom_{D^1}(\Delta^1\times D^1,X\times D^1),\;f(0,0)\in Y,\; f(1,0)\in Y,\;f(-,1)\in (X\backslash Y)\Rightarrow f(x,t)\in (X\backslash Y)\forall t\neq 0\}.
$$
More concretely, this is given by the homotopy pullback (where $U(1)=\mathbb{M}(R\{X,Y\}^\dagger/(XY-1))$ and we use the homotopy equivalence $\Hom(\Delta^1,X)\sim X$)
$$
\array{
    \widetilde{P_Y X} &\to& \Hom_{D^1}(\Delta^1\times D^1,X\times D^1)
    \\
    \downarrow && \downarrow^{\mathrlap{ev_{(0,0)}\times ev_{(1,0)}\times \ev_{(-,U(1))}\times \ev_{(-,1)}}}
    \\
    Y\times Y\times \Hom(\Delta^1\times U(1),(X\backslash Y))\times (X\backslash Y) &\stackrel{i\times j\times k}{\to}& X\times X\times \Hom(\Delta^1\times U(1),X)
\times X}
$$
It has an evident natural projection $t:\widetilde{P_Y X}\to D^1$, and a natural projection $\widetilde{P_Y X}\to Y\times Y$ that makes it a family of groupoids parametrized by $D^1$ and acting on $Y$. There is also a natural projection
$p:\widetilde{P_Y X}\to X$ given by $f\mapsto f(-,1)$. We will denote $s:P_Y X\to \widetilde{P_Y X}$ the fiber $t^{-1}(0)$.

One may complete $\widetilde{P_Y X}$ along the unit of its groupoid structures, to get a family of formal groupoids $\widehat{P_Y X}$ parametrized by $D^1$, whose fiber at $0$ will be the formal loop space $\hat{P}_Y X$ obtained by completing the path groupoid $P_Y X$ acting on $Y$ along its identity morphism, and whose fiber at $1$ will be the formal completion $\hat{X}_Y$ of $X$ along $Y$. There is a natural action of $S^1$ on $\widehat{P_Y X}$. We will still denote $t:\widehat{P_Y X}\to D^1$ the natural projection, $s:\widehat{P}_Y X\to \widehat{P_Y X}$ the fiber $t^{-1}(0)$, and $p:\widehat{P_Y X}\to X$ the natural projection (evaluation at $(-,1)$).

In the particular case of the diagonal embedding $Y=M\hookrightarrow M\times M=X$ over a field of characteristic $0$, the quotient of $X$ by this family of formal groupoids gives back (modulo a convenient [[HKR]] theorem) exactly Simpson's non-abelian Hodge structure, that gives a family of formal stacks over $D^1$ whose fiber at $0$ is the tangent space $TX$ (seen as the quotient of $X$ by the trivial action of $T[-1]X$) and whose fiber at $1$ is the so-called de Rham space $X_{dR}$ of $X$.

## A derived analog of microlocalization

We now work on an arbitrary derived Artin stack $M$, such that the diagonal $Y=M\to M\times M=X$ is an affine closed embedding.

The usual theory of [[microlocalization]] may be adapted to the derived global analytic setting by replacing the deformation to the normal bundle $\widetilde{T_Y X}\to \mathbb{A}^1$ by the formal deformation to the normal bundle $\widehat{P_Y X}\to D^1$.

The derived specialization functor is given on $F\in D^b(X)$ by
$$\nu_Y(F):=s^*p^*F\in D^b(\hat{P}_Y X).$$

Remark that the loop space analog $L^*X$ of the odd cotangent bundle $T^*[1]X$ (that should be dual to the odd tangent bundle $T[-1]X$ (related to $LX$ by the [[HKR]] theorem) is given by $S^1\otimes X$ (external tensor product by the simplicial circle). We will have for every stack $Y$, the canonical equivalence
$$\Hom(S^1\otimes X,Y)\cong \Hom_{SSets}(S^1,\Hom_{dSt}(X,Y)).$$

The derived Fourier-Sato transformation is given on $F\in D^b(\hat{P}_Y X)$ by
$$\Phi(F):=?.$$

The derived microlocalization functor is given on $F\in D^b(M)$ by
$$\mu(F):=\Phi(\nu_Y(F))$$
for $Y=M\hookrightarrow M\times M=X$.

## A derived analytic analog of microlocalization

One may replace the simplicial circle $S^1$, used in the definition of the derived loop space, by the unitary group $U(1)$ of [[overconvergent global analytic geometry]], to get a more analytic theory of microlocalization. In the complex situation, we will have $U(1)\cong S^1$ up to $D^1$-homotopy. Remark that the exponential map $exp(i-):\mathbb{R}\to S^1$ also has a meaning in overconvergent complex analytic geometry, if we see $\mathbb{R}\subset \mathbb{C}$ as a closed subset equipped with its germs of analytic functions.

In the global analytic setting (without imposing homotopy invariance), there is no reason to have $U(1)=D^1\times_{*\coprod *} *$. We thus prefer to use $P^1:=D^1\coprod_{U(1)}D^1$ as a natural global analytic parameter space for paths. We define
$$L^\dagger_Y X:=\{f\in Hom(P^1,X), f(0)\in Y,\; f(\infty)\in Y\}.$$
Remark that up to $D^1_\mathbb{R}$-homotopy, we get
$$L^\dagger_Y X\sim L_Y X.$$

We must now check that the natural morphism

## References

* [[David Ben-Zvi]], [[David Nadler]], _Loops spaces and connections_ ([arXiv:1002.3636](http://arxiv.org/abs/1002.3636))

* [[Damien Calaque]], [[Andrei Caldararu]] and [[Junwu Tu]]: _PBW for an inclusion of Lie algebras_ ([arXiv:1010.0985](http://arxiv.org/abs/1010.0985))

* [[Damien Calaque]], [[Andrei Caldararu]] and [[Junwu Tu]]: _On the Lie algebroid of a derived self-intersection_ ([arXiv:1306.5260](http://arxiv.org/abs/1306.5260))


derived module

Given a group homomorphism, $\varphi:H\to G$, the *derived module* of $\varphi$ is a $\mathbb{Z}[G]$-module, $D_\varphi$, together with a (universal)  $\varphi$-[[derivation on a group|derivation]],

$$\partial_\varphi :H\to D_\varphi,$$

such that, given any $\varphi$-derivation, $f_1:H\to M$, for $M$ a $G$-module, there is a unique $G$-module morphism, $\overline{f_1}: D_\varphi\to M$ such that $f_1= \overline{f_1}\circ \partial_\varphi$.

##Examples
* If $\varphi$ is the identity morphism on $G$ then the augmentation ideal, $I(G)$, together with

$$d_G:G\to I(G)$$

sending $g$ to $g-1$ is the derived module of $id_G$ aka the derived module of $G$.

##References

For the original version of derived module, see

* [[R. H. Crowell]], _The derived module of a homomorphism_, Advances in Math., 5, (1971), 210–238.

For applications in [[arithmetic topology]]

* {#Morishita12} [[Masanori Morishita]], _Knots and Primes: An Introduction to Arithmetic Topology_, 2012 ([web](https://books.google.co.uk/books?id=DOnkGOTnI78C&pg=PA156#v=onepage&q&f=false))

[[!redirects derived modules]]


derived moduli space

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## Idea

A _derived moduli stack_ is a [[moduli space]] in the context of [[derived algebraic geometry]].

The existence of derived moduli stacks was first conjectured by [[Maxim Kontsevich]], see [[hidden smoothness principle]].

## Related concepts

* [[classifying space]], [[classifying stack]], [[moduli space]], [[moduli stack]]

* [[classifying topos]]

* [[universal principal bundle]], [[universal principal infinity-bundle]]

* [[classifying morphism]]

## References

The first examples of [[derived moduli spaces]], using the language of [[dg-schemes]], were constructed in

* [[Ionut Ciocan-Fontanine]], [[Mikhail Kapranov]], _Derived Quot
schemes_, 1999,
[arXiv:math/9905174](http://arxiv.org/abs/math/9905174).
{#CiKa1

* [[Ionut Ciocan-Fontanine]], [[Mikhail Kapranov]], _Derived Hilbert
schemes_, 2000,
[arXiv:math/0005155](http://arxiv.org/abs/math/0005155).
{#CiKa2}

In the language of [[derived algebraic geometry]] they were considered in

* Jacob Lurie, [[Moduli Problems and DG-Lie Algebras]]

* [[J. P. Pridham]], _Representability of derived stacks_, [arxiv/1011.2189](http://arxiv.org/abs/1011.2189)

For the construction of the derived moduli stack of [[perfect complexes]] on a [[smooth scheme|smooth]] [[proper morphism|proper]] [[scheme]], see

* [[Bertrand Toen]], [[Michel Vaquie]], _Moduli of objects in dg-categories_, 2007, [arXiv:math/0503269](http://arxiv.org/abs/math/0503269).

[[!redirects derived moduli spaces]]
[[!redirects derived moduli stack]]
[[!redirects derived moduli stacks]]


derived moduli stack of objects in a dg-category

Given any [[dg-category]] $\mathcal{C}$, there is an associated [[derived moduli stack]] $\mathcal{M}_\mathcal{C}$ which parametrizes the [[pseudo-perfect dg-module|pseudo-perfect]] [[dg-modules]] over $\mathcal{C}^{op}$.  When $\mathcal{C}$ is [[smooth dg-category|smooth]] and [[proper dg-category|proper]], $\mathcal{M}_\mathcal{C}$ classifies the [[objects]] of $\mathcal{C}$.

In the case where $\mathcal{C}$ is the [[dg-category]] [[dg-enhancement|enhancing]] the [[triangulated categories of sheaves|derived category]] of a [[smooth scheme|smooth]] [[proper morphism|proper]] [[scheme]] $X$, the [[derived Artin stack]] $\mathcal{M}_X$ is called the **derived moduli stack of perfect complexes** on $X$.

**Proposition** ([Toen-Vezzosi, 3.4](#TV)).  The [[functor]] $\mathcal{M}_- : Ho(DGCat(k))^{op} \to DSt(k)$ admits a [[left adjoint]]
 $$Pf : \DSt(k) \to \Ho(\DGCat(k))^{op}$$
which associates to a [[derived stack]] its [[dg-category]] of [[perfect complexes]].

## References

* [[Bertrand Toen]], [[Michel Vaquie]], _Moduli of objects in dg-categories_, 2007, [arXiv:math/0503269](http://arxiv.org/abs/math/0503269).
{#TV}


derived Morita equivalence

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## Idea

**Derived Morita equivalence** is a generalization of [[Morita equivalence]] to the "derived" context ([[homotopy theory]] of [[dg-algebras]]). Just as two $k$-[[associative algebra|algebras]] are Morita equivalent if and only if their [[category of modules|categories of left modules]] are [[equivalence of categories|equivalent]], the coarser equivalence relation of derived Morita equivalence holds whenever for two [[differential graded algebras]] their (bounded) [[derived categories]] [[category of modules|of modules]], along with their [[triangulated category]] structure, are equivalent.

The existence of a tilting complex is necessary and sufficient for an equivalence between the unbounded derived categories of two rings. A tilting complex is a special small generator of the derived category. It is a bounded complex $T$ of finitely generated projective $R$-modules which generates the derived category $\mathcal{D}(R)$ and whose graded ring of self maps $\mathcal{D}(R)(T, T)_{\ast}$ is concentrated in dimension zero.

## Examples

A derived Morita equivalence in the context of [[homological mirror symmetry]] appears in ([Okada 09](#Okada09))

## References

Related notions include [[Fourier-Mukai transform]], [[mirror symmetry]]

The classical work in algebra concerning the Morita for derived categories of modules is due Rickard (also called Rickard equivalence)

* Jeremy Rickard, _Morita theory for derived categories_, J. London Math. Soc.  2-39:3 (1989) 436-456 [doi](https://doi.org/10.1112/jlms/s2-39.3.436)

Contemporary generality is outlined in

* [[Stefan Schwede]], _Morita theory in abelian, derived and stable model categories_, In: Structured Ring Spectra (eds. A. Baker, B. Richter) [arXiv:math/0310146](http://arxiv.org/abs/math/0310146)
* B. Shipley, _Morita theory in stable homotopy theory_, Handbook of Tilting Theory, vol. 13 (L. A. Hügel, D. Happel)

In the setting of [[dg-categories]]:

* [[B. Toen]], _The homotopy theory of dg-categories and derived Morita theory_, [arXiv:math/0408337](http://arxiv.org/abs/math/0408337).

In the setting of [[stable (infinity,1)-categories]] (section 4):

* [[Andrew J. Blumberg]], [[David Gepner]], [[Goncalo Tabuada]], _A universal characterization of higher algebraic K-theory_, [arXiv:1001.2282](http://arxiv.org/abs/1001.2282v4).

* {#Okada09} So Okada, _Homological mirror symmetry of Fermat polynomials_ ([arXiv:0910.2014](http://arxiv.org/abs/0910.2014))

For a treatment in terms of bicategories:

* Niles Johnson, _Morita theory For derived categories: a bicategorical perspective_ ([arXiv:0805.3673](http://arxiv.org/abs/0805.3673))


derived noncommutative geometry

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## Idea

In [[noncommutative algebraic geometry]] one represents a [[scheme]] by an [[abelian category]] of [[quasicoherent sheaf|quasicoherent sheaves]] on the [[scheme]], and looks at more general abelian categories as being categories of quasicoherent sheaves on a noncommutative space.

In _derived (higher) noncommutative (algebraic) geometry_ one instead considers the [[derived category]] of [[quasicoherent sheaf|quasicoherent sheaves]], or more precisely its [[enhanced triangulated category|dg-enhancement]] or [[A-infinity-category|A-infinity-enhancement]]; dg-enhancements for the derived categories of quasiprojective smooth varieties are essentially unique by the results of Lunts and Orlov. Taking the derived category instead of the abelian loses a bit of information but sometimes the information is sufficient.

In general one represents complex noncommutative spaces by [[pretriangulated dg-category|pretriangulated dg-categories]].
They may be viewed as models for stable $(\infty,1)$-categories.
Note that accessible stable $(\infty,1)$-categories are quite close to Grothendieck $(\infty,1)$-topoi; more flexibility one gets from
pretriangulated $A_\infty$-categories or, even better,
certain class of [[spectral categories]].
This is well into [[homotopy theory]] area. Quillen [[model category]] structures and [[homotopy limit]]s in the context of dg-categories were studied by a number of people (including the impressive thesis by Tabuada). On the other hand, over a mixed characteristics, the meaning of such representations is less well understood.

Derived noncommutative geometry has been informally introduced by Kapranov-Bondal and later Orlov around 1990; full framework belongs
to [[Kontsevich]], [[Valery Lunts|Lunts]], van den Bergh, Katzarkov, Kuznetsov, Kaledin. Some of the works of [[Bertrand Toen|Toen]], Vaquie, Keller, Cisinski, Tabuada are properly in this area as well.

## Definitions

In [Katzarkov-Kontsevich-Pantev](#KatzarkovKontsevichPantev)
the following definition is given.

+-- {: .un_definition}
###### Definition (graded complex noncommutative space)

A __graded complex nc-space__ is a $\mathbb{C}$-linear [[dg-category|differential graded category]] $C$ which is homotopy complete and cocomplete (has all [[homotopy limit]]s and colimits).

=--

The [[derived category|derived categories]] of [[quasicoherent sheaf|quasicoherent sheaves]] on a [[scheme]] over $Spec(\mathbb{C})$ is one of the examples; another example is the category of dg-modules over a fixed [[dg-algebra]] $A$, which are such that $A$ admits an exhaustive filtration
such that the associated graded is a sum of shifts of $A$.
Call that category $A$-$Mod$.

Kontsevich calls a complex differential $\mathbb{Z}/2$-graded algebra

* __smooth__ if $A$ is a perfect object in the category of $A$-$A$-dg-bimodules (perfect object here means that $Hom(A,-)$ preserves small homotopy colimits);

* __compact__ if the total complex dimension of its cohomology $H^\bullet(A,d_A)$ is finite

The category $A$-$Mod$ is a smooth (resp. compact) nc space if the underlying dg-algebras $A$ is; this notion depends on the category and not on the underlying dg-algebra.

The above definition implies that a category $C$ which represents a nc-space in the sense above is [[triangulated category|triangulated]] and [[Karoubi envelope|Karoubi closed]]. Sometimes this are requirements in another variant of the definition.

+-- {: .un_definition}
###### Definition (noncommutative space (M. Kontsevich))

A __noncommutative space__ $X$ is a small [[triangulated category]] $C_X$ which is [[Karoubi envelope|Karoubi closed]] (=every [[idempotent]] is a [[split idempotent]]) and appropriately [[enriched category|enriched]] over either

* [[spectrum|spectra]] $Hom_{C_X}(E,F[i])=\pi_{-i}\mathbf{Hom}_{C_X}(E,F)$

* [[chain complex|complexes]] of $k$-vector spaces (i.e. is a [[dg-category]]): $Hom(E,F[i])= H^i(\mathbf{Hom}_{C_X}(E,F))$. $C_X$ is $k$-linear over a field $k$ and one writes $X/k$. If instead $k$ is replaced by a ring $R$ then one enriches over complexes of $R$-modules which are [[model structure on chain complexes|cofibrant]].

=--

This receives good motivation from the fact that [[stable (infinity,1)-categories]] with a small set of generators are equivalently the [[categories of modules]] over an $A_\infty$-ring(oid). See [there](stable+(infinity%2C1)-category#AsCategoriesOfModules).

## Some history

In the early works of the Moscow school ([[Kapranov]], [[Alexei Bondal|Bondal]], [[Dmitri Orlov|Orlov]], [[Kontsevich]]) one replaces a [[variety]] by the [[derived category]] of [[coherent sheaves]] (or [[quasicoherent sheaves]] on that variety, or [[dg-category]] (or [[A-infinity category]]) [[enhanced triangulated category|enhancements]] thereof. There are also noncommutative deformations of such derived categories and analogues like the categories corresponding to the so-called [[Landau-Ginzburg model]]s. Therefore [[noncommutative derived algebraic geometry]] (and even [[noncommutative motives]]).

Notice that the [[derived category]] of coherent sheaves on a variety does _not_ remember all the structure of the original variety hence derived geometry loses often some information (sometimes not); thus derived algebraic geometry is sometimes easier than nonderived.

## References

Survey includes

* {#Kontsevich15} [[Maxim Kontsevich]], _Geometry in triangulated categories_, talk at _[[New Spaces for Mathematics and Physics]]_, IHP Paris, Oct-Sept 2015 ([video recording](https://www.youtube.com/watch?v=XnmO9RAbl50))

Original articles include

* L. Katzarkov, [[Maxim Kontsevich]], [[Tony Pantev]], _Hodge theoretic aspects of mirror symmetry_, [arxiv:0806.0107](http://arxiv.org/abs/0806.0107)
{#KatzarkovKontsevichPantev}

and a bit earlier this treatise on formal (infinitesimal in the sense of [[formal schemes]]) aspect as used in the [[deformation theory]] is in

* {#KontsevichSoibelman} [[Maxim Kontsevich]], [[Yan Soibelman]], _Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I_, [math.AG/0606241](http://arxiv.org/abs/math/0606241)

* [[Dmitry Kaledin]], _Homological methods in non-commutative geometry_  (2008) ([pdf](http://imperium.lenin.ru/~kaledin/math/tokyo/final.pdf))

The relations to tropical and symplectic geometry are in recent Kontsevich's talk at 2009 Arbeitstagung:

* [[Maxim Kontsevich]], Mathematische Arbeitstagung 2009, _Symplectic geometry of homological algebra_, preprint MPIM2009-40a, [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=4024)

Homological mirror symmetry is one of the main motivations and statements of the derived noncommutative algebraic geometry

* [[Maxim Kontsevich]], _Homological algebra of mirror symmetry_,   Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994),  120--139, Birkhäuser, Basel, 1995.

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Homological mirror symmetry and torus fibrations_, in Symplectic geometry and mirror symmetry (Seoul, 2000),  203--263, World Sci. Publ., River Edge, NJ, 2001.

* [[Dmitri Orlov]], _Smooth and proper noncommutative schemes and gluing of DG categories_, [arXiv](http://arxiv.org/abs/1402.7364v1).

Algebraic geometry over [[formal duals]] of [[E-n algebras]] is considered in

* [[John Francis]], _Derived algebraic geometry over $\mathcal{E}_n$-Rings_ ([pdf](http://math.northwestern.edu/~jnkf/writ/thezrev.pdf))

* [[John Francis]], _The tangent complex and Hochschild cohomology of $\mathcal{E}_n$-rings_ ([pdf](http://math.northwestern.edu/~jnkf/writ/cotangentcomplex.pdf))

Notice that for $n \geq 2$ the underlying ordinary rings (under $\pi_0$) are commutative. Therefore this has similarities with the [[formal geometry|formal]] [[noncommutative algebraic geometry]] perturbating around abelian schemes that is discussed in

* {#Kapranov98} [[Mikhail Kapranov]], _Noncommutative geometry based on commutator expansions_ ([arXiv:9802041](http://arxiv.org/abs/math/9802041))

For more on this see at _[[Kapranov's noncommutative geometry]]_

[[!redirects derived noncommutative algebraic geometry]]
[[!redirects noncommutative derived algebraic geometry]]


derived representation scheme

Derived representation scheme $DRep_n$ is a (nonabelian) derived functor (or its geometric counterpart) of a functor $Rep_n$ which to an associative algebra $A$ assigns the affine scheme of $n$-dimensional representations of $A$.

There are several formalisms approaching such a derived functor.

* I. Ciocan-Fontanine, M. Kapranov, _Derived Quot schemes_, Ann. Sci. ENS __34__ (2001) 403–440
* [[Yu. Berest]], X. Chen, F. Eshmatov, A. Ramadoss, _Noncommutative Poisson structures, derived representation schemes and Calabi-Yau algebras_, Contemp. Math. __583__ (2012) 219--246 [arXiv:1202.2717](https://arxiv.org/pdf/1202.2717)
* [[Yu. Berest]], [[G. Felder]], A. Ramadoss, _Derived representation schemes and noncommutative geometry_, [arXiv:1304.5314](https://arxiv.org/abs/1304.5314)
* George Khachatryan, _Derived representation schemes and non-commutative geometry_, Cornell PhD thesis under guidance of Yuri Berest [online](https://hdl.handle.net/1813/29138]
* Yuri Berest, George Khachatryan, Ajay Ramadoss, _Derived representation schemes and cyclic homology_, Adv. Math.
__245,__ (2013) 625--689 [arXiv:1112.1449](https://arxiv.org/abs/1112.1449)
* [[Yuri Berest]], [[Giovanni Felder]], Sasha Patotski, Ajay C. Ramadoss, [[Thomas Willwacher]], _Representation homology, Lie algebra cohomology and derived Harish-Chandra homomorphism_, J. Eur. Math. Soc. 19:9 (2017) 2811--2893 [arXiv:1410.0043](https://arxiv.org/abs/1410.0043)

Related notions: Kontsevich-Rosenberg principle


derived scheme

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{:toc}

## Idea

... [[derived algebraic geometry]] ... [[higher algebra]] ...[[generalized scheme]]...

## Definition

Let $k$ be a commutative [[ring]].

A **derived scheme** (over $k$) is a [[generalized scheme]] in the sense of locally affine $\mathcal{G}$-[[structured (infinity,1)-topos]] for $\mathcal{G} = \mathcal{G}_{Zar}(k)$ the Zariski [[geometry (for structured (infinity,1)-toposes)]].

## Special cases

A [[n-truncated structured (infinity,1)-topos|0-trucated]] and [[n-localic (infinity,1)-topos|0-localic]] derived scheme is precisely an ordinary [[scheme]].

More precisely:

+-- {: .un_prop }
###### Proposition ([[Structured Spaces|StSp]], 4.2.9)

Let $Sch_{\leq 0}^{\leq 0}(\mathcal{G}_{Zar}(k)) \subset Sch(\mathcal{G}_{Zar}(k))$ be the full subcategory of all derived schemes on the [[n-truncated structured (infinity,1)-topos|0-trucated]] and [[n-localic (infinity,1)-topos|0-localic]] ones. This is canonically equivalent to the ordinary category $Sch(k)$ of [[scheme]]s over $k$:

$$
  Sch_{\leq 0}^{\leq 0}(\mathcal{G}_{Zar}(k))
  \simeq
  Sch(k)
  \,.
$$

=--

For more comments on this see also

* [[scheme as locally affine structured (infinity,1)-topos]].

## Related concepts

Notice that for [[generalized scheme]]s the Zariski [[geometry (for structured (infinity,1)-toposes)]] $\mathcal{G}_{Zar}(k)$ is not interchangeable with the [[étale (∞,1)-site]] $\mathcal{G}_{et}(k)$. Instead $\mathcal{G}_{et}(k)$-[[generalized scheme]]s are [[derived Deligne-Mumford stack]]s.

##  References

section 4.2 in

* [[Jacob Lurie]], _[[Structured Spaces]]_

[[!redirects derived schemes]]


derived series

Given a group $G$ its derived series is the decreasing (under inclusion order), inductively defined sequence of its subgroups

$$
G = G_0 \supset G_1 \supset G_2\supset G_3\supset \ldots
$$

in which $G_k = [G_{k-1},G_{k-1}]$ is the commutator, that is the subgroup of $G_{k-1}$ generated by all elements of the form $ghg^{-1}h^{-1}$ where $g,h\in G_{k-1}$. A group is __solvable__ iff its derived series terminates with the trivial subgroup after finitely many terms.

Similarly, one defines a derived series for a Lie algebra $L$, and for $\Omega$-[[Omega-group|groups]].


derived smooth manifold

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## Idea

A _derived smooth manifold_ is the generalization of a [[smooth manifold]] in [[derived differential geometry]]: the [[derived geometry]] over the [[Lawvere theory]] for [[smooth algebra]]s ($C^\infty$-rings):

it is a [[structured (∞,1)-topos]] whose [[structure sheaf]] of functions is a [[smooth (∞,1)-algebra]].

## Motivation: correction of limits

According to the general logic of [[derived geometry]], passing from [[smooth manifold]]s to _derived smooth manifold_ serves to _correct_ certain [[limit]]s that do exist in [[Diff]] but do not have the correct [[cohomology|cohomological behaviour]]. This concerns notably [[pullback]]s along [[smooth function]]s that are not [[transversal map]]s.

### Pontrjagin-Thom construction

For $X$ a [[compact space|compact]] [[smooth manifold]], by the [[Pontrjagin-Thom construction]] there is a [[smooth function]] $f : S^n \to M O$ from an $n$-sphere to the [[Thom spectrum]] such that if chosen [[transversal map|transversal]] to the zero-[[section]] $B \hookrightarrow M O$ the [[pullback]] $f^* B$

$$
  \array{
     f^* B &\to& B
     \\
     \downarrow && \downarrow
     \\
     S^n &\stackrel{f}{\to}& M O
  }
$$

is a manifold cobordant to $X$, so that $[X] \simeq [f^* B]$ in the [[cobordism ring]] $\Omega$.

By using derived smooth manifolds instead of ordinary smooth manifolds here, the condition that $f$ be transversal to $B$ could be dropped.

### String topology

(...)

### Floer homology

(...)

## Definition

The following definition characterizes the design criterion for derived smooth manifolds as being objects for which homotopy-intersections

$$
  A \cap_X B := A \times_X^h B
$$

preserve the [[cup product]] in the [[cobordism ring]]

$$
  [A] \smile [B] \simeq [A \cap_X B]
  \,.
$$

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

We say an [[(∞,1)-category]] $C$ supports **derived cup products for cobordisms** if

* it is equipped with a [[full and faithful functor]]

  $$
    i : Diff \hookrightarrow C
  $$

  embedding the category of [[smooth manifold]]s into it;

* for any two submanifolds $A \to X \leftarrow B$ (transversal or not) the [[(∞,1)-pullback]]

  $$
    A \cap_X B := i(A) \times_{i(X)} i(B)
  $$

  exists in $C$;

* if $A \to X \leftarrow B$ happen to be [[transverse map]]s then

  $$
    i(A \times_X B) \simeq i(A) \times_{i(X)} i(B)
    \,,
  $$

  with the image under $i$ of the ordinary [[pullback]] in [[Diff]] on the left;

* $i$ preserves the [[terminal object]];

* (...nice interaction with underlying topological spaces...)

* for each $X \in Diff$ there is a _derived cobordism ring_
  $\Omega(X)$ such that ...

* for any submanifolds $A \to X \leftarrow B$ we have

  $$
    [A] \smile [B] = [A \cap_X B]
  $$

  in $\Omega(X)$

  (...)

=--

A central statement about derived smooth manifolds will be

+-- {: .un_theorem}
###### Theorem

The $(\infty,1)$-category of derived smooth manifolds has derived cup products for cobordisms.

=--

This is ([Spivak, theorem 1.8](#Spivak)).

## Related concepts

* [[dg-manifold]]

## References

The definition of derived smooth manifolds is indicated at the very end of

* [[Jacob Lurie]], _[[Structured Spaces]]_ .

A detailed construction and discjussion in terms of the [[model category]] [[presentable (∞,1)-category|presentation]] by [[homotopy T-algebra]]s is in

* {#Spivak08} {#Spivak} [[David Spivak]], _Derived smooth manifolds_, Duke Math. J. **153** 1 (2010) 22-128 [[arXiv:0810.5174](http://arxiv.org/abs/0810.5174), [doi:10.1215/00127094-2010-021](https://doi.org/10.1215/00127094-2010-021)]

Something similar (derived [[orbifolds]]):

* [[Dominic Joyce]], _D-orbifolds, Kuranishi spaces, and polyfolds_ talk notes (Jan 2010) [[pdf](http://people.maths.ox.ac.uk/joyce/dmtalk.pdf), [[Joyce-DOrbifolds.pdf:file]]]

Seminar notes:

* [[Urs Schreiber]], _[[schreiber:Seminar on derived differential geometry]]_ (2010)

Further developments:

* Dennis Borisov, Justin Noel, _Simplicial approach to derived differential manifolds_ ([arXiv:1112.0033](http://arxiv.org/abs/1112.0033))

* {#CarchediSteffens19} [[David Carchedi]], [[Pelle Steffens]], _On the universal property of derived manifolds_ [[arXiv:1905.06195](https://arxiv.org/abs/1905.06195)]

  > ([[universal property]] of the [[(infinity,1)-category]] of derived smooth manifolds)

* [[David Carchedi]], *Derived Manifolds as Differential Graded Manifolds* [[arXiv:2303.11140](https://arxiv.org/abs/2303.11140)]

* [[Gregory Taroyan]], *Equivalent models of derived stacks* [[arXiv:2303.12699](https://arxiv.org/abs/2303.12699)]

* [[Pelle Steffens]], *Derived $C^\infty$-Geometry I: Foundations* [[arXiv:2304.08671](https://arxiv.org/abs/2304.08671)]

[[!redirects derived smooth manifolds]]

[[!redirects derived manifold]]
[[!redirects derived manifolds]]


derived stack

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{:toc}

## Idea

A _derived stack_ $X$ is an [[∞-stack]] -- an [[(∞,1)-sheaf]] -- on an [[(∞,1)-site]] $C$.  For example, $C$ is most often the [[(∞,1)-category]] of [[simplicial commutative rings]] or [[E-∞ ring spectra]] (with the [[Zariski topology|Zariski]] or [[etale topology|etale]] topology).

One says _derived stack_ in order to distinguish from the more restrictive notion of an [[∞-stack]] on a 1-categorical [[site]], such as for instance described at [[topological ∞-groupoid]].

For recall that a [[sheaf]] is a [[functor]] $F : C^{op} \to Set$ satisfying some [[descent]]-condition.
So there are two steps in which the notion of [[sheaf]] may be [[vertical categorification|categorified]]:

1. the codomain is categorified and the domain remains a 1-category
1. the codomain and the domain are categorified.

The categorification of the codomain leads to the notion of [[stack]]s when [[Set]] replaced by [[Grpd]], and further to [[∞-stack]]s, when sets are replaced by [[∞-groupoid]]s.

But there is no natural reason why the domain should in general remain a 1-category if one passes to an [[infinity-category|∞-categorical]]-context. A _derived stack_ is a generalization of the notion of sheaf where both domain and codomain are taken to be $\infty$-categorical.

Following the general logic of [[models for ∞-stack (∞,1)-toposes]], derived stacks are typically [[model category|modeled]] by the [[model structure on SSet-enriched presheaves]] on an [[SSet-site]] or [[model site]] $C$. In such a model a derived stack is represented by an  [[SSet]]-[[enriched functor]]
$F : C^{op} \to SSet$ from an [[SSet]]-[[enriched category]] $C$ to [[SSet]] that satisfies a [[descent]] condition.

## Central motivation: derived stacks have good limits

One general idea for the use of higher and derived stacks is that

* passing to a higher categorical codomain  -- i.e. from Set-values [[sheaf|sheaves]] to higher [[groupoid]] valued sheaves -- is a means to obtain _good colimits_, [[colimit]]s that do not lose information. For instance

  * in the category [[Diff]] of manifolds the quotient by a non-free action of a group may not exist

  * in [[sheaf|sheaves]] in $[Diff^{op},Set]$ it will exist, but will have the wrong properties in general with respect to some operations such as taking cohomology,

  * while finally in [[stack]]s $[Diff^{op}, Grpd]$ it exists as the corresponding smooth [[action groupoid]] or [[orbifold]] and in this form rembers in terms of the isomorphisms how the quotient was obtained. The cohomology of the stack is then indeed the equivariant cohomology of the original manifold.

* similarly passing to higher categorical domain -- i.e. from presheaves on categories to presheaves on higher categories, is analogously a means to ensure that good [[limit]]s exist.

A detailed illustration and motivation of the need of these "good limits" that don't forget the way they were formed is

* in the introductory section of

  * [[Jacob Lurie]], _[[Structured Spaces]]_

* in the introductory section of

  * David Spivak, _[Quasi-smooth derived manifolds](http://math.berkeley.edu/~dspivak/thesis2.pdf)_

## Examples

* A derived refinement of the ordinary [[site]] [[CRing]]${}^{op}$ of formal duals to commutative tings is the [[SSet-site]] $SCRing^{op}$ of [[simplicial ring]]s. Derived stacks on this site are studied in [[derived algebraic geometry]].

  An introduction to this is for instance in
  [chapter 5](http://www.crm.cat/HigherCategories/hc1.pdf#page=135) of the lecture notes

  * [[Bertrand Toen]], _Simplicial presheaves and derived algebraic geometry_ course on [Simplicial Methods in Higher Categories](http://www.crm.cat/HigherCategories/) ([pdf](http://www.crm.cat/HigherCategories/hc1.pdf#page=99))

  There are various slight variations of this. For instance using the [[monoidal Dold-Kan correspondence]], simplicial rings may be replaced with non-positively graded cochain [[dg-algebra]]s.

  One application of derived stacks on $(dgAlg^-)^{op}$ is the [[BV-BRST formalism]] in [[physics]].

* A derived refinement of the ordinary [[site]] $\mathbb{L}$ of [[smooth loci]] is the [[SSet-site]] $cs \mathbb{L}$ of [[cosimplicial object|cosimplicial]] [[smooth loci]]. Derived stacks on this are the objects in a theory that could be called [[derived synthetic differential geometry]].

## Remarks

### Derived Yoneda embedding

One obvious but notable phenomenon that occurs in derived stacks in general, but not in [[∞-stack]]s over a 1-categorical site, is that under the [[Yoneda lemma for (∞,1)-categories|Yoneda embedding for (∞,1)-categories]] a categorically discrete object, i,e. a [[n-truncated object in an (∞,1)-category|0-truncated object]] may be mapped to a higher categorical object.

Consider specifically an ordinary [[site]] $C$ and let $csC = [\Delta,C]$ be the corresponding [[SSet-site]] of [[cosimplicial object]]s. Write

$$
  \mathbf{Y} : C \hookrightarrow [\Delta,C] \stackrel{\mathbb{R}Y}{\to}
  [[\Delta,C],SSet]
$$

for the composite that first regards objects of $C$ as constant cosimplicial objects and then applies the [[derived functor]] of the [[enriched category theory|enriched]] [[Yoneda embedding]], i.e. the functor

$$
  \mathbb{R}Y = Y(Q(-), P(-))
$$

for $Q$ a fibrant and $P$ a cofibrant replacement functor. Then of course the simplicial presheaf $\mathbf{Y}(U)$ for $U \in C$ may in general take values in [[simplicial set]]s with nontrivial higher [[simplicial homotopy group]]s, to the extent that the [[SSet]]-[[hom-object]]s $\mathbf{Y}(U) : V \mapsto [\Delta,C](V,U)$ in $[\Delta,C]$ are nontrivial.

This cannot happen for [[∞-stack]]s over 1-categorical site. For instance a [[topological space]] regarded as a [[topological ∞-groupoid]] is always an [[n-truncated object of an (∞,1)-category|0-truncated object]] as an [[∞-stack]].

A notable example of this is the case where $C = $ [[CRing]] and $U = Spec R$. While ordinarily this is 0-categorical, when regarded as a derived stack on formal duals of [[simplicial ring]]s this has in general a nontrivial [[free loop space object]], i.e. a nontrivial [[homotopy pullback]] of the form

$$
  \array{
    \Spec \Omega^\bullet_K(R) &\to& Spec R
    \\
    \downarrow && \downarrow^{\mathrlap{(Id,Id)}}
    \\
    Spec R &\stackrel{(Id,Id)}{\to}& Spec R \times Spec R
  }
  \,,
$$

where, as indicated, the free loop space object is something like the [[de Rham space]] of $Spec R$. This means that regarded as a derived stack, the space $Spec R$ becomes an [[∞-groupoid]] whose morphisms are given by [[infinitesimal object|infinitesimal paths]] in the orighinal space.

By the $\infty$-erspective on [[Hochschild cohomology]] (as discussed there) this implies a bunch of nice relations. Details are in

* [[Bertrand Toen]] [[Gabriele Vezzosi]], _$S^1$-Equivariant simplicial algebras and de Rham theory_ ([arXiv:0904.3256](http://arxiv.org/abs/0904.3256))

The fact is also mentioned and used in passing every now and then (e.g. p. 9) in

* [[David Ben-Zvi]], [[David Nadler]], [[John Francis]], _[[geometric infinity-function theory|Integral transforms etc.]]_

But there must be a better reference, somewhere.

## Related concepts

* [[derived manifold]]

* [[infinity-stack]]

## References

An overview is provided in

* [[Bertrand Toën]], _Higher and derived stacks: a global overview_, In: _Algebraic Geometry Seattle 2005_, Proceedings of Symposia in Pure Mathematics, Vol. 80.1, AMS 2009 ([arXiv:math/0604504](http://arxiv.org/abs/math/0604504), [doi:10.1090/pspum/080.1](https://doi.org/10.1090/pspum/080.1))

A set of lecture notes on the [[model structure on simplicial presheaves]] with an eye towrads algebraic sites and derived algebraic geometry is

* [[Bertrand Toën]], _Simplicial presheaves and derived algebraic geometry_ , lecture at [Simplicial methofs in higher categories](http://www.crm.es/HigherCategories/)  ([pdf](http://www.crm.cat/HigherCategories/hc1.pdf))

Details modeled on [[simplicially enriched category|simplicial categories]] have been developed in the series of articles by Toën and Vezossi.

This article generalizes the notion of [[site]] and [[model structure on simplicial presheaves]] from 1-categorical sites to [[simplicially enriched category|simplicial sites]] and hence to a [[model structure on SSet-enriched presheaves]]:

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical Algebraic Geometry I: Topos theory_ ([arXiv](http://arxiv.org/abs/math.AG/0207028))

Of central interest in derived algebraic geometry is the simplicial site of _simplicial algebras_, which generalizes the familiar site of algebra used in algebraic geometry. This is introduced and studied in

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical Algebraic Geometry II: geometric stacks and applications_ ([arXiv](http://arxiv.org/abs/math.AG/0404373))

Further developments in this direction are in

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _From HAG to DAG: derived moduli stacks_ in Axiomatic, enriched
and motivic homotopy theory, 173–216, NATO Sci. Ser. II Math. Phys. Chem., 131, Kluwer
Acad. Publ., Dordrecht, 2004. ([arXiv](http://arxiv.org/abs/math.AG/0210407))

The unifying picture, in particular independent of the choice of model for the [[(infinity,1)-category|(infinity,1)-categories]] is presented in

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

Derived ($\infty$-)stacks are currently mostly, maybe exclusively, studied on algebraic sites $S$, where the category $S^{op} := $ [[Alg]] is replaced with a category of "$\infty$-algebras" of sorts. The theory of these $\infty$-algebras is described in great detail in

* [[Jacob Lurie]], _[[higher algebra|Noncommutative and Commutative algebra]]_

Concretely the need for the site of simplicial ring objects is discussed in the introduction of

* [[Jacob Lurie]], _[[Structured Spaces]]_

and in the introduction of

* [[David Spivak]], _[Quasi-smooth derived manifolds](http://math.berkeley.edu/~dspivak/thesis2.pdf)_ .

The proof that [[simplicial object|simplicial]] [[algebra]]s are Quillen equivalent of [[differential graded algebra]]s -- so that derived stacks on simplicial algebras are the same as derived stacks on DGAs -- is in

* [[Stefan Schwede]], [[Brooke Shipley]], _Equivalences of monoidal model categories_ , Algebr. Geom. Topol. 3 (2003), 287--334 ([arXiv](http://arxiv.org/abs/math.AT/0209342)) .

Further developments:

* [[Gregory Taroyan]], *Equivalent models of derived stacks* [[arXiv:2303.12699](https://arxiv.org/abs/2303.12699)]

[[!redirects derived stacks]]
[[!redirects derived infinity-stack]]
[[!redirects derived infinity-stacks]]
[[!redirects derived ∞-stack]]
[[!redirects derived ∞-stacks]]


derived tensor product

#Contents#
* table of contents
{:toc}

## Idea

The left [[derived functor]] of a [[tensor product]] [[functor]]. In the context of [[triangulated categories]] denoted by $\stackrel{\mathbf{L}}\otimes$ or $\otimes^L$. In the context of classical [[homological algebra]] its components (the classical left derived functors) are traditionally denoted by _[[Tor]]${}_{i}$_.

## Examples

* [[Tor]]

* [[symmetric monoidal smash product of spectra]]

## Related concepts

[[!include homotopy-homology-cohomology]]

[[!redirects derived tensor products]]


derived ∞-Lie algebroid

+-- {: .rightHandSide}
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###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _derived $\infty$-Lie algebroid_ is an [[∞-Lie algebroid]] whose underlying space is a _derived space_ , for instance a [[derived smooth manifold]].

More precisely, let $T$ be any abelian [[Lawvere theory]] and $T \hookrightarrow C \hookrightarrow T Alg^{op}$ a small full subcategory of geometric test objects formally dual to [[algebra over a Lawvere theory|T-algebras]]. Then by the theory of [[function algebras on ∞-stacks]] over $C$ one identifies $\infty LieAlgd$ with the subcategory

$$
  (T Alg^\Delta)^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\to}
  [C^{op}, sSet]
$$

of the [[(∞,1)-category of (∞,1)-sheaves]] on $C$ modeled by [[cosimplicial algebra|cosimplicial T-algebra]]s. Under the [[Dold-Kan correspondence]] applied to the underlying cosimplicial ordinary algbra, this identifies with _non-negatively graded_ [[dg-algebra]]s: the [[Chevalley-Eilenberg algebra]]s of the corresponding dual $\infty$-Lie algebroids.

But in full [[derived geometry]] this setup is further generalized: instead of considering [[∞-stack]]s on just a category of duals of $T$-algebras, one considers [[derived ∞-stack]]s over an [[(∞,1)-site]] of [[simplicial algebra]]s. Let $T \hookrightarrow C^{\Delta^{op}} \hookrightarrow T Alg^{\Delta^{op}}$ be accordingly a small subcategory of simplicial $T$-algebra,  then the above adjunction generalizes to

$$
  ((T Alg^{\Delta^{op}})^\Delta)^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\to}
  [C^{op}, sSet]
  \,,
$$

where now on the left we have _cosimplicial simplicial algebras_ . Under the Dold-Kan correspondence these now identify with _unbounded_ dg-algebras $CE(\mathfrak{a})$. We have that

* the _categorical degree_ $k$ coming from [[k-morphism]]s of the $\infty$-sheaves contributes to positive degrees in $CE(\mathfrak{a})$;

* the _derived_ degree $l$ coming from $l$-morphisms in the $\infty$-function algebras ontribute to negative degree.

This category $(T Alg^{\Delta^{op} \times \Delta})^{op}$ we identify with that of _derived_ $\infty$-Lie algebroids.

## Examples

### BV-BRST complex

In the literature the most familiar example of a derived $\infty$-Lie algebroids -- even if not under this name -- are the [[BV-BRST complex]]es. These are [[action Lie algebroid]]s for actions of [[∞-Lie algebra]]s on [[derived smooth manifold]]s.

## Related concepts

* [[Lie algebra]], [[Lie algebroid]]

* [[L-∞ algebra]], [[L-∞ algebroid]]

[[!redirects derived L-infinity algebra]]
[[!redirects derived L-infinity algebroid]]
[[!redirects derived L-∞ algebra]]
[[!redirects derived L-∞ algebroid]]

[[!redirects derived L-infinity algebras]]
[[!redirects derived L-infinity algebroids]]
[[!redirects derived L-∞ algebras]]
[[!redirects derived L-∞ algebroids]]

[[!redirects derived Lie algebroid]]
[[!redirects derived Lie algebroids]]


Des Catégories Abéliennes

This entry compiles pointers related to:

* [[Pierre Gabriel]]:

  __Des Catégories Abéliennes__

  Bulletin de la Société Mathématique de France

  **90** (1962) 323-448

  [numdam:BSMF_1962__90__323_0](http://www.numdam.org/item?id=BSMF_1962__90__323_0)

The article (based on a thesis, 1960) develops some of the central aspects of the theory of [[abelian categories]] canonized few years earlier by [[Alexander Grothendieck]] in *[[Tohoku]]*. Most important is an advanced theory of [[localization]] in the setting of abelian categories (extending ideas of Serre), and the applications to the study of rings and modules (where an alternative reconstructions via [[Gabriel filter]]s is proposed), as well as of [[quasicoherent sheaf|quasicoherent sheaves]] on schemes. It contains a remarkable reconstruction theorem: for any abelian category Gabriel introduces a spectrum whose points are (isomorphism classes of) indecomposable injectives (there is now a refinement using pure injectives, [[Ziegler spectrum]]). For a reasonable class of commutative schemes this gives a reconstruction of a scheme out of the abelian category of quasicoherent sheaves on the scheme (later extended to all schemes using different kind of spectra and in that generality known as the [[Gabriel-Rosenberg theorem]]). This article is one of the main precursors of a modern, categorically oriented, direction in [[noncommutative algebraic geometry]].

TABLE DES MATIERES.

INTRODUCTION

CHAPITRE 1 : Quelques rappels sur les catégories.

1. Les univers de Grothendieck

2. Définition des catégories

3. Foncteurs

4. Catégories additives

5. Catégories abéliennes

6. Catégories avec générateurs et limites inductives exactes

7. Foncteurs adjoints

8. Equivalences de catégories

9. Catégories abéliennes ayant assez d'injectifs

CHAPITRE II : Foncteurs exacts à gauche et enveloppes injectives.

1. Catégories de foncteurs

2. Foncteurs exacts à gauche à valeurs dans une catégorie abélienne

3. Foncteurs exacts à gauche à valeurs dans les groupes abéliens

4. Catégories nœthériennes

5. Enveloppes injectives dans les catégories abéliennes

6. Catégories avec générateurs et limites inductives exactes

CHAPITRE III : La localisation dans les catégories abéliennes.

1. Catégories quotient

2. Propriétés du foncteur section

3. Catégories avec enveloppes injectives

4. Catégories avec générateurs et limites inductives exactes

5. Quelques exemples de sous-catégories localisantes

CHAPITRE IV : Catégories localement nœthériennes.

1. La dimension de Krull d'une catégorie abélienne

2. La structure des objets injectifs dans une catégorie localement nœthérienne

3. Modules pseudo-compacts

4. Dualité entre catégories localement finies et modules pseudo-compacts

CHAPITRE V : Applications à l'étude des modules.

1. Catégories de modules

2. La localisation

3. Le théorème de Goldie

4. Injectifs indécomposables et idéaux bilatères

5. Stabilité par enveloppes injectives

6. Extensions finies d'anneaux commutatifs nœthériens

7. La dimension de Krull de quelques anneaux

CHAPITRE VI : Applications à l'étude des faisceaux quasi cohérents.

1. Recollement de catégories abéliennes

2. Propriétés d'un recollement de catégories abéliennes

3. Préschémas et catégories abéliennes

BIBLIOGRAPHIE

category: reference

[[!redirects Des catégories abéliennes]]
[[!redirects Des categories abeliennes]]
[[!redirects des catégories abéliennes]]
[[!redirects des categories abeliennes]]
[[!redirects Des Catégories Abéliennes]]
[[!redirects Des Categories Abeliennes]]


Desargues' theorem

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## Idea

(...)

## Related entries

* [[projective geometry]]

* [[projective plane]]

* [[octonionic projective plane]]

## References

* [[Emil Artin]], §II.6 in: *[[Geometric Algebra]]*, Wiley 1957 (1988) [[ISBN:978-1-118-16454-9](https://www.wiley.com/en-ae/Geometric+Algebra-p-9781118164549), Wikipedia entry, [ark:/13960/t4nk37034](https://archive.org/details/geometricalgebra033556mbp/page/n5/mode/2upa)]

See also

* Wikipedia, _[Desargues's theorem](https://en.wikipedia.org/wiki/Desargues%27s_theorem)_

[[!redirects Desargues's theorem]]


descending chain condition

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## Definition

Given a [[category]] $C$ and an object $B:C$, let $\mathrm{Mono}(B)$ be the [[category]] whose [[objects]] are [[subobjects]] of $B$ and whose [[morphisms]] are [[monomorphisms]]. A descending chain of subobjects of $B$ is an [[inverse sequence]] in $\mathrm{Mono}(B)$, a [[sequence]] of subobjects $A:\mathbb{N} \to \mathrm{Mono}(B)$ with the following [[dependent sequence]] of monomorphisms: for [[natural number]] $n \in \mathbb{N}$, a dependent monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$.

Given categories $C$ and $D$ with forgetful functor $F:C \to D$, an object $B$ is said to satisfy the **descending chain condition** on subobjects of $F(B)$ if for every descending chain of subobjects $(A, i_n)$ of $F(B)$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$ is an [[isomorphism]].

Subobjects (considered as isomorphism classes) form a possibly large [[partially ordered set]] (poset). A partially ordered set $(X,\geq)$ is satisfying a descending chain condition if any inverse sequence $x_1\geq x_2\geq x_3\geq\ldots$ eventually stabilizes, that is, there exists $n$ such that $x_n = x_{n+1} =x_{n+2}=\ldots$.

## Examples

There is a [[forgetful functor]] $F:Ring \to BMod$ from the category [[Ring]] of [[rings]] to the category $BMod$ of [[bimodules]], which forgets the multiplicative structure on the bimodule, that the canonical [[left action]] and [[right action]] of each ring $R$ have [[domain]] $R^2$ and are equal to each other and to the multiplicative binary operation of $R$, and that the canonical [[biaction]] of each ring $R$ has domain $R^3$.

Given a ring $R$, a two-sided ideal is a subobject of $F(R)$, a sub-$R$-$R$-bimodule. A [[ring]] $R$ is said to satisfy the descending chain condition on two-sided ideals if for every descending chain of two-sided ideals $(A, i_n)$ of $F(B)$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$ is an [[isomorphism]].

Similarly, there are forgetful functors $G:Ring \to LeftMod$ from $Ring$ to the category $LeftMod$ of [[left modules]] and $H:Ring \to RightMod$ from $Ring$ to the category $RightMod$ of [[right modules]], and one could similarly define the descending chain condition on [[left ideals]] and [[right ideals]] for rings.

## See also

* [[ascending chain condition]]

* [[Artinian bimodule]]

* [[Artinian ring]]

* [[Mittag-Leffler condition]]

[[!redirects descending chain conditions]]