derivations of smooth functions are vector fields

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### Context
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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## Idea

On a [[smooth manifold]] $X$ the smooth [[tangent vector fields]] $v \in \Gamma_X(T X)$ are geometrically defined as smoothly varying collections of smooth [[paths]] $\gamma_x \colon \mathbb{R}^1 \to X$ through every point $x \in X$ (i.e. $\gamma(0) = x$), with two such paths regarded as equivalent if their first [[derivative]] $v_x \in T_x X$ at $x$, seen in any [[chart]], coincides.

By pre-composing a [[smooth function]] $f \colon  X \longrightarrow \mathbb{R}^1$ with such a path, we obtain a function $f\circ \gamma_x \;\colon\; \mathbb{R}^1 \to \mathbb{R}^1$ from the [[real line]] to itself. Therefore its [[derivative]]

$$
  (D_v f)(x) \coloneqq  d (f \circ \gamma)_0
$$

may be identified with a [[real number]] (measuring the rate of change of $f$ along $\gamma$ at $x$ to first order). Since $\gamma_x$, or rather its derivative $v_x \in T_x X$, is assumed to depend smoothly on $x$, this defines a new smooth function

$$
  D_v f \;\colon\; X \longrightarrow \mathbb{R}^1
  \,.
$$

Due to the [[product rule]] of differentiation, this assignment $f \mapsto D_v f$ is such that for $f,g \in C^\infty(X)$ two smooth functions, with pointwise product function $f \cdot g$ then

$$
  D_v(f \cdot g)
  \;=\;
  D_v(f) \cdot g + f \cdot D_v(g)
  \,.
$$

This means that $D_v \;\colon\; C^\infty(X) \to C^\infty(X)$ is a _[[derivation]]_ on the algebra of smooth functions.

Remarkably, _all_ derivations on $C^\infty(X)$ arise this way, and for a unique vector field $v \in \Gamma_X(T X)$:

$$
  \left\{
    \array{
      \text{derivations on}
      \\
      \text{algebra of smooth functions}
      \\
      \text{on}\, X
    }
  \right\}
  \;\simeq\;
  \left\{
     \array{
        \text{smooth}
        \\
        \text{tangent vector fields}
        \\
        \text{on}\, X
     }
  \right\}
$$

This is prop. \ref{Statement} below. What makes this work is the _[[Hadamard lemma]]_, see the [proof](#Proof) below for details.

Notice that the concept of derivations is purely a concept of [[algebra]], with no input from the [[topology]] and [[differential geometry]] that goes into the definition and construction of smooth manifolds and their tangent bundles.
Therefore the identification of smooth vector fields with derivations is an algebraic incarnation of an aspect of differential geometry, an identification  comparable two two other such phenomena:

* the [[embedding of smooth manifolds into formal duals of R-algebras]] -- which identifies smooth manifolds themselves with the [[formal dual]] of their real algebras of smooth functions;

* the [[smooth Serre-Swan theorem]] which identifies [[smooth vector bundles]] over a smooth manifold with the [[projective modules]] of the algebra of smooth functions.

These statements mean that [[differential geometry]] has more in common with [[algebraic geometry]] than is manifest from the traditional definitions. This serves as the bases for the definition of [[formal smooth manifolds]] and the theory of [[synthetic differential geometry]]. For more exposition of this relation see at _[[geometry of physics -- supergeometry]]_.

Beware however that related algebraic properties familiar from [[affine schemes]] may break in differential geometry: For the [[Kähler differential forms]] of $C^\infty(X)$ to come out as the expected smooth [[differential forms]] one needs to refine the plain $\mathbb{R}$-[[commutative algebra]] $C^\infty(X)$ to the structure of a _[[smooth algebra]]_. See at _[[Kähler differential forms]]_ for discussion of this issue.

## Statements

+-- {: .num_theorem #Statement}
###### Theorem

Let $X$ be a [[smooth manifold]].

Write

1. $C^\infty(X)$ for its real [[algebra of functions|algebra]] of [[smooth functions]] $X \to \mathbb{R}^1$ (under pointwise multiplication);

1. $\Gamma_X(T X)$ for the [[real vector space]] of smooth [[tangent vector fields]] on $X$, hence the space of smooth [[sections]] of the [[tangent bundle]] $T X$ of $X$, regarded as a [[smooth vector bundle]].

1, $Der(C^\infty(X))$ for the [[real vector space]] of $\mathbb{R}$-linear [[derivations]] of the $\mathbb{R}$-algebra $C^\infty(X)$

Then:

1. The operation of [[differentiation]] with respect to a [[vector field]]  is a [[derivation]] of $C^\infty(X)$.

1. Every [[derivation]] on $C^\infty(X)$ arises this way, for a unique vector field.

Hence there is an [[isomorphism]]

$$
  D_{(-)}
    \;\colon\;
  \Gamma_X(T X) \stackrel{\simeq}{\longrightarrow} Der(C^\infty(X))
  \,.
$$

=--

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

First to discuss that vector fields induce derivations.
Let $v \in \Gamma_X(T X)$ be a smooth tangent vector field.

This means first of all that for each $x \in X$ there is a [[smooth function]] $\gamma_x \;\colon\; \mathbb{R}^1 \longrightarrow X$ such that

1. $\gamma_x(0) = x$

1. $d \gamma_x_0 = v_x \in T_x X$.

Then for $f \colon X \to \mathbb{R}^1$ a smooth function, define a new function

$$
  D_v(f) \;\colon\; X \longrightarrow \mathbb{R}^1
$$

by

$$
  (D_v(f))(x) \coloneqq d (f \circ \gamma_x)_0 = \frac{d}{d t} (f(\gamma_x(t)))\vert_{t = 0}
  \,.
$$

By the linearity of differentation we have for $c \in \mathbb{R}$ that $D_v(c \cdot f) = c \cdot D_v(f)$. Moreover, by the [[product rule]] for differentiation we have for $f,g \in C^\infty(X)$ two smooth functions, that

$$
  \begin{aligned}
    D_v( f \cdot g)
    & =
    \frac{d}{d t}( (f \cdot g)(\gamma(t)) )\vert_{t = 0}
    \\
    & =
    \frac{d}{ d t}( f(\gamma_x(t)) \cdot g(\gamma_x(t)) )\vert_{t = 0}
    \\
    & =
    \left( \frac{d}{d t}  f(\gamma_x(t))\vert_{t = 0}  \right) \cdot g(\gamma_x(0))
    +
    f(\gamma_x(0)) \cdot
    \left(
      \frac{d}{d t} g(\gamma(t))\vert_{t = 0}
    \right)
    \\
    &
    =
    (D_v(f) \cdot g + f \cdot D_v(g))(x)
  \end{aligned}
  \,.
$$

Therefore it only remains to check that the function $D_v(f)$ is a smooth function in the first place.

Let

$$
  \left\{ \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\to} U_i \subset X \right\}
$$

be an [[atlas]] that exhibits the [[smooth structure]] of $X$, hence an [[open cover]] by subsets equipped with a [[diffeomorphism]] $\phi_i$ from a [[Cartesian space]] $\mathbb{R}^n$ with its standard smooth structure.

By definition of the [[tangent bundle]] $T X$, its restriction to any [[chart]] of this atlas is given by a diffeomorphism $\psi_i$ as in the following diagram

$$
  \array{
    \mathbb{R}^n \times \mathbb{R}^n
      && \underoverset{\simeq}{\psi_i}{\longrightarrow} &&
    T X\vert_{U_i}
    \\
    & {}_{\phi_i \circ \mathllap{pr_1}}\searrow && \swarrow_{p\vert_{U_i}}
    \\
    && U_i
  }
  \,.
$$

For $D_v(f) \colon X \to \mathbb{R}^1$ to be continuous and smooth, it is sufficient to see that its restrictions

$$
  D_v(f) \circ \psi_i
$$

are continuous and smooth, for all $i \in I$. (For continuity this follows by [this example](Top#ClosedSubspacesGluing) and for differentiability this is clear from the fact that derivatives at any point may be computed in an arbitrary open neighbourhood).

With respect to this chart the tangent vector field $v \in \Gamma_X(T X)$ is given by a smooth function

$$
  v_i
  \;\coloneqq\;
  pr_2 \circ \psi_i^{-1}(v \vert_{U_i})
  \;\colon\;
  U_i
    \longrightarrow
  \mathbb{R}^n
$$

By the identification of tangent vectors on $\mathbb{R}^n$ with elements of $\mathbb{R}^n$, these tangent vectors are represented by straight smooth curves ([this remark](Introduction+to+Topology+--+1EuclideanSpaceTangentBundle)):

$$
  \gamma_{x, i}
  \;\colon\;
  t
  \mapsto
  \phi_i^{-1}(x) + t \cdot v_i(x)
  \,.
$$

and for $f_i \coloneqq f \circ \phi_i \;\colon\; \mathbb{R}^n \to \mathbb{R}^1$
the restriction of the given smooth function, we have, by the [[chain rule]]

$$
  \frac{d}{d t} ( f_i(\gamma_{x,i}(t)) )\vert_{t = 0}
  =
  \sum_{k = 1}^n (v_i)^k(x) \cdot \frac{\partial f_i(y)}{\partial y^k}\vert_{y = x}
  \,.
$$

(Here and in the following we are writing $(-)^k$ for the $k$th component of an element of $\mathbb{R}^n$.)

This is the value of a function at $x$ with is the sum over the product of two smooth functions, namely the component functions $(v_i)^k$ of the smooth section
in the given chart, and the derivatives of $f_i$ (which are functions by the assumption that $f$ is smooth).

This shows that $D_v(f)$ is a smooth function.

Now for the converse. Let $D \in Der(\C^\infty(X))$ be a derivation, we need to show that it arises on each chart $\phi_i$ from smooth vector field $v_i$ as above.

The [[Hadamard lemma]] gives that every [[smooth function]] $f_i \colon \mathbb{R}^n \to \mathbb{R}^1$ may be written for each point $x \in \mathbb{R}^n$ as

$$
  f(y) = f(y-x) + \sum_{k = 0}^n (y^k - x^k) \cdot g_{x,k}(y)
$$

for smooth functions

$$
  \{g_k \in C^\infty(\mathbb{R}^n)\}_{k \in \{1,\cdots, n\}}
$$

satisfying

$$
  g_{x,k}(x) = \frac{\partial f(y)}{\partial y^k}\vert_{y = x}
  \,.
$$

Now by the derivation property we have

$$
  \begin{aligned}
    D(f)(x)
    & =
    D
    \left(
       f(0)
       +
       \sum_{k = 1}^n  ((-)^k - x^k) \cdot g_{x,k}(-)
    \right)
    \\
    & =
    \sum_{k = 1}^n
      \big(
        {(D( (-)^k - x^k ))} \cdot g_{x,k}(-)
        +
         ((-)^k - x^k) \cdot D g_{x,k}(-)
      \big)(x)
    \\
    & =
    \sum_{k = 1}^n (v_i)^k(x) \cdot \frac{\partial f(y)}{\partial y^k}\vert_{y = x}
  \end{aligned}
  \,.
$$

Here in the last step we _defined_

$$
  (v_i)^k(x) \coloneqq D( (-)^k )(x)
$$

to be the value at $x$ of the function which the derivation produces when applied to the $k$th coordinate function (which is a smooth function of $x$ by definition of $D$), and we used the above property of the $g_{x,k}$ as well as that $(-)^k - x^k$ vanishes at $x$.

This shows that the derivation $D$ comes, in the chart $\phi_i$, from the vector field $v_i$ as above.

=--

## Related theorems

* [[embedding of smooth manifolds into formal duals of R-algebras]]

* [[smooth Serre-Swan theorem]]

* [[Borel's theorem]]

* the [[Tietze extension theorem]]

* the [[Whitney extension theorem]]

* the [[Steenrod-Wockel approximation theorem]]

## References

* [[W. F. Newns]], [[A. G. Walker]],  _Tangent Planes To a Differentiable Manifold_.  Journal of the London Mathematical Society s1-31:4 (1956), 400–407 ([doi:10.1112/jlms/s1-31.4.400](https://doi.org/10.1112/jlms/s1-31.4.400))

See also

* theorem 3.7 in [this pdf](http://www.math.toronto.edu/mgualt/MAT1300/week9.pdf)

[[!redirects vector fields as derivations]]


derivative

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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#Contents#
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## Idea

In [[analysis]] or [[differential geometry]], a _derivative_ is the result of applying [[differentiation]] to a [[differentiable map]].

## Variants

* [[total derivative]]

* [[partial derivative]]

* [[derivative of a distribution]]

* [[exterior derivative]]

* [[covariant derivative]]

* [[exterior covariant derivative]]

* [[functional derivative]]

* [[Fréchet derivative]]

* [[Lie derivative]]

* [[directional derivative]]

* [[evolutionary derivative]]

* [[Radon-Nikodym derivative]]

* [[Schwarzian derivative]]

## Properties

* [[Stokes theorem]]

* [[derivations of smooth functions are vector fields]]

## Related concepts

* [[integral]]

* [[derivation]]

## References

The [[Kock-Lawvere axiom]] for the [[axiom|axiomatization]] of [[differentiation]] in [[synthetic differential geometry]] was introduced in

* {#Kock77} [[Anders Kock]], _A simple axiomatics for differentiation_,  Mathematica Scandinavica Vol. 40, No. 2 (October 24, 1977), pp. 183-193 ([JSTOR](http://www.jstor.org/stable/24491223))

[[!redirects derivatives]]
[[!redirects second derivative]]
[[!redirects second derivatives]]
[[!redirects third derivative]]
[[!redirects third derivatives]]


derivative of a distribution

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#Contents#
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## Idea

The concept of _derivative of a [[distribution]]_ is the generalization of the concept of [[derivative]] of a [[smooth function]] with distributions thought of as [[generalized functions]]. The concept is uniquely fixed by enforcing the formula for [[integration by parts]] to extend from [[integrals]] against [[compact support|compactly supported]] [[densities]] to distributions.

## Definition

Let $X \subset \mathbb{R}^n$ be an [[open subset]] of [[Euclidean space]] of [[dimension]] $n$. For $\alpha \in \mathbb{N}^n$ a multi-index, write

$$
  \partial_\alpha \coloneqq \frac{\partial^{\alpha_1}}{\partial^{\alpha_1} x^{1}}
  \cdots
  \frac{\partial^{\alpha_n}}{\partial^{\alpha_n} x^{n}}
$$

for the corresponding operation of [[differentiation]] and write

$$
  {\vert \alpha \vert}
  \;\coloneqq\;
  \underoverset{i = 1}{n}{\sum} \alpha_i
$$

for the total degree.

For $u \in \mathcal{D}'(X)$ then its [[derivative]] $\partial_\alpha u \in \mathcal{D}'(X)$ of order $\alpha$ is defined by

$$
  \langle
    \partial_\alpha u, b
  \rangle
  \;=\;
  (-1)^{\vert \alpha \vert} \langle u, \partial_\alpha b\rangle
$$

for any [[bump function]] $b$, where $\partial_\alpha b$ is the ordinary [[derivative]] of [[smooth functions]].

(e.g. [Hörmander 90, def. 3.1.1 with (3.1.1)'](#Hoermander90))

## Properties

+-- {: .num_prop #DerivativeOfDistributionRetainsOrShrinksWaveFrontSet}
###### Proposition
**(derivative of distributions retains or shrinks [[wave front set]])**

Taking derivatives of distributions retains or shrinks the [[wave front set]]:

For $u \in \mathcal{D}'(\mathbb{R}^n)$ a distribution and $\alpha \in \mathbb{N}^n$ a multi-index with $D^\alpha$ denoting the corresponding [[partial derivative]], then

$$
  WF(D^\alpha u) \subset WF(u)
  \,.
$$

=--

([Hörmander 90, (8.1.10), p. 256](#Hoermander90))

## Examples

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

The [[derivative of distributions|distributional derivative]] of the Heaviside distribution $\Theta \in \mathcal{D}'(\mathbb{R})$ is the [[delta distribution]] $\delta \in \mathcal{D}'(\mathbb{R})$:

$$
  \partial \Theta = \delta
  \,.
$$

=--

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

For $b \in C^\infty_c(\mathbb{R})$ any [[bump function]] we compute:

$$
  \begin{aligned}
    \int \partial\Theta(x) b(x) \, d x
    & =
    - \int \Theta(x) \partial b(x)\, dx
    \\
    & =
    - \int_0^\infty \partial b(x) d x
    \\
    & =
    - \left( b(x)\vert_{x \to \infty} - b(0) \right)
    \\
    & =
    b(0)
    \\
    &
    =
    \int \delta(x) b(x) \, dx
    \,.
  \end{aligned}
$$

=--

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

Every [[point-supported distribution]] $u$, hence with $supp(u) = \{p\}$ for some point $p$, is a [[finite set|finite]] [[sum]] of multiplies of derivatives of the [[delta distribution]] at that point:

$$
  u =
  \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq k } }{\sum}  c^\alpha \partial_\alpha \delta(p)
$$

where $\{c^\alpha \in \mathbb{R}\}_\alpha$,
and for $k \in \mathbb{N}$ the [[order of a distribution|order]] of $u$.

=--

(e.g. [Hörmander 90, theorem 2.3.4](#Hoermander90))

## Related concepts

* [[product of a distribution with a smooth function]]

* [[Fourier transform of distributions]]

## References

* {#Hoermander90} [[Lars Hörmander]], _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

[[!redirects derivatives of a distribution]]

[[!redirects derivative of distributions]]
[[!redirects derivatives of distributions]]


derivator

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## Idea

An [[(∞,1)-category]] $C$ can be flattened into a [[1-category]] $ho(C)$, called its [[homotopy category of an (∞,1)-category|homotopy category]], by forgetting higher morphisms.  A _derivator_ is a refinement of $ho(C)$, in the sense that it retains enough information about $C$ for many purposes, like computing [[homotopy colimits]] and [[homotopy limits]].  Roughly speaking, the idea is to retain the data of the [[homotopy category of an (∞,1)-category|homotopy categories]] of all categories of [[diagrams]] in $C$, together with the induced [[functors]] between them.  Since derivators can be studied using only ordinary [[2-category]] theory, they are often practical in situations when one requires more information than the [[homotopy category of an (∞,1)-category|homotopy category]] retains, but not the whole [[(∞,1)-category]] $C$.

Very similar structures were invented independently by [[Grothendieck]] (who introduced the name "derivator"), [[Alex Heller]] (who called his version "homotopy theories"), and Franke (who considered only the [[stable derivator|stable]] case and called his version a "system of triangulated diagram categories").  The definition given below combines elements from the work of all three.

## Motivation

The notion of derivator can be motivated in several ways.

### As a compromise notion

Suppose we start from the perspective that what we really study in [[homotopy theory]] are [[(∞,1)-categories]].  The [[homotopy category of an (∞,1)-category]] (the 1-category obtained by setting all equivalent 1-morphisms equal) is a fairly coarse invariant, but for some purposes it is sufficient.  On the other hand, sometimes we need more than merely the homotopy category, but doing everything with $(\infty,1)$-categories can be technically daunting.  Frequently, all we need from an $(\infty,1)$-category that is not present in its homotopy category is to know that we have well-behaved constructions of [[homotopy limit]]s and [[homotopy colimit]]s.

A *derivator* is thus a compromise notion, containing more information than a homotopy category, but being easier to work with than an $(\infty,1)$-category.  It consists of a homotopy category together with extra structure that enables one to compute with homotopy limits and colimits.  Any $(\infty,1)$-category with sufficiently many limits and colimits has an underlying derivator, and working with these derivators suffices for a surprising number of things we may want to do with $(\infty,1)$-categories.  However, derivators are an essentially 1-categorical notion, so we can study them using ordinary [[2-category]] theory.  Thus derivators provide a "truncated" version of [[higher category theory]], which gives us the language to characterize higher category theory using only usual [[category theory]], without any emphasis on any particular model (in fact, without assuming we even know any such model).

For instance, it turns out that we can also express many convenient [[universal properties]] in terms of derivators. A striking example is the theory of [[triangulated category|triangulated categories]]: if $A$ is an [[abelian category]], its (bounded) [[derived category]] $D^b(A)$ is not defined by a universal property.  A natural statement would be that, given a triangulated category, the category of additive functors $A\to T$ which send [[short exact sequence]]s to distinguished triangles is equivalent to the category of triangulated functors $D^b(A)\to T$, but this statement is false, and in fact, does not even make sense (unless $A$ is [[semisimple category|semi-simple]]). But, in practice, everything behaves as if the above statement were meaningful and true. The 'reason' why this does not work is that the [[mapping cone|cone]] of a map in a triangulated category is not defined by a universal property. On the other hand, the cone of a morphism of complexes $X\to Y$ is canonically defined: this is the [[homotopy colimit]] of the diagram $0\leftarrow X \to Y$.  In the world of derivators, we can remedy this situation and recover a suitable universal property.

### As a motivation for $(\infty,1)$-categories

On the other hand, we may ask: Why do we take for granted that the [[homotopy theory]] of [[topological space|spaces]] provides a good notion (among others) of [[∞-groupoid]]?  And why should we expect everything to be [[enriched category|enriched]] in higher groupoids anyway?  A priori this may seem arbitrary (although it certainly works very well).  Instead we might ask: what is the mathematical structure over which everything is canonically enriched?  How can we even correctly formulate such a question?

The notion of *derivator* provides a way to correctly formulate such a question, and the answer turns out to be mostly what we expect.  Every type of "category theory" (at least, category theory without an *a priori* given [[enriched category|enrichment]]) that we might want to do is automatically and uniquely enriched in [[homotopy types]], i.e. in the homotopy category of [[CW-complexes]] or [[Kan complexes]].  (For ordinary 1-category theory, this enrichment is trivial, i.e. factors through sets considered as discrete homotopy types.)  A derivator is simply a structure with the characteristics of ordinary [[category theory]]: [[category|categories]], [[functor]]s, [[natural transformation]]s, [[Kan extension]]s, [[Grothendieck fibration]]s.  We can then show that any derivator acquires such an enrichment, so that homotopy types are a canonical enriching place for "category theories."

## Definition

### Prederivators

Let [[Cat]] denote the [[2-category]] of [[large category|large]] [[categories]] (not necessarily even [[locally small category|locally small]]), and let $Dia$ be some 2-category of [[small categories]], thought of as **[[diagram]]s**.  One common choice is the 2-category of *all* small categories (which generates $Cat$), but we might also choose the 2-category of finite categories.  A **prederivator** with domain $Dia$ is a strict [[2-functor]]

$$ Dia^{op} \overset{D}{\to} Cat $$

As usual, $(-)^{op}$ denotes the 1-cell dual of a 2-category.  Thus, a prederivator is a "$Cat$-valued [[presheaf]]" on $Dia$.  Prederivators form a 2-category $PDer$ whose morphisms are [[pseudonatural transformations]] and whose 2-cells are [[modification]]s.

Another common convention is to use the double dual $Dia^{coop}$ which reverses both 1-cells and 2-cells, although confusingly sometimes in the literature this double dual is still denoted "$Dia^{op}$".  The motivation for the latter choice seems to be that then $D(A)$ is the category of "[[presheaves]] on $A$ with values in $D$" instead of the category of "[[diagrams]] of shape $A$ in $D$."  We have chosen the convention above since the main purpose of a derivator is a calculus of homotopy limits and colimits, and it is more usual to take limits and colimits of covariant functors.  However, since $Dia^{coop}$ is isomorphic to $Dia^{op}$ via the 2-functor $(-)^{op}$ (as long as $Dia$ is closed under opposite categories in $Cat$), there is really very little difference.

There are two main motivating examples.  Firstly, any category $C$ defines a "representable" prederivator by the assignation $A\mapsto Hom(A,C)$, sending $A\in Dia$ to the category of functors $A\to C$.  This defines an embedding of $Cat$ into $PDer$.

Secondly, if $C$ is any [[category with weak equivalences]], there is a prederivator $Ho(C)$ which sends $A\in Dia$ to the [[localization]]/[[homotopy category]] $Ho(C)(A)$ of $Hom(A,C)$ relative to the [[model structure on functors|objectwise weak equivalences]] (as we allowed categories in $Cat$ to have large [[hom-set]]s, these localization exist).  In general, this is a non-representable prederivator, although of course if the weak equivalences are just the [[isomorphism]]s, it reduces to the representable case.  Note that to construct it, we don't need anything besides ordinary [[2-category|(2-)]][[category theory]].

### Derivators

A **derivator** is a prederivator $D$ which satisfies a list of axioms.  These axioms are of two sorts.

The first set of axioms says that there exist well-behaved homotopy limits and colimits, and more generally [[homotopy Kan extension]]s.  Specifically, we require the following.

* **(Der3)** {#Der3} For any functor $u : X\to Y$ in $Dia$, the [[inverse image]] functor $u^*:D(Y)\to D(X)$ admits a [[left adjoint]] $u_!$ and a [[right adjoint]] $u_*$.  These can be understood as [[homotopy Kan extension]]s

  $$
    (u_! \dashv u^* \dashv u_*)
    :
    D(X)
      \stackrel{
         \overset{u_!}{\to}
      }
      {
        \stackrel{
          \overset{u^*}{\leftarrow}
        }{
          \underset{u_*}{\to}
        }
      }
    D(Y)
  $$

* **(Der4)** For any [[comma square]]
  $$\array{A & \overset{g}{\to} & B \\
  ^f\downarrow &\swArrow& \downarrow^v\\
  C& \underset{u}{\to} & E}$$
  in $Dia$, the [[Beck-Chevalley transformation]]s
  $$ f_! g^* \to u^* v_! \quad \text{and} \quad v^* u_* \to g_* f^* $$
  are [[isomorphism]]s.  Intuitively, this says that the Kan extensions in question are *[pointwise](http://ncatlab.org/nlab/show/Kan+extension#Pointwise)*.  In the presence of the second set of axioms, it suffices to require this when $C$ is the [[terminal category]] (for the first case) and when $B$ is so (for the second).

The second set of axioms are "[[sheaf]]" conditions.  Of course, we cannot assert that $D$ is exactly a sheaf (in the appropriate [[stack|2-categorical sense]]), since the terminal category in $Dia$ is 2-categorically dense and so any sheaf on it is representable (and represented by $D(1)$), whereas we want to also allow non-representable derivators.  But we do need some sheaf-like properties in order to do [[category theory]].  All of the following axioms can be understood as asserting that for some [[cover|covering family]] $\{Y_i \to X\}$ in $Dia$, the canonical map $D(X) \to Desc(D,\{Y_i\})$ from $D(X)$ to the category of [[descent]] data for the covering, while not an [[equivalence of categories|equivalence]], has some weaker good properties.  They can also be understood as 2-categorical [[sketch]] conditions.

The standard axioms are:

* **(Der1)** $D\colon Dia^{coop} \to Cat$ takes [[coproduct]]s to products.  Sometimes we require this only for finite coproducts.  In particular, we have $D(\emptyset) = 1$.

* **(Der2)** For any $X\in Dia$, consider the family of functors $x\colon 1\to X$ determined by the objects of $X$.  Then the induced functor
  $$ D(X) \overset{(x^*)}{\to} \prod_x D(1)$$
  is [[conservative functor|conservative]] (though not generally faithful).  This in turn implies that the same is true for any jointly [[essentially surjective functor|essentially surjective]] family of functors.

* **(Der5)** For any $X\in Dia$, if $I$ denotes the [[interval category]], then the induced functor
  $$ D(X\times I) \to Hom(I,D(X)) $$
  is [[essentially surjective functor|essentially surjective]] and [[full functor|full]] (though again, it is not generally faithful).  Since this functor is also conservative by (Der2), it is then a [[weakly smothering functor]].

There is substantial variation in (Der5).  Sometimes it is convenient to assume this property when $I$ is any (perhaps finite) [[free category]].  Some references do not include (Der5) at all in the definition, instead calling a derivator **strong** if it satisfies (Der5).

+--{: .query}
[[Mike Shulman]]: It's clear to me that these are desirable requirements, which are moreover satisfied by all derivators of the form $Ho(C)$, but I would really like a conceptual explanation for why these axioms are *sufficient*.
=--

It is easy to see that if $C$ has pointwise left and right Kan extensions along all functors in $Dia$, then its representable prederivator is a derivator.  Somewhat more difficult to prove is that if $C$ is a [[model category]] (or more generally, has well-behaved homotopy Kan extensions), then the prederivator $Ho(C)$ is also a derivator.  Thus the derivator encodes the notions of [[homotopy colimit]] and of [[homotopy limit]].  Note again that this way of seeing homotopy (co)limits does not use anything besides usual (2-)category theory.

### Semiderivators {#SemiDerivators}

It may sometimes be useful to consider prederivators which are like derivators, but which "do not have *all* limits and colimits."  Let us say that a **semiderivator** is a prederivator satisfying (Der1), (Der2), and (Der5).

If $D$ is a semiderivator, $x\in D(B)$ is a $B$-shaped diagram, and $v\colon B\to C$ is a functor in $Dia$, then a **pointwise left extension** of $x$ along $v$ is an object "$v_! x$" in $D(C)$, together with a morphism $x \to v^* v_! x$ which is [[initial object|initial]] in the [[comma category]] $(x / v^*)$ (this says that $v_! x$ is a "local" or "partial" value of the left adjoint $v_!$ at $x$, although the entire functor $v_!$ may not exist) which additionally satisfies the [local Beck-Chevalley condition](/nlab/show/Beck-Chevalley+condition#Local) for any comma square as above.  We have a dual notion of **pointwise right extension**.

We say a semiderivator is **complete** (resp. **cocomplete**) if it admits all pointwise right (resp. left) extension.  Clearly a semiderivator is a derivator just when it is both complete and cocomplete.

### Historical remarks

* Grothendieck's definition of a *derivator* included only axioms (Der1), (Der2), (Der3), and (Der4).

* [[Alex Heller]]'s definition of a *homotopy theory* included axioms (Der1), (Der2), (Der3), a weaker form of (Der4), and (Der5) for finite free categories.  His *pointed homotopy theories* add the axiom of a [[pointed derivator]], and his *stable homotopy theories* include a weaker version of the axiom now used for a [[stable derivator]].

* Franke's definition of a *system of triangulated diagram categories* was irreducibly [[pointed derivator|pointed]], taking $Dia$ to consist of categories enriched over pointed sets.  (See [[pointed derivator]] for the relationship of this approach to that of adding a "pointedness" axiom to an unpointed notion of derivator.)  In this context, he assumed axioms analogous to (Der1), (Der2), (Der3), (Der4), and (Der5) for the interval category, plus the [[stable derivator|stability]] axiom.  He also observed that if $Dia$ consists entirely of posets, then (Der4) follows from the other axioms.

Axioms (Der1)--(Der4) are clearly the easiest to motivate and the most obviously necessary.  Axiom (Der5) is used in order to do things with morphisms in $D(*)$, by first lifting them to objects of $D(I)$.  In particular, it is necessary to conclude that a stable derivator gives a [[triangulated category]].

## Computing homotopy Kan extensions

We now describe an "omnibus" theorem which is the main thing enabling us to compute with [[homotopy Kan extension]]s in a derivator.

### $D$-equivalences

For any functor $u\colon I\to J$ in $Dia$, we say it is a $D$-**equivalence** if the induced transformation $(\pi_I)_! (\pi_I)^* \to (\pi_J)_!(\pi_J)^*$ is an isomorphism, where $\pi_I\colon I\to *$ is the projection to the [[terminal category]], and similarly for $\pi_J$.  This means that homotopy colimits of *constant* diagrams of shapes $I$ and $J$ are equivalent.  By the [[Yoneda lemma]], this is equivalent to saying that
$$ D(*)((\pi_J)_!(\pi_J)^* X , Y) \to D(*)((\pi_I)_!(\pi_I)^* X , Y)$$
is an isomorphism for all $X,Y\in D(*)$, and by adjunction this is equivalent to saying that
$$ D(J)((\pi_J)^* X , (\pi_J)^*Y) \to D(I)((\pi_I)^* X , (\pi_I)^*Y)$$
is an isomorphism---i.e. that $u^*$ is fully faithful when restricted to the image of $\pi_J^*$.  In particular, if $u^*$ is fully faithful, then $u$ is a $D$-equivalence.

In a representable derivator (i.e. in ordinary category theory), the colimit of a constant diagram of shape $I$ is a [[copower]] with the set of connected components of $I$.  Thus, in a representable derivator, any functor that induces an isomorphism on sets of connected components will be a $D$-equivalence, and the converse is true as long as the category in question is not a [[preorder]].

By contrast, in the derivator coming from a model category or $(\infty,1)$-category, the colimit of a constant diagram of shape $I$ is a copower with the [[nerve]] of $I$ regarded as an $\infty$-groupoid, so in this case any functor that induces a homotopy equivalence of nerves (a stronger condition) will be a $D$-equivalence.  In fact, one can show:

+-- {: .num_theorem #NerveEquiv}
###### Theorem
**([[Alex Heller]], Cisinski)**
A functor which induces a homotopy equivalence of nerves is a $D$-equivalence for any derivator $D$.
=--

This was proven by [Heller](#Heller) using the canonical [[enriched derivator|enrichment]] of any derivator over $\infty$-groupoids.  It also follows from Cisinski's theorem that the nerve equivalences are the smallest [[basic localizer]], once we verify that the $D$-equivalences are in fact a basic localizer---as we will now proceed to do.

+-- {: .num_lemma}
###### Lemma
Any functor in $Dia$ with a fully faithful left or right adjoint is a $D$-equivalence.
=--
+-- {: .proof}
###### Proof
If $f\dashv g$ in $Dia$, then $g^* \dashv f^*$, and if $f$ is fully faithful, then the unit $1 \to g f$ is an isomorphism, and thus so is the unit $1\to f^* g^*$.  Hence $g^*$ is also fully faithful and so $g$ is a $D$-equivalence.  The other case is dual.
=--

+-- {: .num_lemma}
###### Lemma
Let $W_D$ denote the class of $D$-equivalences in $Dia$.  Then $W_D$ is saturated, in the sense that if $u\colon I\to J$ is a morphism in $Dia$ which becomes an isomorphism in $Dia[W_D^{-1}]$, then $u$ is a $D$-equivalence.
=--
+-- {: .proof}
###### Proof
Fix some $X,Y\in D(*)$ and consider the functor $\Phi\colon Dia \to Set^{op}$ which sends $I$ to $D(I)(\pi_I^*X, \pi_I^*Y)$.  Since $\Phi$ inverts $D$-equivalences, it factors through $Dia[W_D^{-1}]$.  But if $u$ becomes an isomorphism in $Dia[W_D^{-1}]$, then it must be inverted by $\Phi$, but that is the definition of being a $D$-equivalence (as $X$ and $Y$ vary).
=--

+-- {: .num_lemma}
###### Lemma
For any $D$, the class of $D$-equivalences is a [[basic localizer]].
=--
+-- {: .proof}
###### Proof
Saturation gives 2-out-of-3 property and closure under retracts.  If $A$ has a terminal object, then $A\to 1$ has a fully faithful right adjoint and hence is a $D$-equivalence.  Finally, given a triangle
$$\array{ A & & \overset{u}{\to} & & B\\ & _v \searrow & & \swarrow_w \\ & & C, } $$
to show that $u$ is a $D$-equivalence, it suffices to show that the transformation $v_! \pi_A^* \to w_! \pi_B^*$ is an isomorphism.  By (Der2) it suffices to check this for any $c\in C$.  We can then form the comma objects
$$\array{v/c & \overset{g}{\to} & A\\
  ^f\downarrow && \downarrow^v\\
  *& \underset{}{\to} & C} \quad\text{and}\quad
\array{w/c & \overset{k}{\to} & B\\
  ^h\downarrow && \downarrow^w\\
  * & \underset{}{\to} & C}$$
and the transformation $c^* v_! \pi_A^* \to c^* w_! \pi_B^*$ factors as
$$ c^* v_! \pi_A^* \cong f_! g^* \pi_A^* \cong f_! \pi_{v/c}^* \to h_! \pi_{w/c}^* \cong c^* w_! \pi_B^*$$
using the analogous map for the functor $v/c \to w/c$.  Therefore, if $v/c \to w/c$ is a $D$-equivalence, this composite is an isomorphism, and if this holds for all $c$, then by (Der2), $u$ is a $D$-equivalence.
=--

Therefore, since the nerve equivalences are the smallest basic localizer, every nerve equivalence is a $D$-equivalence for any derivator $D$.

### Exact squares {#ExactSquares}

Now suppose given any square
$$\array{L & \overset{p}{\to} & I\\
  ^q\downarrow & \Downarrow^\mu & \downarrow^u\\
  J & \underset{v}{\to} & K}$$
in $Dia$ which commutes up to a specified 2-cell $\mu$.  Given a derivator $D$, we say that this square is $D$-**exact** if the [[Beck-Chevalley transformation]]s
$$ q_! p^* \to v^* u_! \quad \text{and} \quad u^* v_* \to p_* q^* $$
are isomorphisms.  (In fact, if one of these is an isomorphism, so is the other, since they are [[mate|mates]].)  Thus, the derivator axioms say that all comma squares are exact.

Like the notion of [[exact square]] in ordinary category theory, this is a "functional" definition, but we can also give a more explicit characterization, using more or less the same argument.  Given such a square as above and $i\in I$, $j\in J$, we write $(i/L/j)$ for the category whose objects are triples
$$\big(\ell\in L,\, i\overset{\alpha}{\to} p(\ell),\, q(\ell)\overset{\beta}{\to} j\big).$$
The morphisms of $(i/L/j)$ are morphisms $\gamma\colon \ell \to \ell'$ in $L$ which make the evident triangles commute.  Now there is a functor $r\colon (i/L/j) \to K(u(i),v(j))$ (the latter considered as a discrete category), which sends the above triple to the composite
\[ u(i) \overset{u(\alpha)}{\to} u(p(\ell)) \overset{\mu}{\to} v(q(\ell)) \overset{v(\beta)}{\to} v(j) \label{commacomp}\]
in $K$.

+-- {: .num_theorem}
###### Theorem
A square as above is $D$-exact if and only if for all $i\in I$ and $j\in J$, the functor $(i/L/j) \to K(u(i),v(j))$ is a $D$-equivalence.
=--
+-- {: .proof}
###### Proof
It is easy to verify that the horizontal or vertical pasting composite of exact squares is exact; hence if the given square is exact, then so is the composite
$$\array{(L/j) & \overset{}{\to} & L & \overset{p}{\to} & I \\
  \downarrow & \Downarrow & \downarrow^q & \Downarrow & \downarrow^u\\
  * & \underset{j}{\to} & J & \underset{v}{\to} & K}$$
for any $j\in J$, where the left-hand square is a comma square.  Conversely, if all of these squares are exact, then so is the given one, by (Der2).  We play the same game by composing with comma squares on the top to conclude that the given square is exact if and only if all the induced squares
\[\array{(i/L/j) & \overset{f}{\to} & *\\
  ^g\downarrow & \Downarrow & \downarrow^{\mathrlap{u(i)}}\\
  * & \underset{v(j)}{\to} & K}\label{ijsquare}\]
are exact.  But the square
$$\array{K(u(i),v(j)) & \overset{s}{\to} & *\\
  ^t\downarrow & \Downarrow & \downarrow^{\mathrlap{u(i)}}\\
  * & \underset{v(j)}{\to} & K}$$
is exact, since it is a comma square, and by the universal property of a comma square, the square \eqref{ijsquare} factors uniquely through this one by a functor $r\colon (i/L/j) \to K(u(i),v(j))$, which is precisely the functor $r$ defined above.  Specifically, we have $f = s r$ and $g = t r$.  Therefore, the Beck-Chevalley transformation $g_! f^* \to (v j)^* (u i)^*$ is equal to the composite
$$ g_! f^* = t_! r_! r^* s^* \to t_! s^* \to (v j)^* (u i)^* $$
where the second map is the Beck-Chevalley transformation for the comma square, which is an isomorphism.  Thus, \eqref{ijsquare} is exact just when $t_! r_! r^* s^* \to t_! s^*$ is an isomorphism.  But this says exactly that $r$ is a $D$-equivalence.
=--

A square is said to be [[homotopy exact square|homotopy exact]] if it is $D$-exact for all derivators $D$ (or, equivalently, for all $(\infty,1)$-categories, or for all model categories, or simplicially enriched categories).

+-- {: .num_cor}
###### Corollary
A square is homotopy exact if and only if for all $i\in I$ and $j\in J$, the functor $(i/L/j) \to K(u(i),v(j))$ induces a weak homotopy equivalence of nerves.
=--
+-- {: .proof}
###### Proof
"If" follows directly from Theorem \ref{NerveEquiv} and the previous theorem.  Conversely, we can take $D$ to be the derivator of spaces ($\infty$-groupoids), where the $D$-equivalences are precisely the nerve equivalences.
=--

+-- {: .num_cor}
###### Corollary
For each $i\in I$, $j\in J$, and $\varphi\colon u(i) \to v(j)$ in $K$, let $(i/L/j)_\varphi$ denote the subcategory of $(i/L/j)$ consisting of those triples for which the composite \eqref{commacomp} is equal to $\varphi$.  Then the square is homotopy exact if and only if each category $(i/L/j)_\varphi$ has a contractible nerve.
=--

This gives a convenient way to compute many homotopy limits and colimits, which works in any derivator, and *a fortiori* in any $(\infty,1)$-category or model category.  Example applications can be found at [[homotopy exact square]].

## The 2-category of derivators
  {#2CategoryOfDerivators}

We can consider now the 2-category of derivators.  Actually, there are several different ways to define such a 2-category, depending on whether we require morphisms to preserve homotopy colimits, limits, both, or neither.  Let us write $Der_!$ for the 2-category whose morphisms preserve colimits.  Thus:

* its [[object]]s are derivators,

* its 1-[[morphism]]s are [[pseudonatural transformations]] $F:D\to D'$ which commute with the functors $u_!$ (i.e. the canonical comparison maps for these are isomorphisms),

* its 2-cells are the [[modifications]] between these.

We write $Hom_!(D,D') = Der_!(D,D')$ for hom-categories in this 2-category.

### Free cocompletions

Now, given a small category $X$, there is a $2$-functor, defined by evaluating at $X$
$$Der_!\to Cat \quad , \qquad D\mapsto D(X).$$
Note that, for any (pre)derivator $D$, the category $D(X)$ is canonically equivalent to the category $Hom(X,D)$ of morphisms of *prederivators* from $X$ to $D$ (considering $X$ as a representable prederivator).  Of course, $X$ will usually not itself be a derivator, but nevertheless this $2$-functor is *representable* in the 2-category $Der_!$.

Thus, there exists a derivator $\widehat{X}$, endowed with a morphism of prederivators $h:X\to \widehat{X}$ (called the Yoneda embedding), such that, for any derivator $D$, composing with h defines an equivalence of categories
$$Hom_!(\widehat{X},D)\simeq Hom(X,D)=D(X).$$
In other words, the map $h:X\to \widehat{X}$ is the "free completion of $X$ by homotopy colimits" in the sense of derivators.

Furthermore, $\widehat{X}$ can be described rather explicitly: it is the derivator associated to the [[model structure on simplicial presheaves|model category of simplicial presheaves]] on $X$.  In particular, the usual [[model structure on simplicial sets|homotopy theory of simplicial sets]] gives rise to the derivator $\widehat{*}$, where $*$ stands for the [[terminal object|terminal]] category.  Note that in order to conclude this, we didn't take for granted that [[homotopy type]]s should be important: its universal property is formulated purely with ordinary category theory.

From there, you can see that any derivator is canonically enriched in the derivator $\widehat{*}\simeq Ho(SSet)$: as $*$ acts uniquely on any prederivator, $\widehat{*}\simeq Ho(SSet)$ acts uniquely on any derivator (as far as we ask compatibility with homotopy colimits).  Thus the [[homotopy hypothesis]] might be reformulated vaguely as: is there an algebraic model of $\widehat{*}$?  Then one may guess that some notion of higher groupoid might do the job.

## Constructions of derivators

Viewing a derivator as a partway point between an $(\infty,1)$-category and its homotopy category, and recalling that $(\infty,1)$-categories are often presented by [[categories with weak equivalences]] and in particular [[model categories]], we can construct derivators in two general ways.

### From homotopy theory

If $(C,W)$ is a [[category with weak equivalences]], then the representable prederivator defined by $D_C(X) = Cat(X,C) = C^X$ comes equipped with weak equivalences $W^X$ on each category $C^X$.  We define the *homotopy prederivator* $Ho(C)$ by inverting these weak equivalences in each diagram category:
$$Ho(C)(X) = Ho(C^X,W^X) = C^X[(W^X)^{-1}]$$

+--{: .num_theorem}
###### Theorem (Dwyer, Hirschhorn, Kan, and Smith)
If $C$ is a [[homotopical category]] with a functorial [[three-arrow calculus]], then $Ho(C)$ is a prederivator satisfying axioms (Der1), (Der2), and (Der5).
=--

Actually, it is enough to assume that, for every small category $X$, $C^X$ admits a (not necessarily functorial) three-arrow calculus, but in practice this only happens when we have a functorial three-arrow calculus for $C$.

+--{: .num_theorem}
###### Theorem (Cisinski, 2003)
If $C$ is a [[model category]] (complete and cocomplete, but possibly without functorial factorisations), then $Ho(C)$ is a derivator (with the strong version of Der5).
=--

Axioms (Der1) is easy to check, and when $C$ has functorial factorisations, we have a functorial three-arrow calculus. Axiom (Der3) is also easy, since [[homotopy limits]] and colimits in categories with weak equivalences are [[derived functor]]s of the usual limits and colimits, and so they supply left and right adjoints to derived pullback functors.  (This shows moreover that homotopy limits and Kan extensions in a model category coincide with the notions of homotopy limit and Kan extension in its homotopy derivator, so that by working in $Ho(C)$ we really are studying the things we want to study.)  Axiom (Der4) requires a bit of work; there is a proof for combinatorial model categories using the [[injective model structure]] in [(Groth)](#Groth). Axiom Der 5 is discussed in Theorem 9.8.5 in [(Radulescu Banu)](#RadBan06). The theorem above remains true for suitable (co)fibration categories for which a suitable version of the injective and projective structures are always available; see Theorem 9.5.5 in [(Radulescu Banu)](#RadBan06).

Note that if $C$ is any complete and cocomplete 1-category, we can equip it with its [[trivial model structure]] in which the only weak equivalences are isomorphisms.  Then the above derivator $Ho(C)$ is the same as the representable prederivator $D_C(X)$, which can easily be proven to be a derivator directly.

In this connection, it is interesting to point out that a given homotopy category can admit multiple "enrichments" to a derivator.  For instance, the homotopy category of the model category of chain complexes over a field is equivalent to the category of graded modules over that field, which is itself complete and cocomplete.  Thus we have two distinct derivators, which have equivalent underlying homotopy categories $D(*)$.

### From $(\infty,1)$-categories
 {#FromInfinityCategories}

If $C$ is an [[(∞,1)-category]], it has a [[homotopy category of an (∞,1)-category|homotopy category]] $Ho(C)$ obtained by identifying equivalent [[1-morphisms]].  If our $(\infty,1)$-categories are modeled by [[quasicategories]], then $Ho(-)$ is the [[left adjoint]] of the [[nerve]], often denoted $\tau_1$.

Since categories are in particular $(\infty,1)$-categories, for any category $X$ we have a [[functor (∞,1)-category]] $C^X$, and thus a homotopy category $Ho(C^X)$.  We define the *homotopy prederivator* of $C$ by
$$Ho(C)(X) = Ho(C^X).$$
If $C$ has [[limit in an (∞,1)-category|limits]] and colimits in the $(\infty,1)$-categorical sense, then $Ho(C)$ is a derivator (this is an interpretation of the theory of pointwise Kan extensions in $\infty$-categories). A proof of this is sketched in [GPS, Example 2.5](#GPS14). The comparison of the derivator associated to a cofibration category (hence also to any model category) $C$ and of the derivator associated to the Dwyer-Kan localization of $C$ is discussed by [Lenz](#Lenz).

### Exotic examples

There are also examples of derivators not satisfying (Der5) that do not arise from homotopy theory or higher category theory in this way.  For instance, see Remark 5.4 and Example 5.5 in [Lagkas-Nikolos](#LN16).

## Properties

### Presentable derivators and combinatorial model categories
 {#PresentableDerivatorsAndCombinatorialModelCategories}

We discuss precise versions of the idea that derivators indeed constitute a model of [[(∞,1)-category theory]].

+-- {: .num_theorem #RenaudinTheorem}
###### Theorem

The sub-[[2-category]] of "[[locally presentable category|locally presentable]]" derivators with left adjoints as morphisms between them, among the [[2-category]] of all derivators (see [above](#2CategoryOfDerivators)),  is [[equivalence of 2-categories|equivalent]] to the [[2-localization]] of the 2-category of [[left proper model category|left proper]][^1] [[combinatorial model categories]] at the [[Quillen equivalences]].

=--

This is shown in ([Renaudin 2006](#Renaudin06)).  See at [[Ho(CombModCat)]] for more.

Notice that [[locally presentable (∞,1)-categories]] are precisely those [[(∞,1)-categories]] that arise, up to [[equivalence of (∞,1)-categories]] as [[simplicial localizations]] of [[combinatorial model categories]]. Hence this theorem suggests that there is, at least, an [[equivalence of 2-categories]] between the [[2-category]] of presentable derivators and the [[homotopy 2-category]] of the [[(∞,2)-category]] [[Pr(∞,1)Cat]]. However, an actual proof of this seems to be missing.

As a corollary, we obtain a canonical comparison [[functor]]

$$
  Ho(PrDer)
  \longrightarrow
  Ho(Pr(\infty,1)Cat)
$$

from the homotopy category presentable derivators with left adjoint morphisms, to the [[homotopy category of an (∞,1)-category]] of [[Pr(∞,1)Cat]].

This is induced by the [[universal property]] of the [[2-localization of a 2-category|2-localization]] from the fact that the functor $L^H$ that forms [[simplicial localizations]]

$$
  L^H \;\colon\; CombModCat \longrightarrow Ho(Pr(\infty,1)Cat)
$$

sends [[Quillen equivalences]] to [[equivalence of (∞,1)-categories]] (see [there](locally presentable infinity-category#PresentedByCombinatorialSimplicialModelCategories)).

## Related pages

Special kinds of derivators:

* [[pointed derivator]]
* [[stable derivator]]
* [[monoidal derivator]]
* [[enriched derivator]]
* [[finitely complete prederivator]]
* [[regular derivator]]

The calculus of homotopy Kan extensions used in derivators:

* [[homotopy exact square]]

Special limits and structures in derivators:

* [[pullback in a derivator]]
* [[monomorphism in a derivator]]
* [[coend in a derivator]]
* [[orthogonal factorization system in a derivator]]

## References

### General

The term _derivator_ is originally due to [[Grothendieck]], introduced in _[[Pursuing Stacks]]_ .
The first fifteen chapters of a 2000 page manuscript of Grothendieck (in French) about derivators can be found at:

* [[Alexander Grothendieck]],  _[Les Dérivateurs](http://people.math.jussieu.fr/~maltsin/groth/Derivateurs.html)_

Independently, there is a version due to [[Alex Heller]] (who called them "homotopy theories"):

* [[Alex Heller]], _Homotopy theories_ , Memoirs of the American Mathematical Society, Vol. 71, No 383 (1988).
{#Heller}
* Alex Heller, _Stable homotopy theories and stabilization_ , [MR](http://www.ams.org/mathscinet-getitem?mr=1431157) -- see [[pointed derivator]] and [[stable derivator]]

Apparently also independent is the development by Franke, who takes an enriched approach to the pointed case and also assumes stability:

* Jens Franke, _Uniqueness theorems for certain triangulated categories with an Adams spectral sequence_, [K-theory archive](http://www.math.uiuc.edu/K-theory/0139/)

Georges Maltsiniotis has written an introduction to the topic (in French):

* [[Georges Maltsiniotis]], _Introduction à la théorie des dérivateurs, d'après Grothendieck_ , Preprint (2001) [ps](http://www.math.jussieu.fr/~maltsin/ps/m.ps)

He also gave a [course](http://congreso.us.es/htag09/php/index.php?carga=courses) (in English) in Seville, Sep 2010, and part 3 is on derivators:

* [[Georges Maltsiniotis]]

  * [Lecture I (Localizers)](http://people.math.jussieu.fr/~maltsin/Seville/Lecture_I_Localizers.pdf)

    [Lecture II (Model Categories)](http://people.math.jussieu.fr/~maltsin/Seville/Lecture_II_Model_Categories.pdf)

    [Lecture III (Derivators)](http://people.math.jussieu.fr/~maltsin/Seville/Lecture_III_Derivators.pdf)

    [Summary on Derivators](http://people.math.jussieu.fr/~maltsin/Seville/Summary_on_Derivators.pdf)

Part of the above material is adapted from

* [[Denis-Charles Cisinski]], _[blog comment on derivators](http://golem.ph.utexas.edu/category/2010/03/a_perspective_on_higher_catego.html#c032227)_

Cisinski has also written a number of papers on the subject (in French), which can be found at [his homepage](http://www.mathematik.uni-regensburg.de/cisinski/publikationen.html).

* — Images directes cohomologiques dans les catégories de modèles. _Ann. Math. Blaise Pascal_, 10(2):195–244, 2003.

* — Catégories dérivables, Bull. Soc. Math. France, Tome 138 (2010) no. 3, pp. 317-393. [NUMDAM](http://www.numdam.org/item/BSMF_2010__138_3_317_0/)

Derivators were also recently used by [[Gonçalo Tabuada]] in a universal characterization of higher [[algebraic K-theory]]:

* [[Gonçalo Tabuada]], "Higher K-theory via universal invariants" [arXiv](http://arxiv.org/abs/0706.2420).

An introduction to some of the theory of pointed and stable derivators, in English, can be found in the paper:

* [[Denis-Charles Cisinski]] and [[Amnon Neeman]], _Additivity for derivator K-theory_ , [MR](http://www.ams.org/mathscinet-getitem?mr=2382732)

An introductory discussion aimed towards [[stable derivator]]s is also in

* {#Groth} [[Moritz Groth]], _Monoidal derivators and additive derivators_, [arxiv/1203.5071](http://arxiv.org/abs/1203.5071); _Derivators, pointed derivators, and stable derivators_ ([pdf](http://www.math.uni-bonn.de/~mgroth/groth_derivators.pdf))

Other references include:

* {#RadBan06} Andrei Radulescu Banu, Cofibrations in Homotopy Theory, [arXiv:math/0610009](https://arxiv.org/abs/math/0610009), 2006

* {#Lenz} Tobias Lenz, Homotopy (Pre-)Derivators of Cofibration Categories and Quasi-Categories, Algebr. Geom. Topol. 18 No. 6 (2018), pp. 3601–3646. [arXiv:1712.07845](https://arxiv.org/abs/1712.07845)

* {#IN16} Ioannis Lagkas-Nikolos, _Levelwise modules over separable monads on stable derivators_, [arXiv:1608.06340](https://arxiv.org/abs/1608.06340), 2016

* {#Coley20} Coley, Ian, _The theory of half derivators_, [arxiv:2010.12057](https://arxiv.org/abs/2010.12057), 2020

* {#Richardson20} Richardson, James, _Enriched derivators_, [arXiv:2010.07740](https://arxiv.org/abs/2010.07740), 2020

* {#GPS14} [[Moritz Groth]], [[Kate Ponto]], and [[Mike Shulman]], _Mayer-Vietoris sequences in stable derivators_. Homology, Homotopy and Applications 16 (1) 2014, [arxiv:1306.2072](http://arxiv.org/abs/1306.2072)

### Relation to $\infty$-categories

An brief informal discussion of derivators as a 2-categorical tool for studying [[(infinity,1)-categories|$(\infty,1)$-categories]] is contained in

* [[Mike Shulman]], _Squeezing Higher Categories out of Lower Categories_ ([blog](http://golem.ph.utexas.edu/category/2010/05/squeezing_higher_categories_ou.html))

In the paper

* {#Renaudin06} [[Olivier Renaudin]], *Plongement de certaines théories homotopiques de Quillen dans les dérivateurs*, Journal of Pure and Applied Algebra Volume 213, Issue 10, October 2009, Pages 1916-1935
([arXiv:math/0603339](https://arxiv.org/abs/math/0603339), [doi:10.1016/j.jpaa.2009.02.014](https://doi.org/10.1016/j.jpaa.2009.02.014))

it is proven that the [[2-category]] of "[[locally presentable category|locally presentable]]" derivators is equivalent to the [[localization]] of the 2-category of [[combinatorial model categories]] at the [[Quillen equivalences]] (see also at *[[Ho(CombModCat)]]*).  Thus in some sense derivators capture "all the information" about a combinatorial model category, hence also about a [[locally presentable (∞,1)-category]].

Along somewhat similar lines, in

* Kevin Arlin, _On the ∞-categorical Whitehead theorem and the embedding of quasicategories in prederivators_ ([arxiv:1612.06980](https://arxiv.org/abs/1612.06980))

it is shown that $(\infty,1)$-categories can be literally embedded in "prederivators" as long as we allow *strict morphisms* between the latter (which is arguably somewhat against the spirit of derivators, but still interesting), and that at the 2-categorical level the embedding is conservative but generally fails to be full on large $(\infty,1)$-categories.  This is developed further in

* Daniel Fuentes-Keuthan, Magdalena Kedziorek, Martina Rovelli, *A model structure on prederivators for (∞,1)-categories*, [arxiv](https://arxiv.org/abs/1810.06496), 2018

On lifting derivators from the [[homotopy categories of (infinity,1)-categories]] to the [[homotopy 2-categories]] of [[(infinity,2)-categories]]:

* {#DiVittorio23} [[Nicola Di Vittorio]], *2-derivators* (2023). [[arXiv:2309.05216](https://arxiv.org/abs/2309.05216)]

[^1]: The condition of left properness does not appear in the arXiv version of [Renaudin 2006](#Renaudin06), but is added in the published version. By  [[Dugger's theorem]] (see [here](combinatorial+model+category#EveryCombinatorialModelCatQEquivalentToLeftProper)) every combinatorial model category is Quillen equivalent to a left proper one, but it is not immediate that every [[zig-zag]] of Quillen equivalences between left proper combinatorial model categories may be taken to pass through only left proper ones.

[[!redirects derivators]]
[[!redirects prederivator]]
[[!redirects prederivators]]
[[!redirects derivateur]]
[[!redirects dérivateur]]
[[!redirects derivateurs]]
[[!redirects dérivateurs]]
[[!redirects semiderivator]]
[[!redirects semiderivators]]


derived adjunction

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## Stetment

+-- {: .num_prop #QuillenAdjunctionInducesAdjunctionOnHomotopyCategories}
###### Proposition
**([[derived adjunction]])**

For $\mathcal{C} \underoverset{\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{{}_{\phantom{Qu}}\bot_{Qu}}\mathcal{D}$ a [[Quillen adjunction]] between [[model categories]], also the corresponding left and right [[derived functors]] form a pair of [[adjoint functors]]

$$
  Ho(\mathcal{C})
    \underoverset
      {\underset{\mathbb{R}R}{\longrightarrow}}
      {\overset{\mathbb{L}L}{\longleftarrow}}
      {\bot}
  Ho(\mathcal{D})
$$

between the corresponding [[homotopy category of a model category|homotopy categories]].

Moreover, the [[adjunction unit]] and [[adjunction counit]] of this derived adjunction are the images of the [[derived adjunction unit]] and [[derived adjunction counit]] of the original [[Quillen adjunction]].

=--

([Quillen 67, I.4 theorem 3](model+category#Quillen67))

For full proof see also at _[[Introduction to Homotopy Theory]]_ [this Prop.](Introduction+to+Homotopy+Theory#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories), or at _[[geometry of physics -- categories and toposes]]_ [this Prop.](geometry+of+physics+--+categories+and+toposes#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories).

## References

* {#Quillen67} [[Daniel Quillen]], chapter I _Axiomatic homotopy theory_ in:  _[[Homotopical Algebra]]_, Lecture Notes in Mathematics 43, Springer 1967([doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438))

[[!redirects derived adjunctions]]


derived adjunction unit

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## Definition

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

Let $\mathcal{C}$ and $\mathcal{D}$ be [[model categories]], and let

$$
  \mathcal{C}
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}}
      {\bot_{Qu}}
  \mathcal{D}
$$

be a [[Quillen adjunction]]. Then

1. a _derived adjunction unit_ at an [[object]] $d \in \mathcal{D}$ is a [[composition]] of the form

   $$
     Q(d)
       \overset{\eta_{Q(d)}}{\longrightarrow}
     R(L(Q(d)))
       \overset{R( j_{L(Q(d))} )}{\longrightarrow}
     R(P(L(Q(d)))
   $$

   where

   1. $\eta$ is the ordinary [[adjunction unit]];

   1. $\emptyset \underoverset{\in Cof_{\mathcal{D}}}{i_{Q(d)}}{\longrightarrow} Q(d) \underoverset{\in W_{\mathcal{D}} \cap Fib_{\mathcal{D}}}{p_{Q(d)}}{\longrightarrow} d$ is a [[cofibrant resolution]] in $\mathcal{D}$;

   1. $L(Q(d)) \underoverset{\in W_{\mathcal{C}} \cap Cof_{\mathcal{C}}}{j_{L(Q(d))}}{\longrightarrow} P(L(Q(d))) \underoverset{\in Fib_{\mathcal{C}}}{q_{L(Q(d))}}{\longrightarrow} \ast$ is a [[fibrant resolution]] in $\mathcal{C}$;

1. a _derived adjunction counit_ at an object $c \in \mathcal{C}$ is a composition of the form

   $$
     L(Q(R(P(c))))
       \overset{ L( p_{R(P(c))} ) }{\longrightarrow}
     L R(P(c))
       \overset{\epsilon_{P(c)}}{\longrightarrow}
     P(c)
   $$

   where

   1. $\epsilon$ is the ordinary [[adjunction counit]];

   1. $c \underoverset{\in W_{\mathcal{C}} \cap Cof_{\mathcal{C}}}{j_c}{\longrightarrow} P c \underoverset{\in Fib_{\mathcal{C}}}{q_c}{\longrightarrow} \ast$ is a [[fibrant resolution]] in $\mathcal{C}$;

   1. $\emptyset \underoverset{\in Cof_{\mathcal{D}}}{i_{R(P(c))}}{\longrightarrow} Q(R(P(c))) \underoverset{\in W_{\mathcal{D}} \cap Fib_{\mathcal{D}}}{p_{R(P(c))}}{\longrightarrow} R(P(c))$ is a [[cofibrant resolution]] in $\mathcal{D}$.

=--

## Properties

* A [[Quillen adjunction]] is a [[Quillen equivalence]] precisely if both its derived adjunction units and derived adjunction counit are [[weak equivalences]].

* A [[Quillen adjunction]] is a [[Quillen reflection]] precisely if the components of its derived adjunction counit are [[weak equivalences]].

* A [[Quillen adjunction]] is a [[Quillen co-reflection]] precisely if the components of its derived adjunction unit are [[weak equivalences]].

## Related concepts

* [[adjunction unit]]

* [[derived functor]]

* [[Quillen equivalence]], [[Quillen reflection]]

[[!redirects derived adjunction units]]

[[!redirects derived adjunction counit]]
[[!redirects derived adjunction counits]]

[[!redirects derived unit of a Quillen adjunction]]
[[!redirects derived units of Quillen adjunctions]]

[[!redirects derived counit of a Quillen adjunction]]
[[!redirects derived counits of Quillen adjunctions]]


derived affine scheme

A derived affine scheme is a special kind of [[generalized scheme]].

In a version of the theory of [[derived algebraic stack]]s due to Toën, Vezzosi and Vaquie, __the category of derived affine schemes__ is $sComm^{op}$, the [[opposite category|opposite]] of the category of [[simplicial object|simplicial]] [[commutative unital rings]]. The category of [[simplicial presheaves]] on $sComm^{op}$ has several [[model category]] structures. If a [[projective model structure]] is used, this category of simplicial presheaves is denoted $SPr(dAff)$ where weak equivalences and fibrations are defined levelwise. By $dAff^{\hat{}}$ one denotes the left [[Bousfield localization]] of the model category $SPr(dAff)$ with respect to the [[Yoneda embedding|Yoneda images]] $h_X\to h_Y$ of equivalences in $dAff$; this model category $dAff^{\hat{}}$ is called the _model category of prestacks_ over $dAff$. The fibrant objects in $dAff^{\hat{}}$ are the simplicial presheaves $F:dAff^{op}\to sSet$ such that

* (anodyne condition) for all $X\in dAff$, the simplicial set $F(X)$ is fibrant; and

* (prestack condition) for each equivalence $X\to Y$ in $dAff$, the induced morphism $F(Y)\to F(X)$ is a weak equivalence of simplicial sets.

The [[homotopy category]] $Ho(dAff^{\hat{}})$ is naturally equivalent to the [[full subcategory]] of $Ho(SPr(dAff))$ whose objects are the simplicial presheaves satisfying the above prestack condition.

* [[Bertrand Toën|B. Toën]], _Simplicial presheaves and derived algebraic geometry_, lecture notes CRM, Barcelona 2008; ([crm-2008.pdf](http://www.math.univ-toulouse.fr/~toen/crm-2008.pdf))


Derived Algebraic Geometry

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This entry is about the Ph.D. thesis

* [[Jacob Lurie]], _Derived Algebraic Geometry_, Ph.D. thesis, [pdf](http://www.math.harvard.edu/~lurie/papers/DAG.pdf).

which develops the theories of [[homotopical algebraic geometry]] and in particular [[derived algebraic geometry]].

See also the sequels, which are concerned with [[spectral algebraic geometry]].

* [[Jacob Lurie]], _[[Spectral Schemes]]_

* [[Jacob Lurie]], _[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]_

* [[Jacob Lurie]], _[[Closed Immersions]]_

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

* [[Jacob Lurie]], _[[Proper Morphisms, Completions,
and the Grothendieck Existence Theorem]]_

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

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


derived algebraic geometry

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## Idea

_Derived algebraic geometry_ is the specialization of [[higher geometry]] and [[homotopical algebraic geometry]] to the [[(infinity,1)-category]] of [[simplicial commutative rings]] (or sometimes, coconnective commutative [[dg-algebras]]).
Hence it is a generalization of ordinary [[algebraic geometry]] where instead of [[commutative rings]], [[derived schemes]] are locally modelled on [[simplicial commutative rings]].

Derived algebraic geometry is the correct setting for certain problems arising in [[algebraic geometry]] that involve [[intersection theory]] and [[deformation theory]] (see below).

## Terminology

Sometimes the term _derived algebraic geometry_ is also used for the related subject of [[spectral algebraic geometry]], where [[commutative ring spectra]] are used instead of [[simplicial commutative rings]].  Sometimes it may also refer to the subject of [[derived noncommutative algebraic geometry]].

## Motivation

There are several motivations for the study of derived algebraic geometry.

1.  The [[hidden smoothness principle]] of [[Maxim Kontsevich]], which conjectures that in classical [[algebraic geometry]], the non-[[smoothness]] of certain [[moduli spaces]] is a consequence of the fact that they are in fact truncations of [[derived moduli stacks]] (which are quasi-[[smooth]]).

1.  [[intersection theory|Intersection theory]]: a [[geometric]] interpretation of the [[Serre intersection formula]] for non-transverse [[intersections]].

1.  [[deformation theory|Deformation theory]]: a [[geometric]] interpretation of the [[cotangent complex]]: the "total space" of the cotangent complex $\mathbb{L}_X$ of $X$ makes sense as an object of derived algebraic geometry (a [[derived stack]]), namely it can be regarded as a derived vector bundle which specializes to the [[cotangent bundle]] $T^*X$ when $X$ is smooth.

For more detail on the final two points, see [(Vezzosi, 2011)](#VezzosiWhatIs).

## History

The idea of developing a theory of schemes where structure sheaves are sheaves of dg-algebras was first proposed by [[Maxim Kontsevich]]. The first rigorous approach to such a theory was via [[dg-schemes]], introduced by  [[Ionut Ciocan-Fontanine]] and [[Mikhail Kapranov]], who used it to construct the first examples of [[derived moduli spaces]] (derived [[Hilbert scheme]] and derived [[Quot scheme]]).  However, this theory was relatively restrictive because dg-schemes were equipped with an embedding into an ambient smooth scheme by definition (hence in particular one could not model the correct notion of morphisms of derived schemes in this language), and because dg-algebras need to be replaced by [[simplicial commutative rings]] in positive or mixed characteristic.

[[Bertrand Toen]] and [[Gabriele Vezzosi]] developed [[homotopical algebraic geometry]], which is [[algebraic geometry]] in any [[HAG context]], i.e. over any [[symmetric monoidal model category]] satisfying certain assumptions.  As special cases they recover the [[algebraic geometry]] of [[Grothendieck]] and the [[higher stacks]] of [[Carlos Simpson]], and also develop new theories of [[derived algebraic geometry]], [[complicial algebraic geometry]], and [[brave new algebraic geometry]].

In his thesis [[Jacob Lurie]] also developed fundamentals of derived algebraic geometry, using the language of [[structured (infinity,1)-toposes]] where [[Bertrand Toen|Toen]]-[[Gabriele Vezzosi|Vezzosi]] used [[model toposes]].  He also developed a version of derived algebraic geometry which is locally modelled on [[E-∞ rings]], called [[spectral algebraic geometry]].

## Terminology "derived" versus "$\infty$-"

The adjective "derived" means pretty much the same as the "$\infty$-" in [[∞-category]], so this is higher algebraic geometry in the sense being locally represented by higher algebras. The word stems from the use of "derived" as in [[derived functor]]. This came from the study of derived moduli problem. Namely to parametrize the moduli, one first looks at some space of "cochains" which are candidates for structures to parametrize. then one cuts those which indeed satisfy the axioms ("equations") for the structures. Satisfying these equations is a limit-type construction, hence left exact and one is lead to right derived functors to improve;
exactness on the right; this leads to use cochain complexes.
The obtained moduli is too big as there are many isomorphic structures, so one needs to quotient by the automorphisms; this is a colimit type construction hence right exact. The improved quotient is the left derived functor, what is obtained by passing to stacks.

## Definitions

### structured $(\infty,1)$-toposes ##

In [Lurie, Structured spaces](#LurieStructured) a definition of [[derived algebraic scheme]] and [[derived Deligne-Mumford stack]] is given in the wider context of [[generalized scheme]]s realized as locally affine [[structured (∞,1)-topos]]es.

See these links for more details.

+-- {: .num_remark}
###### Remark (derived scheme are $(\infty,1)$-toposes)

This definition is based on the observation that it is a deficiency of the ordinary definition of [[scheme]] to demand that underlying a scheme is a [[topological space]] and that a better definition is obtained by demanding it to have an underlying [[locale]]. But a [[locale]] is a [[0-topos]]. This motivates then the definition of a [[generalized scheme]] as a (locally affine, [[structured (infinity,1)-topos|structured]]) [[(∞,1)-topos]].

=--

## Relation to derived noncommutative geometry
 {#RelationToDerivedNoncommutativeGeometry}

Under some conditions, [[derived schemes]] $X$ in the sense of ([[Structured Spaces|Lurie, Structured Spaces]]) are faithfully encoded by their [[stable (∞,1)-categories]] $QCoh(X)$ of [[quasicoherent sheaves]]. This is the content of [[Tannaka duality for geometric stacks]], ([[Quasi-Coherent Sheaves and Tannaka Duality Theorems|Lurie, Quasi-Coherent Sheaves and Tannaka Duality Theorems]]). Therefore one can turn this around and declare that a suitable [[stable (∞,1)-category]] $\mathcal{A}$ which is not of the form $QCoh(X)$ for an actual [[derived scheme]] $X$ represents a generalized, "noncommutative" derived scheme. This is much like a [[2-category theory]] or rather [[(∞,2)-category]] theory analog of how in [[algebraic geometry]] the opposite of the category of [[monoids]] (algebras) is regarded as a category of generalized spaces. This might be (and has been) called _[[2-algebraic geometry]]_.

Accordingly, one can decide to regard the [[opposite (∞,1)-category]] of suitable (e.g. [[monoidal (∞,1)-category|monoidal]]) [[stable (∞,1)-categories]]  as being a category of "noncommutative derived schemes". This is effectively the perspective on [[noncommutative algebraic geometry]] that [[Maxim Kontsevich]] has been promoting.

Often and traditionally, all this is expressed in terms of certain presentations for these [[stable (∞,1)-categories]] by [[triangulated category|triangulated]] [[derived categories]] or better, [[dg-enhancement|enhancements]] as [[dg-categories]].

In this fashion then in [[derived noncommutative algebraic geometry]], a [[space]] is by definition a [[dg-category]] that is smooth and proper in an appropriate sense.  The relation between [[noncommutative algebraic geometry]] and [[derived algebraic geometry]] may then be summed up by the adjunction
  $$ Pf : \DSt(k)^{op} \rightleftarrows NCSp(k) : \mathcal{M}_- $$
where $Pf(X)$ denotes the [[dg-category]] of [[perfect complexes]] on the [[derived stack]] $X$, and $\mathcal{M}_\mathcal{C}$ denotes the [[derived moduli stack of objects in a dg-category|derived moduli stack of objects]] in the [[dg-category]] $\mathcal{C}$.  See [[derived moduli stack of objects in a dg-category]] for details.

## Related concepts

* [[homotopical algebraic geometry]]
* [[E-∞ geometry]]
* [[dg-geometry]]
* [[spectral algebraic geometry]]
* [[brave new algebraic geometry]]
* [[higher stacks]]
* [[higher differential geometry]]
* [[derived category of coherent sheaves]]
* [[derived noncommutative algebraic geometry]]

## References

(For references on [[dg-schemes]], the historical precursor to [[derived schemes]], see there.)

The main references are

* [[Jacob Lurie]], [[Derived Algebraic Geometry]]

* {#LurieStructured} [[Jacob Lurie]], _[[structured (∞,1)-topos|Structured Spaces]]_

* {#ToenVezzosi04} [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical algebraic geometry II: geometric stacks and applications_, Memoirs of the AMS **193** (2008) [[arXiv:math/0404373](http://arxiv.org/abs/math/0404373), [ams:memo-193-902](https://bookstore.ams.org/memo-193-902)]

* {#TV-HAG-to-DAG} [[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 (2004) [[math.AG/0210407](http://arxiv.org/abs/math/0210407)]

The following notes deal with the theory modelled on coconnective [[commutative dg-algebras]].

* [[Dennis Gaitsgory]], _Notes on geometric Langlands: stacks_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/Stackstext.pdf).

* J. Eugster and J. P. Pridham. _An introduction to derived (algebraic) geometry_.
[arXiv](http://arxiv.org/abs/2109.14594).

The following notes deal with the theory modelled on [[E-infinity ring spectra]] ([[E-infinity geometry]]):

* [[Clark Barwick]], _Applications of derived algebraic geometry to homotopy theory_, lecture notes, mini-course in Salamanca, 2009, [pdf](https://dl.dropboxusercontent.com/u/1741495/papers/salamanca.pdf).

See also the surveys

* [[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))

* {#VezzosiWhatIs} [[Gabriele Vezzosi]], _What is a derived stack?_, 2011, [pdf](http://www.ams.org/notices/201107/rtx110700955p.pdf).

* [[Bertrand Toën]], _Derived algebraic geometry_, [arxiv/1401.1044](http://arxiv.org/abs/1401.1044)

Discussion of [[derived noncommutative algebra]] over [[E-n algebras]] is 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))


derived analytic geometry

#Contents#
* table of contents
{:toc}

## Idea

Derived analytic geometry is the study of derived analogs of analytic spaces in various context, such as [[complex analytic geometry]], [[non-archimedean analytic geometry]] and [[global analytic geometry]].

The main motivation for the introduction of derived analytic spaces is to have a proper functorial setting for deformation theory and the cotangent complex in analytic geometry, to prove an Artin/Lurie representability theorem, that characterizes Artin stacks among higher derived stacks.

One also gets, using these methods, a derived construction of the Chern character and an analytic version of derived de Rham cohomology.

Derived analytic methods may also be useful to study intersection theory and virtual fundamental classes on some analytic moduli spaces.

## Related concepts

* [[derived complex analytic space]]

* [[derived global analytic geometry]]

* [[overconvergent global analytic geometry]]

## References

* [[Jacob Lurie]], _[[Closed Immersions]]_ (DAG IX).

* [[Mauro Porta]], [[Tony Yue Yu]], _Higher analytic stacks and GAGA theorems_, _Derived non-archimedean analytic spaces_, Selecta Math. New Ser. 24, 609-665 (2018) [arXiv:1601.00859](https://arxiv.org/abs/1601.00859) [doi](https://doi.org/10.1007/s00029-017-0310-1)

* [[Mauro Porta]], [[Tony Yue Yu]], _Derived non-archimedean analytic spaces_, Selecta Math. New Ser. 24, 609-665 (2018) [arXiv:1601.00859](https://arxiv.org/abs/1601.00859) [doi](https://doi.org/10.1007/s00029-017-0310-1)

* [[Mauro Porta]], _Derived complex analytic geometry I: GAGA theorems_, [arXiv:1506.09042](https://arxiv.org/abs/1506.09042)

* [[Mauro Porta]], _Derived complex analytic geometry II: square-zero extensions_, [arXiv:1507.06602](https://arxiv.org/abs/1507.06602)

* [[Frédéric Paugam]], _Overconvergent global analytic geometry_, [arXiv:1410.7971](https://arxiv.org/abs/1410.7971).


derived blow-up

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## Idea

The notion of *derived blow up* is the generalization to [[derived algebraic geometry]] and generally to [[derived geometry]] of the notion of *[[blow-up]]* in ordinary [[algebraic geometry|algebraic]] [[geometry]].

## References

* [[Adeel Khan]], David Rydh, _Virtual Cartier divisors and blow-ups_ ([arXiv:1802.05702](https://arxiv.org/abs/1802.05702)).

* [[Jeroen Hekking]], _Graded algebras, projective spectra and blow-ups in derived algebraic geometry_ [[arXiv:2106.01270](https://arxiv.org/abs/2106.01270)]

* [[Jeroen Hekking]], [[Adeel Khan]], David Rydh, _Deformation to the normal cone and blow-ups via derived Weil restrictions_, forthcoming.

* [[Oren Ben-Bassat]], [[Jeroen Hekking]], _Blow-ups and normal bundles in connective and nonconnective derived geometries_ [[arXiv:2303.11990](https://arxiv.org/abs/2303.11990)]

[[!redirects derived blow-ups]]


derived category

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{:toc}

## Idea

The _derived (infinity,1)-category_ or _derived category_ of an [[abelian category]] $\mathcal{A}$ is the setting for [[homological algebra]] in $\mathcal{A}$: the [[(infinity,1)-categorical localization]] of the [[category of chain complexes]] in $\mathcal{A}$ at the class of [[quasi-isomorphisms]]. The derived category is a fundamental example of a [[stable (infinity,1)-category]]. In the case that $\mathcal{A} \simeq$ [[Mod|$R Mod$]] (cf. the [[Freyd-Mitchell embedding theorem]]), the [[stable Dold-Kan correspondence]] says that the derived $\infty$-category of $\mathcal{A}$ is equivalently the [[stable (infinity,1)-category of spectra|stable $\infty$-category of]] [[Eilenberg-MacLane spectrum|$H R$]]-[[module spectra]].

The derived $\infty$-category is presented by various [[dg-model structures]] on the [[category of chain complexes]], as described at [[model structures on chain complexes]]. As such it has also a natural incarnation as a [[pretriangulated dg-category]], which might be called the _derived dg-category_.

Like any [[stable (infinity,1)-category]], the [[homotopy category of an (infinity,1)-category|homotopy category]] of the derived (infinity,1)-category admits a canonical [[triangulated category]] structure. Often in the literature, the term _derived category_ refers to the [[homotopy category of an (infinity,1)-category|homotopy category]], viewed only as a [[triangulated category]]. The loss of information can often be problematic, but for many purposes is not important.

In what follows, we will describe only the homotopy category. See [[(infinity,1)-category of chain complexes]] for the full [[(infinity,1)-category]].

Associated to $\mathcal{A}$ is

* the [[category of chain complexes]] $Ch_\bullet(\mathcal{A})$ in $\mathcal{A}$ which is naturally a [[homotopical category]];

* the "[[homotopy category of chain complexes]]" $K(\mathcal{A})$;

* the [[stable ∞-category]] $K_\infty(\mathcal{A})$ of [[chain complexes]] in $C$.

The  _derived category_ $D(C)$ of $C$ is equivalently

* the [[homotopy category|1-categorical homotopy category]] of $Ch_\bullet(\mathcal{A})$ with respect to the [[quasi-isomorphisms]];

* the [[homotopy category of an (infinity,1)-category|(∞,1)-categorical homotopy category]] of $K_\infty(\mathcal{A})$.

In either case, this means that under the canonical [[localization]] functor

$$
  Q : Ch_\bullet(\mathcal{A}) \to D(\mathcal{A})
$$

the [[quasi-isomorphisms]] of [[chain complexes]] become true [[isomorphisms]]
and that $D(\mathcal{A})$ is [[universal property|universal]] with respect to this property.

Hence the derived category is an approximation to the full [[simplicial localization]] of $K(\mathcal{A})$. It is or can be equipped with several further [[properties]] and [[structure]] that give a more accurate approximation. Notably every derived category is a _[[triangulated category]]_, which is a way of remembering the [[suspension]] and de-suspension operations on its objects -- the [[suspension of chain complexes]] -- hence its "[[stable (infinity,1)-category|stability]]".

## History

Derived categories were introduced by [[Jean-Louis Verdier]] in his thesis under the supervision of [[Alexandre Grothendieck]].  It was originally used to extend [[Serre duality]] to a relative context.  See [[Robin Hartshorne|Hartshorne]]'s lecture notes "Residues and duality".

## Definition

Let $\mathcal{A}$ be an [[abelian category]] and $K(\mathcal{A})$ its
[[category of chain complexes]] modulo [[chain homotopy]] (the "[[homotopy category of chain complexes]]").

Equip $K(\mathcal{A})$ with the structure of a [[homotopical category]]
by declaring the [[weak equivalences]] to be the
**quasi-isomorphisms**: those morphisms
$f : V \to W$
which induce [[isomorphisms]] in [[homology]],
$H(f) : H(V) \stackrel{\simeq}{\to} H(W)$.

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

The **derived category** $D(\mathcal{A})$ is the [[homotopy category]] of
$K(\mathcal{A})$ with respect to these weak equivalences.

=--

Analogously, for $K^{+,-,b}(\mathcal{A})$ denoting the [[full subcategory]] on the chain complexes bounded above, bounded below, or bounded, respectively (see at _[[category of chain complexes]]_), one writes

$$
  D^{+,-,b}(\mathcal{A}) \hookrightarrow D(\mathcal{A})
$$

for the correspponding full subcategory of the derived category.

## Equivalent characterizations and constructions
 {#Properties}

There are various ways to construct or express the derived category more explicitly in terms of various special objects or morphisms in the [[category of chain complexes]].

* _[In terms of localization at a null system](#AsLOcalizationAtNullSystem)_

* _[In terms of resolutions](#InTermsOfResolutions)_.

### In terms of localization at a null system
 {#AsLOcalizationAtNullSystem}

The "[[homotopy category of chain complexes]]" $K(\mathcal{A})$ is already a [[triangulated category]].  The derived category can be obtained as the construction of a [[homotopy category]] of a [[triangulated category]] with respect to a [[null system]].

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

Let

$$
  N(\mathcal{A}) \subset K(\mathcal{A})
$$

and analogously

$$
  N^{+,-n,b}(\mathcal{A}) \subset K^{+,-,b}(\mathcal{A})
$$

be the [[full subcategory]] of $K(C)$ or on $K^{+,-,b}$, respectively, on those [[chain complexes]] $V$ whose  [[chain homology]] vanishes in every degree, $H_\bullet(V) = 0$.

=--

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

A [[chain map]] $f_\bullet : V_\bullet \to W_\bullet$ is a
[[quasi-isomorphism]] precisely there exists a [[distinguished triangle]] in $K(\mathcal{A})$ of the form

$$
  V \stackrel{f}{\to} W \to cone(f)
$$

with the [[mapping cone]] $cone(f) \in N(\mathcal{C})$.

=--

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

The derived category is equivalently the localization of $K(\mathcal{A})$ at the [[null system]] $N(\mathcal{A})$.

$$
  D(\mathcal{A}) \simeq K(\mathcal{A})/N(\mathcal{A})
  \,.
$$

=--

This perspective is discussed in ([Kashiwara-Schapira, section 13](#KashiwaraSchapira)) and ([Schapira, section 6.2, 72](#Schapira)).

### In terms of injective and projective resolutions
 {#InTermsOfResolutions}

In the case that the underlying [[abelian category]] $\mathcal{A}$ has [enough injectives](injective%20object#EnoughInjectives) or [enough projectives](projective%20object#EnoughInjectives), the [[hom sets]] in the derived category may equivalently be obtained as [[homotopy]]-classes of [[chain maps]] from [[projective resolutions]] to [[injective resolutions]] of chain complexes.

In view of the existence of the injective and projective [[model structure on chain complexes]] this is a special case of the general fact that [[homotopy categories]] of [[model categories]] may be obtained by forming homotopy classes of maps in the model category from  [[cofibrant resolutions]] to [[fibrant resolutions]]. But here we spell out an direct discussion of this fact for chain complexes.

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

Write $K^+(\mathcal{I}_{\mathcal{A}}) \hookrightarrow K^+(\mathcal{A})$
for the [[full subcategory]] of the [[homotopy category of chain complexes]] bounded above on those that are degreewise [[injective objects]].

Dually, let $K^-(\mathcal{P}_{\mathcal{A}}) \hookrightarrow K^-(\mathcal{A})$
for the [[full subcategory]] of the [[homotopy category of chain complexes]] bounded below on those that are degreewise [[projective objects]].

=--

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

If $\mathcal{A}$ has [enough injectives](http://ncatlab.org/nlab/show/injective%20object#EnoughInjectives) then the canonical functor

$$
  K^+(\mathcal{I}_{\mathcal{A}}) \to D^+(\mathcal{A})
$$

is an [[equivalence of categories]].

Dually, if $\mathcal{A}$ has [enough projectives](http://ncatlab.org/nlab/show/projective%20object#EnoughProjectives) then the canonical functor

$$
  K^-(\mathcal{P}_{\mathcal{A}}) \to D^-(\mathcal{A})
$$

is an [[equivalence of categories]].

=--

For instance ([Schapira, cor. 7.2.3](#Schapira)).

## Related concepts

* [[derived functor]]
* [[triangulated categories]]

## References

The original reference:

* [[Verdier, Jean-Louis]], _Des Catégories Dérivées des Catégories Abéliennes_, Astérisque **239** (1996) [[doi:10.24033/ast.364](https://smf.emath.fr/publications/des-categories-derivees-des-categories-abeliennes), [numdam:AST_1996__239__R1_0](http://www.numdam.org/issues/AST_1996__239__R1_0), [pdf](http://www.numdam.org/item/AST_1996__239__R1_0.pdf)]

* {#GelfandManin96} [[Sergei Gelfand]], [[Yuri Manin]], Section III of: *[[Methods of homological algebra]]*,  transl. from the 1988 Russian (Nauka Publ.) original, Springer (1996, 2002) [[doi:10.1007/978-3-662-12492-5](https://doi.org/10.1007/978-3-662-12492-5)]

Systematic discussion from the point of view of [[localization]] and [[homotopy theory]]:

* {#KashiwaraSchapira} [[Masaki Kashiwara]], [[Pierre Schapira]], section 13 of  *[[Categories and Sheaves]]* (2006)

and, similarly, in section 7 of

* [[Pierre Schapira]], _Categories and homological algebra_ (2011) ([pdf](http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf))
 {#Schapira}

A detailed treatment of derived categories (including of DG modules over DG
rings), with applications to noncommutative algebra, is in the book

* [[Amnon Yekutieli]], *Derived Categories*, Cambridge University Press (2019) [[doi:10.1017/9781108292825]( https://doi.org/10.1017/9781108292825), [arXiv:1610.09640](https://arxiv.org/abs/1610.09640)]

A pedagogical introduction:

* R. P. Thomas, _Derived categories for the working mathematician_ ([arXiv:math.AG/0001045](http://arxiv.org/abs/math.AG/0001045))

A good survey of the more general topic of derived categories is

* [[Bernhard Keller]], _Derived categories and their uses_ ([ps](http://www.google.de/url?sa=t&source=web&ct=res&cd=6&ved=0CC8QFjAF&url=http%3A%2F%2Fwww.math.jussieu.fr%2F~keller%2Fpubl%2Fdcu.ps&rct=j&q=derived+category&ei=Ib76SsSdAsjb-QaAw7moDw&usg=AFQjCNGIgXLHlprAoR70bGLWQmyKGHDjTQ))

See in particular also the list of references given there.

Other lecture notes include

* Theo Bühler, _An introduction to the derived category_ ([pdf](https://ncatlab.org/nlab/files/intro-derived.PDF)).

and for applications to [[coherent sheaves]],

* [[Franco Rota]], _An introduction to the derived category of coherent sheaves_ ([pdf](https://drive.google.com/file/d/12fhshN_lFOi54pjBdjVEPzUb1XBxz-TW/view))

For a discussion in the context of [[(∞,1)-category|(∞,1)-categories]] and in particular [[stable (∞,1)-category|stable (∞,1)-categories]] see [section 13, p. 53](http://www.math.harvard.edu/~lurie/papers/DAG-I.pdf#page=53) of

* [[Jacob Lurie]], _[[Stable ∞-Categories]]_

For the applications of derived categories in [[algebraic geometry]], see

* [[Dmitri Orlov]], _Derived categories of coherent sheaves and equivalences between them_, Uspekhi Mat. Nauk, 2003, Vol. 58, issue 3(351), pp. 89–172, [English translation (PDF)](http://www.mi.ras.ru/~orlov/papers/Uspekhi2003.pdf)

* [[Aleksei Bondal]], [[Dmitri Orlov]], _[Reconstruction of a variety from the derived category and groups of autoequivalences](http://www.mi.ras.ru/~orlov/papers/Compositio2001.pdf)_, Compositio Mathematica 125 (03), 327-344.  See also [[Bondal-Orlov reconstruction theorem]].

* [[Daniel Huybrechts]], _Fourier-Mukai Transforms in Algebraic Geometry_, Oxford University Press, USA, 2006.

* [[Igor Dolgachev]], [_Derived categories_](http://www.math.lsa.umich.edu/~idolga/derived9.pdf).

* Andrei Caldararu, _Derived categories of coherent sheaves: a skimming_.  Lecture notes from _Algebraic Geometry: Presentations by Young Researchers_ in Snowbird, Utah, July 2004.  Available on [arXiv](http://arxiv.org/abs/math/0501094).

[[!redirects derived categories]]
[[!redirects bounded derived category]]
[[!redirects bounded derived categories]]

[[!redirects derived (infinity,1)-category]]
[[!redirects derived (infinity,1)-categories]]
[[!redirects derived (∞,1)-category]]
[[!redirects derived (∞,1)-categories]]
[[!redirects derived dg-category]]
[[!redirects derived dg-categories]]


An analogue of a derived category of differential graded $R$-modules, where $R$ is an associative ring can be found when the ring is replaced by a more general algebra over an operad.

* Clemens Berger, Ieke Moerdijk, *On the derived category of an algebra over an operad*, Georgian Math. J. 1. **16** (2009), n.1, 13--28 [pdf at gmj site](http://www.heldermann-verlag.de/gmj/gmj16/gmj16002.pdf) [arxiv:0801.2664](http://arxiv.org/abs/0801.2664)

*Abstract.* We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the [[enveloping operad]] of an algebra over an operad.


derived category of coherent sheaves

#Contents#
* table of contents
{:toc}

## Idea

The [[derived category]] of a category [[quasi-coherent sheaves]].

## Related concepts

* [[derived category]]

* [[coherent sheaves]]

* [[Bondal-Orlov reconstruction theorem]]

* [[Fourier-Mukai transform]]

* [[triangulated categories of sheaves]]

* [[Bridgeland stability condition]]

## References

### General

* [[Alexei Bondal|A. I. Bondal]], D. O. Orlov, _Derived categories of coherent sheaves_, Proc. Internat. Congress of Mathematicians vol. II 46--54 (Beijing, 2002)
[djvu:134K](http://www.mathunion.org/ICM/ICM2002.2/Main/icm2002.2.0047.0056.ocr.djvu) [pdf:881K](http://www.mathunion.org/ICM/ICM2002.2/Main/icm2002.2.0047.0056.ocr.pdf)

* V. A. Lunts, [[Dmitri Orlov|D. O. Orlov]], _Uniqueness of enhancement for triangulated categories_, J. Amer. Math. Soc. __23__  (2010), 853-908, [journal](http://www.ams.org/journals/jams/2010-23-03/S0894-0347-10-00664-8/home.html), [arXiv:0908.4187](http://arxiv.org/abs/0908.4187), [doi](http://dx.doi.org/10.1090/S0894-0347-10-00664-8)
* [[Dmitri Orlov]], _Derived categories of coherent sheaves and motives_.
[math.AG/0512620](http://arxiv.org/abs/math/0512620)
* [[Dmitri Orlov]], _Derived categories of coherent sheaves and triangulated categories of singularities_, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 503--531, Progr. Math., 270, Birkhauser, Boston, Inc., Boston, MA, 2009 [math.AG/0503632](http://arxiv.org/abs/math/0503632)
* A. N. Kapustin, D. O. Orlov, _Lectures on mirror symmetry, derived categories, and D-branes_ (Russian) Uspekhi Mat. Nauk __59__ (2004), no. 5 (359), 101--134; English translation in Russian Math. Surveys __59__ (2004), no. 5, 907--940 [math.AG/0308173](http://arxiv.org/abs/math/0308173)

* A. I. Bondal, D. O. Orlov, _Reconstruction of a variety from the derived category and groups of autoequivalences_, Compos. Math. 125 (2001), 327–344 [doi:10.1023/A:1002470302976](http://dx.doi.org/10.1023/A:1002470302976)

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

* [[Valery Lunts]], Olaf M. Schnuerer, _New enhancements of derived categories of coherent sheaves and applications_, [arXiv](http://arxiv.org/abs/1406.7559).

Introduction and review:

* [[Franco Rota]], _The derived categories of coherent sheaves_, lecture notes [[pdf](https://drive.google.com/file/d/12fhshN_lFOi54pjBdjVEPzUb1XBxz-TW/view), [[Rota-DerivedCategories.pdf:file]]]

### In Rozansky-Witten theory

On [[Rozansky-Witten weight systems]] as [[Lie algebra weight systems]] for [[Lie algebra objects]] in the [[derived category of quasi-coherent sheaves]], and unified [[Wheels theorem]]:

* {#RobertsWillerton} [[Justin Roberts]], [[Simon Willerton]], _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))

## Related concepts

* [[B-model]] [[topological string]]

## References

On [[t-structures]] on [[derived categories of coherent sheaves]]:

* [[Dan Abramovich]], [[Alexander Polishchuk]], *Sheaves of t-structures and valuative criteria for stable complexes*, J. reine angew. Math. **590** (2006) 89-130 [[arXiv:math/0309435](https://arxiv.org/abs/math/0309435), [doi:10.1515/CRELLE.2006.005](https://doi.org/10.1515/CRELLE.2006.005)]

* [[Alexander Polishchuk]], *Constant families of t-structures on derived categories of coherent sheaves*, Moscow Math. J. __7__ (2007) 109-134 [[arXiv:math/0606013](https://arxiv.org/abs/math/0606013)]

[[!redirects derived categories of coherent sheaves]]
[[!redirects derived category of quasi-coherent sheaves]]
[[!redirects derived category of quasicoherent sheaves]]
[[!redirects derived categories of quasi-coherent sheaves]]
[[!redirects derived categories of quasicoherent sheaves]]


derived complex analytic space

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

#Contents#
* table of contents
{:toc}

## Idea

Complex [[derived analytic geometry]] is the refinement of the notion of [[complex analytic space]] from [[geometry]] to [[higher geometry]]/[[higher algebra]].

## Related concepts

* [[higher complex analytic geometry]]

* [[derived analytic geometry]]

## References

* [[Jacob Lurie]], sections 11 and 12 of _[[Closed Immersions]]_

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

* [[Mauro Porta]], _Derived complex analytic geometry I: GAGA theorems_ ([arXiv:1506.09042](http://arxiv.org/abs/1506.09042))
* [[Mauro Porta]], _[[Derived Analytic Rings]]_ (preprint including a good theory of modules over analytic rings)

* [[Frédéric Paugam]], _[[Overconvergent global analytic geometry]]_

[[!redirects derived complex analytic spaces]]

[[!redirects derived complex analytic spaces]]