derived critical locus

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

#Contents#
* table of contents
{:toc}

## Idea

The concept of _derived critical locus_ is the refinement of the notion of [[critical locus]] from [[geometry]] to [[derived geometry]].

The formal duals to derived critical loci are described by [[BV-BRST formalism]].

## Details

The following is a basic setup in [[dg-geometry]] aimed at exhibiting the (formal dual of) the [[BV-BRST complex]] as the derived critical locus of (the formal dual of) the [[BRST complex]] (the example [below](#ComparisonToBVBRSTExample))

### Basic dg-geometry
 {#BasicDgGeometry}

Let $k$ be a [[field]] of [[characteristic zero]].

Write $dgcAlg_k$ for the [[category]] of unbounded cochain [[differential graded-commutative algebras]] (dgc-algebras) over $k$.

An object in the [[opposite category]]

$$
  C \in cdgAlg_k^{op}
$$

we may regard as an affine space in [[dg-geometry]], and hence we write

$$
  \mathcal{O}(C) \in cdgAlg_k
$$

for the corresponding dgc-algebra.

Let $\mathcal{O}(C)$[[Mod]] be the [[category of dg-modules]] over $\mathcal{O}(C)$ equipped with the standard [[model structure on dg-modules]].

+-- {: .num_defn #OCDGCAlgebras}
###### Definition
**([[dgc-algebras]] over $\mathcal{O}(C)$)**

Write

$$
  \begin{aligned}
    cdgAlg_{\mathcal{O}(C)}
      & \coloneqq
    CMon(\mathcal{O}(C) Mod)
    \\
      & \simeq
    \mathcal{O}(C)/_{cdgAlg_k}
  \end{aligned}
$$

for the [[category of monoids|category of commutative monoids]] in $\mathcal{O}(C)$-modules, equivalently the [[coslice category]] of $cdgAlg_k$ under $\mathcal{O}(C)$.

=--

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

There is a [[model category]] structure on $cdgAlg_{\mathcal{O}(C)}$ (def. \ref{OCDGCAlgebras}) whose fibrations and weak equivalences are those of the underlying $\mathcal{O}(C)$-modules such that the [[free-forgetful adjunction]]

$$
  cdgAlg_{\mathcal{O}(C)}
   \underoverset
     {\underset{U}{\longrightarrow}}
     {\overset{Sym_{\mathcal{O}(C)}}{\longleftarrow}}
     {\bot}
  \mathcal{O}(C) Mod
$$

is a [[Quillen adjunction]].

This is

1. [[combinatorial model category|combinatorial]];

1. [[proper model category|proper]].

=--

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

This follows with the general discussion at _[[dg-geometry]]_. We indicate how to see it directly.

We observe that the adjunction exhibits the [[transferred model structure]] on the left. By the statement discussed there, it is sufficient to check that

1. $\mathcal{O}(C) Mod$ is a [[cofibrantly generated model category]].

   This follows because the [[nLab:model structure on dg-modules]] (as discussed there) is itself transferred along

   $$
     U' \colon \mathcal{O}(C) Mod \to Ch^\bullet(k)
   $$

   from the cofibrantly generated [[model structure on chain complexes|model structure on cochain complexes]].

1. $U$ preserves [[filtered colimits]].

   This follows from the general fact $U : CMon(\mathcal{C}) \to \mathcal{C}$ creates filtered colimits for $\mathcal{C}$ closed symmetric monoidal (see [there](category+of+monoids#FilteredColimits)) and that $A Mod$ is [[closed monoidal category|closed]] [[symmetric monoidal category|symmetric monoidal]] (see [there](model+structure+on+dg-modules#Properties)).

   To check this explicitly:

   Let $A_\bullet \colon D \to cdgAlg_k$ be a [[filtered category|filtered diagram]]. We claim that there is a unique way to lift the underlying colimit $\lim_\to U A_\bullet$ to a dg-algebra cocone: for $a \in A_i \to \lim_\to U A_\bullet$ and $b \in A_j \to \lim_\to U A_\bullet$ there is by the assumption that $D$ is filtered a $A_i \to A_l \leftarrow A_j$. Therefore in order for the cocone component $U A_l \to \lim_{\to} U A_\bullet$ to be an algebra homomorphism the product of $a$ with $b$ in $\lim_\to U A_\bullet$ has to be the image of this product in $A_l$. This defines the colimiting cocone $A_l \to \lim_\to A_\bullet$.

1. The left hand has functorial fibrant replacement (this is trivial, since every object is fibrant) and functorial [[path objects]].

   This follows by the same argument as for the path object in $cdgAlg_k$ ([here](model+structure+on+dg-algebras#PathObjectsForUnboundedCommutative)) this can be taken to be $(-)\otimes_k \Omega^\bullet_{poly}(\Delta[1])$.

=--

### Formal cotangent bundle

Given $C \in dgcAlg_k^{op}$ as [above](#BasicDgGeometry) we want to consider  its _formal [[cotangent bundle]]_ $T^\ast_f c$, i.e. the [[infinitesimal neighbourhood]] around the [[zero section]] of the would-be actual [[cotangent bundle]]

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

Write

$$
  Der(\mathcal{O}(C)) \in \mathcal{O}(C) Mod
$$

for the [[automorphism ∞-Lie algebra]] of $A$ whose underlying cochain complex is

$$
  \array{
    \cdots
      \to
    Der(\mathcal{O}(C))_k
      \overset{[d_{\mathcal{O}(C)},-]}{\longrightarrow}
    Der(\mathcal{O}(C))_{k+1}
      \longrightarrow
    \cdots
  }
  \,.
$$

where $Der(\mathcal{O}(C))_k$ is the module of [[derivations]]

$$
  v \colon \mathcal{O}(C)^\bullet \to \mathcal{O}(C)^{\bullet + k}
$$

of degree $k$ and $[d_{\mathcal{O}(C)}, -]$ is the graded commutator of derivations with the [[differential]] of $\mathcal{O}(C)$ regarded as a degree 1 derivation $d_{\mathcal{O}(C)} \colon \mathcal{O}(C) \to \mathcal{O}(C)$.

We say that $\mathcal{O}(C)$ is _smooth_ if $Der(\mathcal{O}(C))$ is cofibrant as an object on $\mathcal{O}(C) Mod$.

Write

$$
  \mathcal{O}(T^*_f C)
    \coloneqq
  Sym_{\mathcal{O}(C)} Der(\mathcal{O}(C))
   \in
  cdgAlg_{\mathcal{O}(C)}
$$

for the free $\mathcal{O}(C)$-algebra over $Der(\mathcal{O}(C))$.

We write

$$
  T^*_f C \in cdgAlg^{op}_k/_C
$$

for its formal dual.

=--

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

Every $S \in \mathcal{O}(C)$ defines a morphism

$$
  d S \colon C \to T^*_f C
$$

in $dgcAlg_k^{op}$ which is dually given by

$$
  \mathcal{O}(C)
    \leftarrow
   Sym_{\mathcal{O}(C)} Der(\mathcal{O}(C))
    \;\colon\;
   Sym_{\mathcal{O}(C)} [\hat S , -]
  \,,
$$

where $\hat S : \mathcal{O}(C) \to \mathcal{O}(C)$ is the $k$-linear multiplication operator defined by $S$ and where for
$v \in Der(\mathcal{O}(C))$ we set

$$
  [\hat S, v] = v(S)
  \,,
$$

which may be regarded as the multiplication operator given by the commutator of $k$-linear endomorphisms of $\mathcal{O}(C)$ as indicated.

=--

### Derived critical locus

+-- {: .num_defn #DerivedCriticalLocusWithFormalCotangentBundle}
###### Definition
**(derived critical locus)**

The **derived critical locus** of a morphism  $S \colon C \to \mathbb{A}^1$ in dgcAlg_k
is the [[homotopy pullback]] $C_{\{d S = 0\}}$ in $cdgAlg^{op}/_{C}$

$$
  \array{
    C_{\{d S = 0\}} &\to& C
    \\
    \downarrow &\swArrow& \downarrow^{\mathrlap{0}}
    \\
    C &\stackrel{d S}{\to}& T^*_f C
  }
  \,.
$$

=--

+-- {: .num_prop #PresentationByFreeDGCAlegbraOnMappingCone}
###### Proposition
**(presentation by free dgc-algebra on mapping cone)**

If $C$ is smooth in the sense that $Der(\mathcal{O}(C)) \in \mathcal{O}(C) Mod$ is cofibrant, then the derived critical locus (def. \ref{DerivedCriticalLocusWithFormalCotangentBundle}) is presented by

$$
  \mathcal{O}(C_{\{d S = 0\}})
   \simeq
  Sym_{\mathcal{O}(C)}
  \left(
     Cone\left(
       Der(\mathcal{O}(C)) \stackrel{[\hat S , -]}{\to} \mathcal{O}(C)
    \right)
  \right)
    \underset{Sym_{\mathcal{O}(C)}(\mathcal{O}(C))  }{\otimes}
  \mathcal{O}(C)
  \,,
$$

where on the right we have the free $\mathcal{O}(C)$-algebra over the [[mapping cone]] of $[\hat S, -]$ with [[extension of scalars]] along $\mathcal{O}( C \overset{(id,0)}{\to} C \times \mathbb{A}^1 )$.

=--

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

Using the [[pasting law]] we may decompose the homotopy pullback into a [[pasting]] of two homotopy pullback squares as follows:

$$
  \array{
    C_{\{d S = 0\}} &\longrightarrow& &\longrightarrow& C
    \\
    \downarrow &\swArrow& \downarrow &\swArrow& \downarrow^{\mathrlap{0}}
    \\
    C
      &\underset{(id,0)}{\longrightarrow}&
    C \times \mathbb{A}^1 &\underset{}{\longrightarrow}& T^*_f C
  }
  \,.
$$

First consider the square on the right:

By prop \ref{SymOCAdjunction} the functor $Sym_{\mathcal{O}(C)}$ is left Quillen. Hence if $Der(\mathcal{O}(C))$ is cofibrant in $\mathcal{O}(C) Mod$ then the homotopy pushout corresponding to the square on the right may be computed as the image under $Sym_{\mathcal{O}(C)}$ of the homotopy pushout in $\mathcal{O}(C) Mod$.

By the disucssion at [[model structure on dg-modules]], for these the homotopy cofibers are given by the ordinary [[mapping cone]] construction for chain complexes.

$$
  \array{
    Cone\left(
      Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to} \mathcal{O}(C)
    \right)
    &\leftarrow&
    Cone\left(
      Der(\mathcal{O}(C)) \stackrel{Id}{\to} Der(\mathcal{O}(C))
    \right)
    \\
    \uparrow && \uparrow
    \\
    \mathcal{O}(C)
    &\stackrel{[\hat S, -]}{\leftarrow}&
    Der(\mathcal{O}(C))
  }
  \,.
$$

More in detail, write

$$
  Cone\left(
    Der(\mathcal{O}(C)) \stackrel{Id}{\to} Der(\mathcal{O}(C))
  \right)
   \in
  \mathcal{O}(C) Mod
$$

for the [[mapping cone]] on the identity:

$$
  \array{
     \cdots
     &
     Der(\mathcal{O}(C))_k &\stackrel{-[d_{\mathcal{O}(C)}, -]}{\to}&
     Der(\mathcal{O}(C))_{k+1}
     \\
     & \oplus &\searrow^{\pm \mathrlap{Id}}& \oplus & \cdots
     \\
     \cdots
     & Der(\mathcal{O}(C))_{k-1}
        &\stackrel{[d_{\mathcal{O}(C)}, -]}{\to}&
     Der(\mathcal{O}(C))_k
     &
     \cdots
  }
  \,.
$$

Then the [[mapping cone]] $Cone\left(Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to}  \mathcal{O}(C) \right)$ is

\[
  \label{TheBVComplex}
  \array{
     \cdots
     &
     Der(\mathcal{O}(C))_k &\stackrel{-[d_{\mathcal{O}(C)}, -]}{\to}&
     Der(\mathcal{O}(C))_{k+1}
     \\
     & \oplus &\searrow^{\pm [\hat S, -]}& \oplus & \cdots
     \\
     \cdots
     & \mathcal{O}(C)_{k-1}
        &\stackrel{[d_{\mathcal{O}(C)}, -]}{\to}&
     \mathcal{O}(C)_k
     &
     \cdots
  }
  \,.
\]

If we extend the graded commutators in the evident way we may write the [[nLab:differential]] in $Cone(Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to} \mathcal{O}(C))$ as

$$
  d
   =
  \left[
    \hat S
      +
    d_{\mathcal{O}(C)}
    \;,\;
    -
  \right]
  \,.
$$

Here the second term will be the differential of the [[nLab:BRST-complex]] of $\mathfrak{c}$, whereas the sum is of the type of a differential in a [[nLab:BRST-BV complex]].

For that to happen, however the two copies of $\mathcal{O}(C)$ in $Sym_{\mathcal{O}(C)}(\mathcal{O}(C))$ need to be identified, this is achived by the remaining homotopy pushout corresponding to the square on the left

$$
  \array{
    \mathcal{O}(C_{\{d S = 0\}})
    &\longleftarrow&
    Sym_{\mathcal{O}(C)}
    \left(
      Cone\left(
        Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to} \mathcal{O}(C)
      \right)
    \right)
    \\
    \uparrow && \uparrow
    \\
    \mathcal{O}(C)
       &\underset{}{\longleftarrow}&
    Sym_{\mathcal{O}(C)}(\mathcal{O}(C))
  }
  \,.
$$

Since here the morphism on the right is the pushout of a cofibration, it is itself still a cofibration, and by assumption $Sym_{\mathcal{O}(C)}(\mathcal{O}(C))$ is cofibrant. Therefore this homotopy pushout is given by the ordinary pushout, and that yields the [[tensor product]] as in the claim.

=--

### BV-BRST complex
 {#ComparisonToBVBRSTExample}

Traditionally the [[BV-BRST complex]] of a [[Lagrangian field theory]] is obtained by

1. choosing a [[Koszul-Tate complex]] $s_{KT}$ resolving the [[shell]];

1. choosing a [[BRST complex]] $s_{BRST}$ exhibiting the [[gauge invariance]]

1. appealing to [[homological perturbation theory]], for extending the sum of the two differentials to a unified [[BV-BRST differential]]

   $$
     s_{BV} = s_{KT} + s_{BRST} + more
   $$

(e.g. [Henneaux 90. around (50)](BV-BRST+formalism#Henneaux90))

Vie the concept of the derived critical locus this process is systematized: Given just $s_{BRST}$ and the Lagrangian, both $s_{KT}$ and "more" follows (if $s_{BRST}$ does capture all the relevant gauge symmetries, that is) and the appearance of the [[antibracket]] finds its conceptual explanation.

Let $\mathfrak{a}$ be a [[Lie algebroid]] over a space $X$, with [[Chevalley-Eilenberg algebra]]

$$
  \mathcal{O}(\mathfrak{a})
  \;=\;
  \left(
    Sym_{\mathcal{O}(X)}(\langle c^a\rangle)
    \;,\;
    d_{\mathfrak{a}}
  \right)
$$

with differential given by

$$
  d_{\mathfrak{a}}
    \;:\;
  f \mapsto c^a R^i_a \frac{\partial}{\partial x^i} f
$$

$$
  d_{\mathfrak{a}} \;:\;
   c^a
    \mapsto
   \frac{1}{2} C^{a}{}_{b c} c^b \wedge c^c
  \,.
$$

for functions $f \in \mathcal{O}(X)$, [[infinitesimal gauge symmetries]] $R^i_a \frac{\partial}{\partial x^i}$, gauge symmetry structure functions $C^{a}{}_{b c}$ and [[ghost]] generators $c^a$.

The "algebra of vector fields/[[derivations]]" $Der(\mathcal{O}(\mathfrak{a}))$ on $\mathfrak{a}$ is the [[automorphism ∞-Lie algebra]]  whose underlying cochain complex is

$$
  \array{
    \left\langle
      \frac{\partial}{\partial c^a}
    \right\rangle
    &
    \overset{[d_{\mathfrak{a}}, -]}{\to}
    &
    \left\langle
      \frac{\partial}{\partial x^i}
    \right\rangle
    \oplus
     \left\langle
       c^a \frac{\partial}{\partial c^b}
     \right\rangle
    \\
    -1 && 0
  }
  \,.
$$

We check on generators that

$$
  \begin{aligned}
    \left[d_{\mathfrak{a}}, \frac{\partial}{\partial c^a} \right]
     =
     R_a^i \frac{\partial}{\partial x^i}
      +
     C^b{}_{a c} c^c \frac{\partial}{\partial c^b}
  \end{aligned}
$$

and

$$
  \begin{aligned}
    \left[
      d_{\mathfrak{a}}, \frac{\partial}{\partial x^i}
    \right]
     =
    c^a \frac{\partial R_a^j}{\partial x^i} \frac{\partial}{\partial x^j}
  \end{aligned}
  \,.
$$

Now let

$$
  S \;\colon\; \mathfrak{a} \longrightarrow \mathbb{R}
$$

be a morphism, dually a dgc-algebra homomorphism of the form

$$
  \mathcal{O}(\mathfrak{a})
    \longleftarrow
   \mathcal{O}(\mathbb{R})
     \;\colon\;
   S^*
  \,.
$$

This is equivalently a function

$$
  S \;\colon\; X \longrightarrow \mathbb{R}
$$

which is [[gauge invariance|gauge invariant]]

$$
  \begin{aligned}
    d_{\mathfrak{a}} S
     & =
    c^a R_a^i \frac{\partial}{\partial x^i} S
    \\
     & = 0
   \end{aligned}
  \,.
$$

We have a contraction homomorphism of $\mathcal{O}(\mathfrak{a})$-[[modules]]

$$
  \iota_{d S}
  \;\colon\;
  Der(\mathcal{O}(\mathfrak{a}))
    \longrightarrow
  \mathcal{O}(\mathfrak{a})
  \,.
$$

and may form its [[mapping cone]] (eq:TheBVComplex)

$$
  \array{
    \left\langle
      \frac{\partial}{\partial c^a}
    \right\rangle
       &\stackrel{[d_{\mathfrak{a}}, -]}{\longrightarrow}&
    \left\langle
       c^a \frac{\partial}{\partial c^b}
    \right\rangle
      \oplus
    \left\langle
       \frac{\partial}{\partial x^i}
    \right\rangle
    \\
      &&
      &\searrow^{\mathrlap{\iota_{d S}}}&
    \\
      &&
      &&
    \mathcal{O}(X)
      &\stackrel{d_{\mathfrak{a}}}{\to}&
    \left\langle
      c^a
    \right\rangle
   \\
   -2 && -1 && 0 && 1
  }
  \,.
$$

On the free algebra of this

$$
  Sym_{\mathcal{O}(X)}
  \left(
    Der(\mathcal{O}(\mathfrak{a}))[-1] \stackrel{\iota_{d S}}{\to} \mathcal{O}(X)\oplus \langle c^a\rangle)
  \right)
$$

we have the differential given on generators by

$$
  \frac{\partial}{\partial c^a}
   \mapsto
  R_a^i \frac{\partial}{\partial x^i} + C^b{}_{a c} c^c
  \frac{\partial}{\partial c^b}
$$

$$
  \frac{\partial}{\partial x^i} \mapsto \frac{\partial S}{\partial x^i}
    +
    c^a \frac{\partial R_a^j}{\partial x^i} \frac{\partial}{\partial x^j}
$$

$$
  x^i \mapsto c^a R_a^i
$$

$$
  c^a \mapsto \frac{1}{2}C^a{}_{b c} c^b \wedge c^c
$$

and similarly after tensoring in order to identify the extra copy of $\mathcal{O}(X)$ with the base $\mathcal{O}(X)$.

If $\langle R_a\rangle$ is the full kernel of $\iota_{d S} : Der(C^\infty(X)) \to C^\infty(X)$  and there are no further relations, then this is the full [[BRST-BV complex]] of $S$.

## References

The above material is adapted from

* [[Urs Schreiber]]: _[[schreiber:derived critical locus]]_ Seminar notes, March 2011

(taking into account a correction provided by [[Vincent Schlegel]])

aimed at providing proof for the claim in

* {#CostelloGwilliam} [[Kevin Costello]], [[Owen Gwilliam]], section 4.8.1 of _[[Factorization algebras in perturbative quantum field theory]]_  ([pdf](https://pdfs.semanticscholar.org/0eb5/652340bb414869764cf5b30feed90597558d.pdf))

See also

* [[Gabriele Vezzosi]], _Derived critical loci I - Basics_, [arxiv/1109.5213](http://arxiv.org/abs/1109.5213)

* [[Tony Pantev]], [[Bertrand Toen]], M. Vaquie, G. Vezzosi, _Quantization and derived moduli spaces I: shifted symplectic structures_, [arxiv/1111.3209](http://arxiv.org/abs/1111.3209)

* [[Kevin Costello]], _Notes on supersymmetric and holomorphic field theories in dimension 2 and 4_ ([pdf](http://www.math.northwestern.edu/~costello/sullivan.pdf))

[[!redirects derived critical loci]]


derived deformation theory

## See also

* [[deformation theory]]
* [[cotangent complex]]

## References

* [[Luc Illusie]], _Complexe cotangent et déformations I_, Lec. Notes Math. __239__, Springer 1971, xv+355 pp.; _II_, LNM __283__, Springer 1972. vii+304 xv+355 pp.

* [[Jacob Lurie]], _Derived algebraic geometry_, thesis, [pdf](http://www.math.harvard.edu/~lurie/papers/DAG.pdf).

* [[Jacob Lurie]], [[Higher Algebra]].

* [[Jacob Lurie]], [[Deformation Theory]].

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

* [[Vladimir Hinich]], _DG coalgebras as formal stacks_, J. Pure Appl. Algebra 162 (2001), 209-250. [arXiv](https://arxiv.org/abs/math/9812034).

* [[Marco Manetti]], _Extended deformation functors_, Int. Math. Res. Not., (14):719–
756, 2002. [arXiv](https://arxiv.org/abs/math/9910071).

* J. P. Pridham, _Unifying derived deformation theories_, Adv. Math., 224(3):772–
826, 2010. [arXiv](https://arxiv.org/abs/math/0705.0344).

* Ai Guan, Andrey Lazarev, Yunhe Sheng, and Rong Tang. _Review of deformation theory II: a homotopical approach_. [arXiv](http://arxiv.org/abs/1912.04028).

* [[Mauro Porta]], [[Gabriele Vezzosi]], _Infinitesimal and square-zero extensions of simplicial algebras_, [arXiv](http://arxiv.org/abs/1310.3573).

* [[Timo Schuerg]], _Deformation theory of commutative algebras_, lecture notes by Gregor Bruns, Humboldt Universitaet zu Berlin, 2013, [[schuergdeformation.pdf:file]].

* [[Dennis Gaitsgory]], _Lie theory from the point of view of derived algebraic geometry_, Lecture notes from a mini-course, Nantes, 2014, [web](http://www.math.harvard.edu/~gaitsgde/Nantes14/).

* [[Mauro Porta]], _Derived formal moduli problems_, master thesis, [pdf](http://algant.eu/documents/theses/porta.pdf).


derived Deligne-Mumford stack

#Contents#
* table of contents
{:toc}

## Idea

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

...[[E-∞ scheme]], [[locally representable structured (∞,1)-topos]]

## Definition

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

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

## Special cases

A [[n-localic (infinity,1)-topos|1-localic]] derived Deligne-Mumford stack is an ordinary [[Deligne-Mumford stack]]. See there for more details.

## Properties

### Relation to derived algebraic spaces
 {#RelationToDerivedAlgebraicSpaces}

([[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProp, theorem 1.3.8]]):

spectral Deligne-Mumford stack is quasi-compact, quasi-separated [[E-∞ algebraic space]] precisely if it admits a [[scallop decomposition]].

### Characterization as $(\infty,1)$-presheaves on $E_\infty$-rings

The [[(∞,1)-presheaves]] on [[E-∞ rings]] which are [[representable functor|represented]] by spectral Deligne-Mumford stacks are described by the [[Artin-Lurie representability theorem]].

## Related concepts

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

## References

* [[Jacob Lurie]], section 4.3 _[[Structured Spaces]]_

In the context of [[E-infinity geometry]] (spectral Deligne-Mumford stacks):

* [[Jacob Lurie]], section 1 of _[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]_

* [[Jacob Lurie]], section 1 of _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_

[[!redirects spectral Deligne-Mumford stack]]
[[!redirects spectral Deligne-Mumford stacks]]


derived dg-category of a dg-category

## Idea

The _derived dg-category_ of a [[dg-category]] is the [[dg-localization]] of the [[dg-category]] of [[dg-modules]] at the class of [[quasi-isomorphisms]]. The result is a [[pretriangulated dg-category]]. However an important caveat is that, even if the original [[dg-category]] is [[pretriangulated dg-category|pretriangulated]], it is strictly contained inside its derived dg-category.

Despite the terminology, the derived dg-category is not quite analogous to the [[derived category]] of an [[abelian category]]. Rather it should be thought of as the analogue of the [[simplicially enriched category]] of [[simplicial presheaves]], hence a presentation of the [[(infinity,1)-category of (infinity,1)-presheaves]].

## Definition

Let $T$ be a [[dg-category]].

+-- {: .num_defn}
###### Definition
The **derived dg-category** of $T$, denoted $D(T)$, is the [[dg-localization]] of the category of [[dg-modules]] over $T$ at the class of objectwise [[quasi-isomorphisms]]:
  $$ D(T) = dg-mod_T[qis^{-1}]. $$
=--

$D(T)$ can also be described as the [[dg-category]] [[dg-category presented by a dg-model category|presented]] by the projective [[dg-model structure on dg-modules]].

## Properties

+-- {: .num_lemma}
###### Lemma
The [[dg-category]] $D(T)$ is [[pretriangulated dg-category|pretriangulated]].
=--

+-- {: .num_lemma}
###### Lemma
There exists a fully faithful functor of [[dg-categories]]
  $$ T \hookrightarrow D(T) $$
called the [[dg-Yoneda embedding]].
=--

## Related concepts

* [[dg-module]]
* [[dg-Yoneda embedding]]
## References

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

[[!redirects derived category of a dg-category]]

[[!redirects derived dg-category]]
[[!redirects derived dg-categories]]


derived differential geometry

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

#Contents#
* table of contents
{:toc}

## Idea

_Derived differential geometry_ is [[higher differential geometry]] in an ambient [[(∞,1)-topos]] which is not [[n-localic (∞,1)-topos|1-localic]]. This is the [[derived geometry]] corresponding to [[differential geometry]]. Typically this is specifically taken to be the derived geometry induced by the [[Lawvere theory]] for [[smooth algebras]] ($C^\infty$-rings):

See

* [[smooth (∞,1)-algebra]], [[simplicial C-∞-ring]]

* [[derived smooth manifold]]

* [[derived critical locus]]

## Related concepts

* [[synthetic differential geometry]]

* [[D-geometry]]

## References

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

* [[Dominic Joyce]], _D-manifolds and d-orbifolds: a theory of derived differential geometry_ ([arXiv:1208.4948](https://arxiv.org/abs/1208.4948), [web](http://people.maths.ox.ac.uk/~joyce/dmanifolds.html), [pdf](http://people.maths.ox.ac.uk/joyce/dmbook.pdf))

* [[Dominic Joyce]], _Aarhus Masterclass on Derived Differential Geometry_ ([videos](http://qgm.au.dk/video/mc/derived-differential-geometry/))

Discussion via ([[differential cohesion|differential]]) [[cohesive (infinity,1)-topos|cohesive $(\infty,1)$-topos]] theory:

* [[Luigi Alfonsi]], [[Charles A. S. Young]], *Towards non-perturbative BV-theory via derived differential cohesive geometry* [[arXiv:2307.15106](https://arxiv.org/abs/2307.15106)]

exposition:

* [[Luigi Alfonsi]]: *Derived $n$-plectic geometry: towards non-perturbative BV-BFV quantisation and M-theory*, [talk at](M-Theory+and+Mathematics#Alfonsi2023) *[[M-Theory and Mathematics]]* (Jan 2023) [slides: [[Alfonsi-MTheoryAndMath2023.pdf:file]], video: [YT](https://www.youtube.com/watch?v=O7pQkyqCiA4)]


derived direct image

#Contents#
* table of contents
{:toc}

## Idea

The [[derived functor]] of a [[direct image]] functor.

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

## Related concepts

* [[abelian sheaf cohomology]]

* [[derived inverse image]]

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

* [[six operations]]

[[!redirects derived direct images]]

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


derived elliptic curve

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

# Derived elliptic curve#
* table of contents
{:toc}

## Idea ##

A _derived elliptic curve_ is an object in [[higher geometry]] that is to an [[E-∞ ring]] as an ordinary [[elliptic curve]] is to an ordinary [[commutative ring]].

The ordinary [[moduli stack of elliptic curves]] when equipped with the [[structure sheaf]] of [[E-∞ rings]] which assigns to each elliptic curve its [[elliptic spectrum]] becomes a [[spectral Deligne-Mumford stack]] which as such is what modulates derived elliptic curves (see [below](#ModuliStack)).

## Definition ##

Let $A$ be an [[E-∞ ring]]. A **derived elliptic curve** over $A$ is [[derived group scheme]] $E \to Spec A$ with the property that the underlying morphism $\bar E \to Spec \pi_0 A$ is an ordinary [[elliptic curve]].

## Properties

### Moduli stack of derived elliptic curves
 {#ModuliStack}

By the [[Goerss-Hopkins-Miller theorem]] the [[structure sheaf]] $\mathcal{O}$ of the [[moduli stack of elliptic curves]] (ordinary [[elliptic curves]]) lifts to a sheaf $\mathcal{O}^{top}$ of [[E-∞ rings]] which over a given [[elliptic curve]] is the corresponding [[elliptic spectrum]].

By ([[A Survey of Elliptic Cohomology|Lurie (Survey), theorem 4.1]]), this yields a  [[spectral Deligne-Mumford stack]] refinement $(\mathcal{M}_{ell}, \mathcal{O}^{top})$ which is the moduli stack of derived elliptic curves, in that there is a [[natural equivalence]] in [[E-∞ rings]] $A$ of the form

$$
  Hom(Spec(A), (\mathcal{M}_{ell},\mathcal{O}^{top}))
  \simeq
  E(A)
  \,,
$$

where on the left we have maps of [[structured (∞,1)-toposes]] and on the right the [[∞-groupoid]] of derived elliptic curves over $A$.

This is based on the representability theorem ([[A Survey of Elliptic Cohomology|Lurie (Survey), prop. 4.1]], [[Representability Theorems|Lurie (Representability)]]). An ingredient in the proof is the [essential uniqueness of lifts of etale morphisms](étale+morphism+of+E-∞+rings#EssentiallyUniqueLiftsFromOrdinaryRings) from commutative rings to $E_\infty$-rings.

## References

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

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

[[!redirects derived elliptic curves]]


derived functor

> This entry discusses the concept of derived functors in full generality. For the dedicated discussion of the traditional case see at [[derived functors in homological algebra]].

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _derived functor_ is a [[functor]] in [[homotopy theory]] induced from, "derived from" or _[[presentable (infinity,1)-category|presented by]]_ an ordinary [[functor]] on a [[category with weak equivalences]].

Historically the concept first arose in the special context of [[homological algebra]] on [[categories of chain complexes]] and is often still understood by default in this special sense. The relation to the general case is discussed below in the section _[In homological algebra](#InHomologicalAlgebra)_. For a dedicated discussion of this case see the entry _[[derived functor in homological algebra]]_.

### General idea

A [[category with weak equivalences]] $C$ serves as a presentation for an [[(∞,1)-category]] $\mathbf{C}$ by [[simplicial localization]]. Accordingly, a [[functor]] $F : C \to D$ should induce an [[(∞,1)-functor]] between the corresponding [[(∞,1)-categories]] $\mathbf{C} \to \mathbf{D}$. From the [[nPOV]], this is a **derived functor**.

If $F$ is a [[homotopical functor]] in that it respects the weak equivalences in $C$ and $D$, then by the [[universal property]] of [[simplicial localization]] it extends to a functor of [[(∞,1)-category|(∞,1)-categories]] and this is the corresponding _derived functor_ .

However, typically functors of interest do not respect weak equivalences and hence do not uniquely or even naturally give rise to an [[(∞,1)-functor]]. In general, they contain too little information to accomplish this. Notably, to objects $x, y \in C$ that are [[homotopy equivalence|equivalent]] in $\mathbf{C}$ but not [[isomorphism|isomorphic]] in $C$, the functor will in general not assign objects $F(x)$ and $F(y)$ that are equivalent in $\mathbf{D}$, as an [[(∞,1)-functor]] would. So it matters on which representatives of a $\mathbf{C}$-[[equivalence class]] of objects the functor $F$ is applied.

Remembering that by Dwyer-Kan [[simplicial localization]] the morphisms in $\mathbf{C}$ and $\mathbf{D}$ are [[zig-zags]] of morphisms in $C$ and $D$, a very general notion of derived functor therefore takes a derived functor of $F$ to be a functor $\mathbb{D}F : \mathbf{C} \to \mathbf{D}$ induced from the [[universal property]] of the localization by a functor of the form $F \circ Q : C \to D$, where $Q : C \to C$ is an [[endofunctor]] which is naturally connected to the identity by a [[zig-zag]] of [[weak equivalences]]:

$$
  X \stackrel{\simeq}{\leftarrow}
  X_1
   \stackrel{\simeq}{\to}
  X_2
  \cdots
  \stackrel{\simeq}{\leftarrow}
  Q X
  \,.
$$

Here if this zig-zag consists just of one morphism to the left one would speak of a **left derived functor**. If it consists of just one morphism to the right, one would speak of a **right derived functor**. In general, it is just a derived functor.

### On model categories

In highly structured situations where $C$ and $D$ are equipped not just with weak equivalences but with the full structure of a [[model category]] and if $F$ is a left or right [[Quillen adjunction|Quillen functor]] with respect to these model structures, there are accordingly more structured ways to solve this problem:

The **left derived functor** $\mathbb{L}F : \mathbf{C} \to \mathbf{D}$ of a [[left Quillen functor]] $F : C \to D$ is obtained by applying $F$ to _[[cofibrant objects]]_ of $C$. Similarly a **right derived functor** $\mathbb{R}G : \mathbf{D} \to \mathbf{C}$ of a [[right Quillen functor]] $G : D \to C$ is obtained by applying $G$ to _[[fibrant objects]]_.

Recalling that the [[(∞,1)-category]] [[presentable (∞,1)-category|presented]] by a [[simplicial model category]] $C$ may be identified with the full [[sSet]]-[[subcategory]] $C^\circ$ of fibrant-cofibrant objects, this may be understood as ensuring that the derived functor indeed respects the $(\infty,1)$-categorical structure. More precisely, for

$$
  (F \dashv G) : C \stackrel{\leftarrow}{\to} D
$$

an [[sSet]]-[[enriched category theory|enriched]] [[Quillen adjunction]] between [[simplicial model categories]], combining $F$ and $G$ with cofibrant-fibrant replacement induces a pair of [[adjoint (∞,1)-functors]]

$$
  \mathbb{L}F
    \colon
    \mathbf{C} \stackrel{\longleftarrow}{\longrightarrow} \mathbf{D}
  \colon
  \mathbb{R}G
$$

between [[quasi-category|quasi-categories]] $\mathbf{C} = N(C^\circ)$, $D = N(D^\circ)$, where $N$ is the [[homotopy coherent nerve]] functor.

Often a simplified version of this situation is considered, where instead of the [[(∞,1)-categories]] $\mathbf{C}$ and $\mathbf{D}$ only [[homotopy category of an (∞,1)-category|their homotopy categories]] are remembered, equivalently the [[homotopy category of a model category|homotopy categories of the model categories]]  $C$ and $D$. The above [[adjoint (∞,1)-functors]] restrict to functors

$$
  L F \colon
    Ho(C) \stackrel{\longleftarrow}{\longrightarrow} Ho(D)
  \colon
  R G
$$

on [[homotopy category of a model category|homotopy categories]], and often it is these functors that are called ([[total derived functor|total]]) **derived functors** in the literature. For more on this see at _[[homotopy category of a model category]]_ the section _[derived functors](homotopy+category+of+a+model+category#DerivedFunctors)_.

More generally, derived functors in this sense may be considered in situations where less than the above extra structure is available (no [[model category]] structure or not [[Quillen adjunction]]).

### On homotopy categories

If one forgets the [[nPOV]] and that a [[category with weak equivalences]] should be regarded as presentation for an [[(∞,1)-category]], then it might seem as if all one wants when deriving a [[homotopical functor]] $f \colon C \longrightarrow D$ is to [[extension|extend]] it to a diagram

$$
  \array{
    C &\stackrel{F}{\to}& D
    \\
    \downarrow^{\mathrlap{Q_C}} &(?)& \downarrow^{\mathrlap{Q_D}}
    \\
    Ho(C) &\to& Ho(D)
  }
  \,,
$$

where $Q_C \colon C \longrightarrow Ho(C)$ is the universal morphism characterizing the [[homotopy category]] and similarly for $Q_D$.

There is a general method of ordinary [[category theory]] to solve such problems universally: one may take $Ho(C) \to Ho(D)$ to be either the left or right [[Kan extension]] of $Q_d \circ F$ along $Q_C$.

In the literature this is often taken as the _definition_ of [[total derived functor|total]] left or right derived functors. Unfortunately, it is not clear how this definition by Kan extension relates to what should be the right [[(∞,1)-category theory|(∞,1)-category theoretic]] situation above. Moreover, the examples of derived functors that play a role in practice are effectively always constructed instead rather by combining $F$ with cofibrant/fibrant or similar replacement functors. It is then but a happy byproduct that the functors so obtained also happen to be left or right Kan extensions.

## Definition

We first give the [[decategorification|decategorified]] definition of total derived functors on [[homotopy category|homotopy categories]] in

* [As functors on homotopy categories](#OnHomotopyCat)

and then the [[(∞,1)-category]]-version in

* [As functors on (∞,1)-categories](#OnInftyCats).

The special case of derived functors in the context of [[homological algebra]] is discussed from this general perspective in

* [In homological algebra](#InHomologicalAlgebra).

A dedicated discussion of this case is at [[derived functors in homological algebra]].

### As functors on homotopy categories
 {#OnHomotopyCat}

For $Core(C) \hookrightarrow W \hookrightarrow C$ a [[category with weak equivalences]], then for $F : C \to D$ any functor, the __left derived functor__ $L F$ of $F$ is the _right_ [[Kan extension]] of $F$ along the projection $p : C \to Ho_C$ to the [[homotopy category]]

$$
 \array{
   C &&\stackrel{F}{\to}&& D
   \\
   \downarrow^p &\Uparrow& \nearrow_{L F}
   \\
   Ho_C
 }
$$

(if it exists).  Dually, the __right derived functor__ $R F$ of $F$ is its _left_ Kan extension along $p$.  Note the reversal of handedness; this is unfortunate but unavoidable.

More generally, if $D$ is itself a category with weak equivalences, then by derived functors of $F$ we often mean derived functors of the composite

$$ C \stackrel{F}{\to} D \to Ho_D $$

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

By the universal property of $Ho_C$, functors $Ho_C \to D$ are equivalent to functors $C\to D$ which take weak equivalences to isomorphisms.  If $F$ itself takes weak equivalences to isomorphisms, then its left and right derived functors are both (isomorphic to) its unique extension along $p$.  In general, however, $L F$ and $R F$ are not extensions of $F$ even up to isomorphism.

=--

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

In practice, derived functors are usually computed using fibrant and cofibrant [[resolution]] replacements (see the entries on [[homotopy theory]] and [[model category]]) or, more generally, [[deformation retract of a homotopical category|deformation retracts]].

=--

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

If the codomain admits sufficiently many [[limits]] and [[colimits]], a [[Kan extension]] can be computed in terms of those, and that such Kan extensions are called *[pointwise](Kan%20extension#Pointwise)*.  Homotopy categories generally do not admit even small limits and colimits, and moreover the domains of the functors in question are generally large, so such a *construction* of a derived functor is not possible.

  However, when derived functors are constructed using fibrant and cofibrant replacements, as above, it turns out *a posteriori* that they are actually pointwise: they are preserved by all representable functors, and hence their individual object values have the universal property of the (generally large) limits that would have been used to compute them, even though not all limits exist in the homotopy category.  In fact, derived functors constructed in this way are actually [[absolute colimit|absolute]] Kan extensions: preserved by any functor whatsoever.

=--

### As functors on $(\infty,1)$-categories
  {#OnInftyCats}

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

Let $C$ and $D$ by [[simplicial model category|simplicial model categories]] and let

$$
  (F \dashv G) : C \stackrel{\overset{G}{\leftarrow}}{\underset{F}{\to}} D
$$

be an [[sSet]]-[[enriched functor|enriched]] [[Quillen adjunction]]. Then there is an [[adjoint (∞,1)-functor|(∞,1)-adjunction]]

$$
  (\mathbb{L}F \dashv \mathbb{R}G) :
   N(C^\circ)
    \stackrel{\overset{\mathbb{R} G}{\leftarrow}}{\underset{\mathbb{L}F}{\to}}
    N(D^\circ)
$$

between [[quasi-categories]] $N(C^\circ)$ and $N(D^\circ)$ otained as the [[homotopy coherent nerve]]s of the full [[sSet]]-[[subcategories]] of fibrant-cofibrant objects. Their image on the [[homotopy category|homotopy categories]] produces the notion of total derived functor between homotopy categories discussed above

$$
  (L F \dashv R G) :
  Ho(C)
   \stackrel{\overset{R G}{\leftarrow}}{\underset{L F}{\to}}
  Ho(D)
  \,.
$$

=--

This is prop. 5.2.4.6 and remark 5.2.4.7 in ([Lurie](#Lurie)). For more along these lines see also at _[Quillen adjunction - Associated infinity-adjunction](Quillen+adjunction#AssociatedInfinityAdjunction)_.

### In homological algebra
 {#InHomologicalAlgebra}

Often and traditionally, the concept of derived functors is considered in [[homological algebra]] exclusively in the context of [[categories of chain complexes]] $Ch_\bullet(\mathcal{A})$ in an [[abelian category]] $\mathcal{A}$. The definitions in this case are disucssed in detail at

* **[[derived functor in homological algebra]]**

Here put that special case a bit more into the general perspective.

By taking [[quasi-isomorphism]]s as weak equivalences, $Ch_\bullet(\mathcal{A})$ is naturally a [[category with weak equivalences]]. In much of the literature on homological algebra, the refinement of this structure to a projective or injective [[model structure on chain complexes]] is implicit.  For instance, an injective [[resolution]] of chain complexes is nothing but a fibrant replacement in the injective model structure.  Dually, a projective resolution is a cofibrant replacement in the projective model structure.  (Note, though, that hypotheses on $\mathcal{A}$ are required in order for these model structures to exist.)

Now, any ordinary [[additive functor]] $F\colon \mathcal{A} \to \mathcal{B}$ between [[abelian categories]] induces a functor $Ch_\bullet(F)\colon Ch_\bullet(\mathcal{A}) \to Ch_\bullet(\mathcal{B})$ between [[category of chain complexes|categories of chain complexes]].  We can therefore ask about derived functors of $Ch_\bullet(F)$.

Note first that $Ch_\bullet(F)$ automatically preserves [[chain homotopies]], and therefore also preserves [[chain homotopy equivalence]]s.  Since the projective (resp. injective) model structure on chain complexes has the property that weak equivalences (that is, quasi-isomorphisms) between cofibrant (resp. fibrant) objects are chain homotopy equivalences, it follows that $Ch_\bullet(F)$ automatically preserves weak equivalences between projective-cofibrant objects, and also between injective-fibrant objects.  Thus, it has a left derived functor if the projective model structure on $Ch_\bullet(\mathcal{A})$ exists, and a right derived functor if the injective model structure exists.

In the homological algebra literature, what is called the $p$th right derived functor

$$
  R^p F \colon \mathcal{A} \to \mathcal{B}
$$

is the composite

$$
  \mathcal{A} \stackrel{\mathbf{B}^p (-)}{\hookrightarrow} Ch_\bullet(\mathcal{A})
  \stackrel{\mathbb{R} Ch_\bullet(F)}{\to}
  Ch_\bullet(\mathcal{B})
  \stackrel{H^0(-)}{\to}
  \mathcal{B}
  \,,
$$

1. The first map sends an object $A \in \mathcal{A}$ to the corresponding [[Eilenberg-MacLane object]] $\mathbf{B}^p A$: the cochain complex $A[p]$ concentrated on $A$ in degree $p$.

1. The second map is the actual right derived functor $\mathbb{R}Ch_\bullet(F)$ of $Ch_\bullet(F)$ in the sense used previously on this page.  Thus, this is itself the composite

   $$
     \mathbb{R}F : Ch_\bullet(\mathcal{A}) \stackrel{P}{\to}
      Ch_\bullet(\mathcal{A})
      \stackrel{Ch_\bullet(F)}{\to}
     Ch_\bullet(\mathcal{B})
     \,,
   $$

   where $P$ denotes a fibrant resolution functor in the injective [[model structure on chain complexes]].  Applied to an Eilenberg-MacLane object, this amounts to the usual [[injective resolutions]] seen in the homological algebra literature.

1. The last morphism computes the [[cochain cohomology]] of the resulting cochain complex in degree 0.

Of course, it is equivalent to instead regard $A$ as concentrated in degree $0$, and then take the $p$th homology group at the last step.  Left derived functors are dual, using the projective model structure.

The first and the last steps are traditionally included, but are not really necessary:

1. Instead of applying the first step and restricting attention to arguments that are chain complexes concentrated in a single degree, one can evaluate $\mathbb{R} Ch_\bullet(F)$ on all chain complexes (and then, if desired, take homology groups). In homological algebra one then speaks of [[hyper-derived functors]].

1. The last step of taking cohomology groups serves to extract invariant and computable information. It also destroys the simple composition law of functors, though. But there is a computational tool that can be used to recover the derived functor -- in this homological sense -- of the composite of two functors from their individual derivations: this is the [[spectral sequence]] called the  Grothendieck spectral sequence.

#### Long exact sequences

Traditionally, in homological algebra, one only takes left derived functors of right exact functors, and right derived functors of left exact ones.  As we saw above, both left and right derived functors can be *defined* without these hypotheses, but it is only in the presence of these hypotheses that we obtain [[long exact sequences]].

Specifically, suppose we have a [[short exact sequence]]
$$ 0 \to A \to B \to C \to 0$$
in $\mathcal{A}$.  Assuming $\mathcal{A}$ has [[enough projectives]], we may then find [[projective resolutions]] $Q A$, $Q B$, and $Q C$ of $A$, $B$, and $C$, respectively, such that
$$ 0 \to Q A \to Q B \to Q C \to 0 $$
is a short exact sequence of chain complexes.  But since $Q C$ is projective, this short exact sequence is split, and therefore preserved by any additive functor.  Thus we have another short exact sequence
$$ 0 \to F Q A \to F Q B \to F Q C \to 0 $$
which therefore gives rise to a long exact sequence in homology:
$$ \cdots \to H_1(F Q A) \to H_1(F Q B) \to H_1(F Q C) \to H_0 (F Q A) \to H_0(F Q B) \to H_0(F Q C). $$
Of course, these homology groups are precisely the left derived functors of $F$, in the traditional homological algebra sense, applied to $A$, $B$, and $C$.

All of this works without hypothesis on $F$.  However, if $F$ is [[right exact functor|right exact]], then it preserves the exactness of the sequence
$$ Q A_1 \to Q A_0 \to A \to 0 $$
(and the analogous ones for $B$ and $C$).  This implies that $F A \cong H_0 (F Q A)$ and so on, so that the above long exact sequence actually finishes
$$ \cdots \to H_1(F Q A) \to H_1(F Q B) \to H_1(F Q C) \to F A \to F B \to F C \to 0. $$
This is how derived functors are traditionally introduced in homological algebra: as a way to continue the right half of a short exact sequence preserved by a right exact functor into a long exact sequence.  The case of left exact functors and right derived functors is dual.

## Examples

* The (total) derived functor of the [[limit]] functor is the [[homotopy limit]].  The functors $lim^{(i)}$ often called the derived functors of Lim are then given by the (co)homology of that 'total' form.

* More generally, the total derived functor of [[Kan extension]] is [[homotopy Kan extension]].

* In the context of a [[model structure on chain complexes]] of [[module]]s the left and right derived functors of the [[tensor product]] functor and the [[hom-functor]] are called [[Tor]]-functor and [[Ext-functor]], respectively.

* A [[derived direct image]] functor computes [[abelian sheaf cohomology]]. See also _[[derived inverse image]]_.

## Functoriality

Passage to left derived functors is a [[pseudofunctor]] from a [[2-category]] of model categories, left Quillen functors, and natural transformations to [[Cat]], and similarly for right derived functors.  These can be combined into a [[double pseudofunctor]] from the [[double category]] [[double category of model categories|of model categories]] to the double category of [[quintets]] in [[Cat]], which implies that some [[mates]] are also preserved by deriving, even when they relate composites of left and right Quillen functors; see [(Shulman)](#Shulman).

## Related concepts

* [[hyper-derived functor]], [[total derived functor]]

* [[delta-functor]]

* [[adjoint (∞,1)-functor]]

* [[derived adjunction unit]]

* [[homotopy limit]], [[homotopy colimit]], [[homotopy Kan extension]]

* [[derived category]]

* [[satellite]]

## References

### General

The notion of derived functors is due to:

* {#CartanEilenberg56} [[Henri Cartan]], [[Samuel Eilenberg]], *[[Homological Algebra]]*, Princeton Univ. Press (1956), Princeton Mathematical Series **19** (1999) [[ISBN:9780691049915](https://press.princeton.edu/books/paperback/9780691049915/homological-algebra-pms-19-volume-19), [doi:10.1515/9781400883844](https://doi.org/10.1515/9781400883844), [pdf](http://www.math.stonybrook.edu/~mmovshev/BOOKS/homologicalalgeb033541mbp.pdf)]

General discussion of derived functors in [[homotopy theory]]:

* {#Adams74} [[Frank Adams]], part III, section 8 of _[[Stable homotopy and generalised homology]]_, 1974

* [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]],  [[Jeff Smith]], _[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]_, volume 113 of Mathematical Surveys and Monographs

* [[Bruno Kahn]], [[Georges Maltsiniotis]], _[[Structures de Dérivabilité]]_

and with an eye towards [[abelian sheaf cohomology]]:

* [[Ugo Bruzzo]], _Derived functors and sheaf cohomology_, Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes: Volume 2 ([doi:10.1142/11473](https://doi.org/10.1142/11473))

Discussion in the context of [[(∞,1)-categories]] is in section 5.2.4 of

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

The double-categorical functoriality is in

* {#Shulman} [[Mike Shulman]], _Comparing composites of left and right derived functors_, [NYJM](http://nyjm.albany.edu/j/2011/17-5.html)

Review:

* [[Emily Riehl]], Chapter 2 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

A more modern definition with better theoretical properties and with emphasis on the role of correspondences in $\infty$-categorical setup is given in

* [[Vladimir Hinich]], _So, what is a derived functor?_,
[arXiv:1811:12255](https://arxiv.org/abs/1811.12255)

### In homological algebra

A standard textbook introduction to [[derived functors in homological algebra]] is in

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_.

A systematic discussion of this case from the point of view of [[localization]] and [[homotopy theory]] is in section 13 of

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_

and, similarly, in section 7 of

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

[[!redirects derived functors]]
[[!redirects left derived functor]]
[[!redirects right derived functor]]
[[!redirects left derived functors]]
[[!redirects right derived functors]]

[[!redirects left derived]]
[[!redirects right derived]]


derived functor in homological algebra

***

This entry discusses the general notion of _[[derived functor]]_ specified to the special context of [[homological algebra]], hence to functors between [[categories of chain complexes]].

In the literature this is often understood to be the default case of derived functors. For more discussion of how the following relates to the more general concepts of derived functors see at _[derived functor -- In homological algebra](derived+functor#InHomologicalAlgebra)_.

***

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

#Contents#
* table of contents
{:toc}

## Idea

The general concept of [[derived functor]] is in homological algebra usually called the _[[total derived functor|total]] [[hyper-derived functor]]_, with just "derived functor" being reserved for a more restrictive case. In this tradition, we consider the special case first and then generalize it in stages. The relation between all these notions is discussed [below](#Properties).

## Definition

Let $\mathcal{A}$ be an [[abelian category]]. Without essential loss of generality, we may assume that $\mathcal{A} = R Mod$ is the [[category of modules]] over some [[ring]], and we will often speak in terms of this case.

The category $\mathcal{A}$ embeds into its [[nLab:derived category]], the category of degreewise [[nLab:injective object|injective]] [[nLab:cochain complexes]]

$$
  P : \mathcal{A} \to \mathcal{D}^\bullet(\mathcal{A}) = \mathcal{K}^\bullet(\mathcal{I}_{\mathcal{A}})
$$

or degreewise [[nLab:projective object|projective]] [[nLab:chain complexes]]

$$
 Q : \mathcal{A} \to \mathcal{D}_\bullet(\mathcal{A}) =  \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}})
$$

modulo [[nLab:chain homotopy]]. This construction of the derived category naturally gives rise to the following notion of [[nLab:derived functors]], which we now discuss.

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

For $\mathcal{A}, \mathcal{B}$ two [[nLab:abelian categories]] (e.g. $R$[[nLab:Mod]] and $R'$[[nLab:Mod]]), a [[nLab:functor]]

$$
  F \colon \mathcal{A} \to \mathcal{B}
$$

is called an **[[nLab:additive functor]]** if

1. $F$ maps [[nLab:generalized the|the]] [[nLab:zero object]] to the zero object, $F(0) \simeq 0 \in \mathcal{B}$;

1. given any two [[nLab:objects]] $x, y \in \mathcal{A}$, there is an [[nLab:isomorphism]] $F(x \oplus y) \cong F(x) \oplus F(y)$, and this respects the inclusion and projection maps of the [[nLab:direct sum]]:

$$
\array { x &          &            &          & y \\
           & {}_{\mathllap{i_x}}\searrow &            & \swarrow_{\mathrlap{i_y}} \\
           &          & x \oplus y \\
           & {}^{\mathllap{p_x}}\swarrow &            & \searrow^{\mathrlap{p_y}} \\
         x &          &            &          & y }
\quad\quad\stackrel{F}{\mapsto}\quad\quad
\array { F(x) &          &                                      &          & F(y) \\
              & {}_{\mathllap{i_{F(x)}}}\searrow &   & \swarrow_{\mathrlap{i_{F(y)}}} \\
              &          & F(x \oplus y) \cong F(x) \oplus F(y) \\
              & {}^{\mathllap{p_{F(x)}}}\swarrow &                                      & \searrow^{\mathrlap{p_{F(y)}}} \\
         F(x) &          &                                      &          & F(y) }
$$

=--

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

Given an [[nLab:additive functor]] $F : \mathcal{A} \to \mathcal{A}'$, it canonically induces a functor

$$
  Ch_\bullet(F) \colon Ch_\bullet(\mathcal{A}) \to Ch_\bullet(\mathcal{A}')
$$

between [[nLab:categories of chain complexes]] (its "prolongation") by applying it to each [[nLab:chain complex]] and to all the diagrams in the definition of a [[nLab:chain map]]. Similarly it preserves [[nLab:chain homotopies]] and hence it passes to the quotient given by the strong [[nLab:homotopy category of chain complexes]]

$$
  \mathcal{K}(F) \colon \mathcal{K}(\mathcal{A}) \to \mathcal{K}(\mathcal{A}')
  \,.
$$

=--

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

If $\mathcal{A}$ and $\mathcal{A}'$ have [[nLab:projective object|enough projectives]], then their [[nLab:derived categories]] are

$$
  \mathcal{D}_\bullet(\mathcal{A})
   \simeq
  \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}})
$$

and

$$
  \mathcal{D}^\bullet(\mathcal{A}) \simeq \mathcal{K}^\bullet(\mathcal{I}_{\mathcal{A}})
$$

etc. One wants to accordingly _derive_ from $F$ a functor $\mathcal{D}_\bullet(\mathcal{A}) \to \mathcal{D}_\bullet(\mathcal{A}')$ between these derived categories. It is immediate to achieve this on the domain category, there we can simply precompose and form

$$
  \mathcal{A}
    \to
  \mathcal{D}_\bullet(\mathcal{A})
   \simeq
  \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}})
   \hookrightarrow
  \mathcal{K}_\bullet(\mathcal{A})
   \stackrel{\mathcal{K}_\bullet(F)}{\to}
  \mathcal{K}_\bullet(\mathcal{A}')
  \,.
$$

But the resulting composite lands in $\mathcal{K}_\bullet(\mathcal{A}')$ and in general does not factor through the inclusion $\mathcal{D}_\bullet(\mathcal{A}') = \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}'}) \hookrightarrow \mathcal{K}_\bullet(\mathcal{A}')$.

In a more general abstract discussion than we present here, one finds that by applying a projective resolution functor _on chain complexes_, one can enforce this factorization. However, by definition of [[nLab:resolution]], the resulting chain complex is [[nLab:quasi-isomorphism|quasi-isomorphic]] to the one obtained by the above composite.

This means that if one is only interested in the "weak chain homology type" of the chain complex in the image of a [[nLab:derived functor]], then forming [[nLab:chain homology]] groups of the chain complexes in the images of the above composite gives the desired information. This is what def. \ref{RightDerivedFunctorOfLeftExactFunctor} and def. \ref{LeftDerivedFunctorOfRightExactFunctor} below do.

=--

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

Let $\mathcal{A}, \mathcal{A}'$ be two [[nLab:abelian categories]], for instance $\mathcal{A} = R $[[nLab:Mod]] and $\mathcal{A}' = R'$[[nLab:Mod]]. Then a [[nLab:functor]] $F \colon \mathcal{A} \to \mathcal{A}'$
which preserves [[nLab:direct sums]] (and hence in particular the [[nLab:zero object]]) is called

* a **[[nLab:left exact functor]]** if it preserves [[nLab:kernels]];

* a **[[nLab:right exact functor]]** if it preserves [[nLab:cokernels]];

* an **[[nLab:exact functor]]** if it is both left and right exact.

=--

Here to "preserve kernels" means that for every morphism $X \stackrel{f}{\to} Y$ in $\mathcal{A}$ we have an [[nLab:isomorphism]] on the left of the following [[nLab:commuting diagram]]

$$
  \array{
    F(ker(f)) &\to& F(X) & \stackrel{F(f)}{\to} & F(Y)
    \\
    \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{=}}
    \\
    ker(F(f)) &\to& F(X) &\stackrel{F(f)}{\to}& F(Y)
  }
  \,,
$$

hence that both rows are [[nLab:exact sequence|exact]]. And dually for right exact functors.

We record the following immediate consequence of this definition (which in the literature is often taken to be the definition).

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

If $F$ is a left exact functor, then for every [[nLab:exact sequence]]  of the form

$$
  0 \to A \to B \to C
$$

also

$$
  0 \to F(A) \to F(B) \to F(C)
$$

is an exact sequence. Dually, if $F$ is a right exact functor, then for every [[nLab:exact sequence]] of the form

$$
  A \to B \to C \to 0
$$

also

$$
  F(A) \to F(B) \to F(C) \to 0
$$

is an exact sequence.

=--

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

If $0 \to A \to B \to C$ is exact then $A \hookrightarrow B$ is a [[monomorphism]].
But then the statement that $A \to B \to C$ is exact at $B$ says
precisely that $A$ is the [[kernel]] of $B \to C$.
So if $F$ is left exact then by definition also $F(A) \to F(B)$
is the kernel of $F(B) \to F(C)$ and so is in particular also
a monomorphism.
Dually for right exact functors.

=--

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

Proposition \ref{LeftRightExactFunctorsCharacterizedByExactSequences} is
clearly the motivation for the terminology in def. \ref{LeftRightExactFunctor}: a functor is left exact if it preserves short exact sequences to the left, and right exact if it preserves them to the right.

=--

Now we can state the main two definitions of this section.

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

Let

$$
  F : \mathcal{A} \to \mathcal{A}'
$$

be a [[left exact functor]] between [[nLab:abelian categories]] such that $\mathcal{A}$ has [[enough injectives]]. For $n \in \mathbb{N}$ the **$n$th [[right derived functor]]** of $F$ is the composite

$$
  R^n F
    \;\colon\;
  \mathcal{A}
   \stackrel{P}{\to}
  \mathcal{K}^\bullet(\mathcal{I}_{\mathcal{A}})
   \stackrel{\mathcal{K}^\bullet(F)}{\to}
  \mathcal{K}^\bullet(\mathcal{A}')
    \stackrel{H^n(-)}{\to}
  \mathcal{A}'
  \,,
$$

where

* $P$ is an [[injective resolution]] functor;

* $\mathcal{K}^\bullet(F)$ is a prolongation of $F$ analogous to def. \ref{ProlongationOfFunctorToChainComplexes};

* $H^n(-)$ is the $n$-[[chain homology]] functor. Hence

$$
  (R^n F)(X^\bullet) \coloneqq H^n(F(P(X)^\bullet))
  \,.
$$

=--

Dually:

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

Let

$$
  F \colon \mathcal{A} \to \mathcal{A}'
$$

be a [[right exact functor]] between [[abelian categories]] such that $\mathcal{A}$ has [[enough projectives]]. For $n \in \mathbb{N}$ the **$n$th [[left derived functor]]** of $F$ is the composite

$$
  L_n F
   \;\colon\;
  \mathcal{A}
   \stackrel{Q}{\to}
  \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}})
   \stackrel{\mathcal{K}_\bullet(F)}{\to}
  \mathcal{K}_\bullet(\mathcal{A}')
    \stackrel{H_n(-)}{\to}
  \mathcal{A}'
  \,,
$$

where

* $Q$ is a [[projective resolution]] functor;

* $\mathcal{K}_\bullet(F)=\mathcal{K}(F)$ is the prolongation of $F$ according to def. \ref{ProlongationOfFunctorToChainComplexes};

* $H_n(-)$ is the $n$-[[chain homology]] functor. Hence

$$
  (L_n F)(X_\bullet) \coloneqq H_n(F(Q(X)_\bullet))
  \,.
$$

=--

## Properties
 {#Properties}

### Basic properties

The following proposition says that in degree 0 these derived functors coincide with the original functors.

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

Let $F \colon \mathcal{A} \to \mathcal{B}$ a [[left exact functor]], def. \ref{LeftRightExactFunctor} in the presence of [[enough injectives]]. Then for all $X \in \mathcal{A}$ there is a [[nLab:natural isomorphism]]

$$
  R^0F(X) \simeq F(X)
  \,.
$$

Dually, if $F$ is a [[nLab:right exact functor]] in the presence of [[enough projectives]], then

$$
  L_0 F(X) \simeq F(X)
  \,.
$$

=--

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

We discuss the first statement, the second is formally dual.

By [this](/nlab/show/injective+resolution#InjectiveResolutionInComponents) remark an injective resolution $X \stackrel{\simeq_{qi}}{\to} X^\bullet$ is equivalently an [[nLab:exact sequence]] of the form

$$
  0 \to X \hookrightarrow X^0 \to X^1 \to \cdots
  \,.
$$

If $F$ is left exact then it preserves this exact sequence by definition of left exactness, and hence

$$
  0 \to F(X) \hookrightarrow F(X^0) \to F(X^1) \to \cdots
$$

is an exact sequence. But this means that

$$
  R^0 F(X) \coloneqq ker(F(X^0) \to F(X^1)) \simeq F(X)
  \,.
$$

=--

The following immediate consequence of the definition is worth recording:

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

Let $F$ be an [[nLab:additive functor]].

* If $F$ is [[nLab:right exact functor|right exact]] and $N \in \mathcal{A}$ is a [[nLab:projective object]], then

  $$
    L_n F(N) = 0 \;\;\;\; \forall n \geq 1
    \,.
  $$

* If $F$ is [[nLab:left exact functor|left exact]] and $N \in \mathcal{A}$ is a [[nLab:injective object]], then

  $$
    R^n F(N) = 0 \;\;\;\; \forall n \geq 1
    \,.
  $$

=--

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

If $N$ is projective then the chain complex $[\cdots \to 0 \to 0 \to N]$ is already a [[projective resolution]] and hence by definition $L_n F(N) \simeq H_n(0)$ for $n \geq 1$. Dually if $N$ is an injective object.

=--

For proving the basic property of derived functors below in prop. \ref{LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence} which continues these basis statements to higher degree, in a certain way, we need the following technical lemma.

+-- {: .num_lemma #ProjectiveResolutionOfExactSequenceByExactSequence}
###### Lemma

For $0 \to A \stackrel{i}{\to} B \stackrel{p}{\to} C \to 0$ a [[nLab:short exact sequence]]
in an [[nLab:abelian category]] with [[nLab:enough projectives]],
there exists a [[nLab:commuting diagram]] of [[nLab:chain complexes]]

$$
  \array{
    0 &\to& A_\bullet &\to& B_\bullet &\to& C_\bullet &\to& 0
    \\
      &&
    \downarrow^{\mathrlap{f_\bullet}}
      &&
    \downarrow^{\mathrlap{g_\bullet}}
      &&
    \downarrow^{\mathrlap{h_\bullet}}
    \\
    0 &\to& A &\stackrel{i}{\to}& B &\stackrel{p}{\to}& C &\to& 0
  }
$$

where

* each vertical morphism is a [[nLab:projective resolution]];

and in addition

* the top row is again a [[nLab:short exact sequence]] of chain complexes.

=--

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

By prop. \ref{ExistenceOfInjectiveResolutions} we can choose $f_\bullet$ and $h_\bullet$. The task is now to construct the third resolution $g_\bullet$ such as to obtain a short exact sequence of chain complexes, hence degreewise a short exact sequence, in the two row.

To construct this, let for each $n \in \mathbb{N}$

$$
  B_n \coloneqq A_n \oplus C_n
$$

be the [[nLab:direct sum]] and let the top horizontal morphisms be the canonical inclusion and projection maps of the direct sum.

Let then furthermore (in [[nLab:matrix calculus]] notation)

$$
  g_0 =
   \left(
     \array{
        (g_0)_A
        &
        (g_0)_B
      }
  \right)
 : A_0 \oplus C_0 \to B
$$

be given in the first component by the given composite

$$
 (g_0)_A : A_0 \oplus C_0 \stackrel{}{\to} A_0 \stackrel{f_0}{\to} A \stackrel{i}{\hookrightarrow} B
$$

and in the second component we take

$$
  (g_0)_C : A_0 \oplus C_0 \to C_0 \stackrel{\zeta}{\to} B
$$

to be given by a lift in

$$
  \array{
    && B
    \\
    & {}^{\mathllap{\zeta}}\nearrow & \downarrow^{\mathrlap{p}}
    \\
    C_0 &\stackrel{h_0}{\to}& C
  }
  \,,
$$

which exists by the [[nLab:left lifting property]] of the [[nLab:projective object]]
$C_0$ (since $C_\bullet$ is a projective resolution) against the [[nLab:epimorphism]] $p : B \to C$ of the [[nLab:short exact sequence]].

In total this gives in degree 0

$$
  \array{
    A_0 &\hookrightarrow& A_0 \oplus C_0 &\to& C_0
    \\
    {}^{\mathllap{f_0}}\downarrow
      &&
    {}^{\mathllap{((g_0)_A, (g_0)_C)}}\downarrow &\swarrow_{\zeta}& \downarrow^{\mathrlap{h_0}}
    \\
    A &\stackrel{i}{\hookrightarrow}& B &\stackrel{p}{\to}& C
  }
  \,.
$$

Let then the [[nLab:differentials]] of $B_\bullet$ be given by

$$
  d_k^{B_\bullet}
    =
  \left(
    \array{
      d_k^{A_\bullet} & (-1)^k e_k
      \\
      0 & d_k^{C_\bullet}
    }
  \right)
  :
  A_{k+1} \oplus C_{k+1}
  \to
  A_k \oplus C_k
  \,,
$$

where the $\{e_k\}$ are constructed by [[nLab:induction]] as follows. Let $e_0$ be a lift in

$$
  \array{
    & && A_0
    \\
    & & {}^{\mathllap{e_0}}\nearrow & \downarrow^{\mathrlap{f_0}}
    \\
    \zeta \circ d^{C_\bullet}_0 \colon & C_1 &\stackrel{}{\to}& A &\hookrightarrow B&
  }
$$

which exists since $C_1$ is a [[nLab:projective object]] and $A_0 \to A$ is an epimorphism by $A_\bullet$ being a projective resolution. Here we are using that by exactness the bottom morphism indeed factors through $A$ as indicated, because the definition of $\zeta$ and  the chain complex property of $C_\bullet$ gives

$$
  \begin{aligned}
    p \circ \zeta \circ d^{C_\bullet}_0
    &=
    h_0 \circ d^{C_\bullet}_0
    \\
    & = 0 \circ h_1
    \\
    & = 0
  \end{aligned}
  \,.
$$

Now in the induction step, assuming that $e_{n-1}$ has been been found satisfying the chain complex property, let $e_n$ be a lift in

$$
  \array{
    & && A_n
    \\
    & & {}^{\mathllap{e_{n}}}\nearrow & \downarrow^{\mathrlap{d^{A_\bullet}_{n-1}}}
    \\
    e_{n-1}\circ d_n^{C_\bullet} \colon & C_{n+1} &\stackrel{}{\hookrightarrow}& ker(d^{A_\bullet}_{n-2}) = im(d^{A_\bullet}_{n-1})) &\to& A_{n-1}
  }
  \,,
$$

which again exists since $C_{n+1}$ is projective. That the bottom morphism factors as indicated is the chain complex property of $e_{n-1}$ inside $d^{B_\bullet}_{n-1}$.

To see that the $d^{B_\bullet}$ defines this way indeed squares to 0 notice that

$$
  d^{B_\bullet}_{n}
  \circ
  d^{B_\bullet}_{n+1}
  =
  \left(
    \array{
       0 &  (-1)^{n}\left(e_{n} \circ d^{C_\bullet}_{n+1} - d^{A_\bullet}_n \circ e_{n+1} \right)
       \\
       0 & 0
     }
  \right)
  \,.
$$

This vanishes by the very commutativity of the above diagram.

This establishes $g_\bullet$ such that the above diagram commutes and the bottom row is degreewise a short exact sequence, in fact a [[nLab:split exact sequence]], by construction.

To see that $g_\bullet$ is indeed a quasi-isomorphism, consider the [[nLab:homology long exact sequence]] associated to the short exact sequence of cochain complexes $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$. In positive degrees it implies that the chain homology of $B_\bullet$ indeed vanishes. In degree 0 it gives the short sequence $0 \to A \to H_0(B_\bullet) \to B\to 0$ sitting in a commuting diagram

$$
  \array{
    0 &\to& A &\hookrightarrow& H_0(B_\bullet) &\to& C &\to& 0
    \\
    \downarrow && \downarrow^{\mathrlap{=}} && \downarrow && \downarrow^{\mathrlap{=}} && \downarrow
    \\
    0 &\to& A &\hookrightarrow& B &\to& C &\to& 0
    \,,
  }
$$

where both rows are exact. That the middle vertical morphism is an [[nLab:isomorphism]] then follows by the [[nLab:five lemma]].

=--

The formally dual statement to lemma \ref{ProjectiveResolutionOfExactSequenceByExactSequence} is the following.

+-- {: .num_lemma #InjectiveResolutionOfExactSequenceByExactSequence}
###### Lemma

For $0 \to A \to B \to C \to 0$ a [[nLab:short exact sequence]]
in an [[abelian category]] with [[injective object|enough injectives]],
there exists a [[nLab:commuting diagram]] of cochain complexes

$$
  \array{
    0 &\to& A &\to& B &\to& C &\to& 0
    \\
      &&
    \downarrow^{\mathrlap{}}
      &&
    \downarrow^{\mathrlap{}}
      &&
    \downarrow^{\mathrlap{}}
    \\
    0 &\to& A^\bullet &\to& B^\bullet &\to& C^\bullet &\to& 0
  }
$$

where

* each vertical morphism is an [[nLab:injective resolution]];

and in addition

* the bottom row is again a [[nLab:short exact sequence]] of cochain complexes.

=--

### Long exact sequence of a derived functor
 {#LongExactSequence}

The central general fact about derived functors to be discussed here is now the following.

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

Let $\mathcal{A}, \mathcal{B}$ be [[abelian categories]] and assume that $\mathcal{A}$ has [[injective object|enough injectives]].

Let $F : \mathcal{A} \to \mathcal{B}$ be a [[left exact functor]] and let

$$
  0 \to A \to B \to C \to 0
$$

be a [[short exact sequence]] in $\mathcal{A}$.

Then there is a [[long exact sequence]] of images of these objects under the right derived functors $R^\bullet F(-)$ of def. \ref{RightDerivedFunctorOfLeftExactFunctor}

$$
  \array{
    0 &\to& R^0F (A)  &\to&  R^0 F(B)  &\to&  R^0 F(C)
     &\stackrel{\delta_0}{\to}&
    R^1 F(A) &\to& R^1 F(B) &\to& R^1F(C)
     &\stackrel{\delta_1}{\to}&
    R^2 F(A) &\to& \cdots
    \\
    && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}}
    \\
    0 &\to& F(A) &\to& F(B) &\to& F(C)
  }
$$

in $\mathcal{B}$.

=--

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

By lemma \ref{InjectiveResolutionOfExactSequenceByExactSequence} we can find an injective resolution

$$
  0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0
$$

of the given exact sequence which is itself again an exact sequence of cochain complexes.

Since $A^n$ is an [[nLab:injective object]] for all $n$, its component sequences $0 \to A^n \to B^n \to C^n \to 0$ are indeed [[nLab:split exact sequences]] (see the discussion there). Splitness is preserved by any functor $F$ (and also since $F$ is [[nLab:additive functor|additive]] it even preserves the [[nLab:direct sum]] structure that is chosen in the proof of lemma \ref{ProjectiveResolutionOfExactSequenceByExactSequence}) and so it follows that

$$
  0 \to F(\tilde A^\bullet) \to F(\tilde B^\bullet) \to F(\tilde C^\bullet) \to 0
$$

is a again short exact sequence of cochain complexes, now in $\mathcal{B}$. Hence we have the corresponding [[nLab:homology long exact sequence]] from prop. \ref{HomologyLongExactSequence}:

$$
  \cdots
   \to
  H^{n-1}(F(A^\bullet)) \to H^{n-1}(F(B^\bullet)) \to H^{n-1}(F(C^\bullet))
    \stackrel{\delta}{\to}
  H^n(F(A^\bullet)) \to H^n(F(B^\bullet)) \to H^n(F(C^\bullet))
    \stackrel{\delta}{\to}
  H^{n+1}(F(A^\bullet)) \to H^{n+1}(F(B^\bullet)) \to H^{n+1}(F(C^\bullet))
   \to
  \cdots
  \,.
$$

By construction of the resolutions and by def. \ref{RightDerivedFunctorOfLeftExactFunctor}, this is equal to

$$
  \cdots
   \to
  R^{n-1}F(A) \to R^{n-1}F(B) \to R^{n-1}F(C)
   \stackrel{\delta}{\to}
  R^{n}F(A) \to R^{n}F(B) \to R^{n}F(C)
    \stackrel{\delta}{\to}
  R^{n+1}F(A) \to R^{n+1}F(B) \to R^{n+1}F(C)
   \to
  \cdots
  \,.
$$

Finally the equivalence of the first three terms with $F(A) \to F(B) \to F(C)$ is given by prop. \ref{BasicPropertiesOfDerivedFunctors}.

=--

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

Prop. \ref{LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence} implies that one way to interpret $R^1 F(A)$ is as a "measure for how a [[nLab:left exact functor]] $F$ fails to be an [[nLab:exact functor]]". For, with $A \to B \to C$ any [[nLab:short exact sequence]], this proposition gives the exact sequence

$$
  0 \to F(A) \to F(B) \to F(C) \to R^1 F(A)
$$

and hence $0 \to F(A) \to F(B) \to F(C) \to 0$ is a short exact sequence itself precisely if $R^1 F(A) \simeq 0$.

Dually, if $F$ is [[nLab:right exact functor]], then $L_1 F (C)$ "measures how $F$ fails to be exact" for then

$$
  L_1F (C) \to F(A) \to F(B) \to F(C) \to 0
$$

is an exact sequence and hence is a short exact sequence precisely if $L_1F(C) \simeq 0$.

=--

Notice that in fact we even have the following statement (following directly from the definition).

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

Let $F$ be an [[additive functor]] which is an [[exact functor]]. Then its left and right derived functors vanish in positive degree:

$$
  R^{\geq 1} F = 0
$$

and

$$
  L_{\geq 1} F = 0
  \,.
$$

=--

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

Because an [[exact functor]] preserves all [[exact sequences]]. If $Y_\bullet \to A$ is a projective resolution then also $F(Y)_\bullet$ is exact in all positive degrees, and hence $L_{n\geq 1} F(A) ) H_{n \geq}(F(Y)) = 0$. Dually for $R^n F$.

=--

Conversely:

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

Let $F \colon \mathcal{A} \to \mathcal{B}$ be a left or right exact additive functor. An object $A \in \mathcal{A}$ is called an $F$-**[[nLab:acyclic object]]** if all positive-degree right/left derived functors of $F$ are zero on $A$.

=--

### Via acyclic resolutions
 {#ViaAcyclicResolutions}

We now discuss how the derived functor of an additive functor $F$ may also be computed not necessarily with genuine injective/projective resolutions as in def. \ref{RightDerivedFunctorOfLeftExactFunctor}, but with (just) "$F$-injective"/"$F$-projective resolutions".

While projective resolutions in $\mathcal{A}$ are _sufficient_ for computing _every_ [[nLab:left derived functor]] on $Ch_\bullet(\mathcal{A})$ and injective resolutions are sufficient for computing _every_ [[nLab:right derived functor]] on $Ch^\bullet(\mathcal{A})$, if one is interested just in a single functor $F$ then such resolutions may be more than _necessary_. A weaker kind of resolution which is still sufficient is then often more convenient for applications. These _$F$-projective resolutions_ and _$F$-injective resolutions_, respectively, we discuss now. A special case of both are _$F$-[[nLab:acyclic resolutions]]_.

Let $\mathcal{A}, \mathcal{B}$ be [[nLab:abelian categories]] and let $F \colon \mathcal{A} \to \mathcal{B}$ be an [[nLab:additive functor]].

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

Assume that $F$ is [[nLab:left exact functor|left exact]]. An [[nLab:additive category|additive]] [[nLab:full subcategory]] $\mathcal{I} \subset \mathcal{A}$ is called **$F$-injective** (or: consisting of $F$-injective objects) if

1. for every object $A \in \mathcal{A}$ there is a [[nLab:monomorphism]] $A \to \tilde A$ into an object $\tilde A \in \mathcal{I} \subset \mathcal{A}$;

1. for every [[nLab:short exact sequence]] $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with $A, B \in \mathcal{I} \subset \mathcal{A}$ also $C \in \mathcal{I} \subset \mathcal{A}$;

1. for every [[nLab:short exact sequence]] $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with $A\in \mathcal{I} \subset \mathcal{A}$ also $0 \to F(A) \to F(B) \to F(C) \to 0$ is a short exact sequence in $\mathcal{B}$.

=--

And dually:

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

Assume that $F$ is [[nLab:right exact functor|right exact]]. An [[nLab:additive category|additive]] [[nLab:full subcategory]] $\mathcal{P} \subset \mathcal{A}$ is called **$F$-projective** (or: consisting of $F$-projective objects) if

1. for every object $A \in \mathcal{A}$ there is an [[nLab:epimorphism]] $\tilde A \to A$ from an object $\tilde A \in \mathcal{P} \subset \mathcal{A}$;

1. for every [[nLab:short exact sequence]] $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with $B, C \in \mathcal{P} \subset \mathcal{A}$ also $A \in \mathcal{P} \subset \mathcal{A}$;

1. for every [[nLab:short exact sequence]] $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with $C\in \mathcal{P} \subset \mathcal{A}$ also $0 \to F(A) \to F(B) \to F(C) \to 0$ is a short exact sequence in $\mathcal{B}$.

=--

With the $\mathcal{I},\mathcal{P}\subset \mathcal{A}$ as above, we say:

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

For $A \in \mathcal{A}$,

* an **$F$-injective resolution** of $A$ is a [[cochain complex]] $I^\bullet \in Ch^\bullet(\mathcal{I}) \subset Ch^\bullet(\mathcal{A})$ and a [[nLab:quasi-isomorphism]]

  $$
    A \stackrel{\simeq_{qi}}{\to} I^\bullet
  $$

* an **$F$-projective resolution** of $A$ is a [[chain complex]] $Q_\bullet \in Ch_\bullet(\mathcal{P}) \subset Ch^\bullet(\mathcal{A})$ and a [[nLab:quasi-isomorphism]]

  $$
    Q_\bullet \stackrel{\simeq_{qi}}{\to} A
    \,.
  $$

=--

Let now $\mathcal{A}$ have [[enough projectives]] / [[enough injectives]], respectively.

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

For $F \colon \mathcal{A} \to \mathcal{B}$ an [[nLab:additive functor]],
let $Ac \subset \mathcal{A}$ be the [[nLab:full subcategory]] on the $F$-[[nLab:acyclic objects]], def. \ref{AcyclicObject}. Then

* if $F$ is [[nLab:left exact functor|left exact]], then $\mathcal{I} \coloneqq Ac$ is a subcategory of $F$-injective objects;

* if $F$ is [[nLab:right exact functor|right exact]], then $\mathcal{P} \coloneqq Ac$ is a subcategory of $F$-projective objects.

=--

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

Consider the case that $F$ is left exact. The other case works dually.
Then the first condition of def. \ref{FInjectives} is satisfied because every [[nLab:injective object]] is an $F$-[[nLab:acyclic object]] and by assumption there are enough of these.

For the second and third condition of def. \ref{FInjectives} use that there is the [[nLab:long exact sequence]] of [[nLab:derived functors]] prop. \ref{LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence}

$$
  0 \to F(A) \to F(B) \to F(C) \to R^1 F(A) \to R^1 F(B) \to R^1 F(C) \to R^2 F(A) \to R^2 F(B) \to R^2 F(C) \to \cdot
  \,.
$$

For the second condition, by assumption on $A$ and $B$ and definition of $F$-[[nLab:acyclic object]] we have $R^n F(A) \simeq 0$ and $R^n F(B) \simeq 0$ for $n \geq 1$ and hence short exact sequences

$$
  0 \to 0 \to R^n F(C) \to 0
$$

which imply that $R^n F(C)\simeq 0$ for all $n \geq 1$, hence that $C$ is acyclic.

Similarly, the third condition is equivalent to $R^1 F(A) \simeq 0$.

=--

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

The $F$-projective/injective resolutions (def. \ref{FProjectivesResolution}) by [[acyclic objects]] as in example \ref{FAcyclicObjectsAreFProjectiveObjects} are called **$F$-acyclic resolutions**.

=--

Let $\mathcal{A}$ be an [[nLab:abelian category]] with [[nLab:injective object|enough injectives]].
Let $F \colon \mathcal{A} \to \mathcal{B}$ be an [[nLab:additive functor|additive]] [[nLab:left exact functor]] with [[nLab:right derived functor]] $R_\bullet F$, def. \ref{RightDerivedFunctorOfLeftExactFunctor}. Finally let $\mathcal{I} \subset \mathcal{A}$ be a subcategory of $F$-injective objects, def. \ref{FInjectives}.

+-- {: .num_lemma #FPreservesNullnessOfFInjectiveComplexes}
###### Lemma

If a [[nLab:cochain complex]] $X^\bullet \in Ch^\bullet(\mathcal{I}) \subset Ch^\bullet(\mathcal{A})$ is [[nLab:quasi-isomorphism|quasi-isomorphic]] to 0,

$$
  X^\bullet \stackrel{\simeq_{qi}}{\to} 0
$$

then also $F(X^\bullet) \in Ch^\bullet(\mathcal{B})$ is quasi-isomorphic to 0

$$
  F(X^\bullet) \stackrel{\simeq_{qi}}{\to} 0
  \,.
$$

=--

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

Consider the following collection of [[nLab:short exact sequences]] obtained from the [[nLab:long exact sequence]] $X^\bullet$:

$$
  0 \to X^0 \stackrel{d^0}{\to} X^1 \stackrel{d^1}{\to} im(d^1) \to 0
$$

$$
  0 \to im(d^1) \to X^2 \stackrel{d^2}{\to} im(d^2) \to 0
$$

$$
  0 \to im(d^2) \to X^3 \stackrel{d^3}{\to} im(d^3) \to 0
$$

and so on. Going by [[nLab:induction]] through this list and using the second condition in def. \ref{FInjectives} we have that all the $im(d^n)$ are in $\mathcal{I}$. Then the third condition in def. \ref{FInjectives} says that all the sequences

$$
  0 \to F(im(d^n)) \to F(X^n+1) \to F(im(d^{n+1})) \to 0
$$

are [[nLab:short exact sequence|exact]]. But this means that

$$
  0 \to F(X^0)\to F(X^1) \to F(X^2) \to \cdots
$$

is exact, hence that $F(X^\bullet)$ is quasi-isomorphic to 0.

=--

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

For $A \in \mathcal{A}$ an object with $F$-injective resolution $A \stackrel{\simeq_{qi}}{\to} I_F^\bullet$, def. \ref{FProjectivesResolution}, we have for each $n \in \mathbb{N}$ an [[nLab:isomorphism]]

$$
  R^n F(A) \simeq H^n(F(I_F^\bullet))
$$

between the $n$th right derived functor, def. \ref{RightDerivedFunctorOfLeftExactFunctor} of $F$ evaluated on $A$ and the [[nLab:cochain cohomology]] of $F$ applied to the $F$-injective resolution $I_F^\bullet$.

=--

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

By [this prop.](https://ncatlab.org/schreiber/show/Introduction+to+Homological+algebra#ExistenceOfInjectiveResolutions) we may also find an injective resolution $A \stackrel{\simeq_{qi}}{\to} I^\bullet$. By [this prop](https://ncatlab.org/schreiber/show/Introduction+to+Homological+algebra#InjectiveResolutionOfCodomainRespectsMorphisms) there is a lift of the identity on $A$ to a [[chain map]] $I^\bullet_F \to I^\bullet$ such that the [[diagram]]

$$
  \array{
     A &\stackrel{\simeq_{qi}}{\to}& I_F^\bullet
     \\
     \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}}
     \\
     A &\stackrel{\simeq_{qi}}{\to}& I^\bullet
  }
$$

[[nLab:commuting diagram|commutes]] in $Ch^\bullet(\mathcal{A})$. Therefore by the [[nLab:2-out-of-3]] property of [[nLab:quasi-isomorphisms]] it follows that $f$ is a quasi-isomorphism

Let $Cone(f) \in Ch^\bullet(\mathcal{A})$ be the [[nLab:mapping cone]] of $f$ and let $I^\bullet \to Cone(f)$ be the canonical [[nLab:chain map]] into it. By the explicit formulas for mapping cones, we have that

1. there is an [[nLab:isomorphism]] $F(Cone(f)) \simeq Cone(F(f))$;

1. $Cone(f) \in Ch^\bullet(\mathcal{I})\subset Ch^\bullet(\mathcal{A})$ (because $F$-injective objects are closed under [[nLab:direct sum]]).

The first implies that we have a [[nLab:homology exact sequence]]

$$
  \cdots \to H^n(I^\bullet) \to H^n(I_F^\bullet) \to H^n(Cone(f)^\bullet)
   \to H^{n+1}(I^\bullet) \to H^{n+1}(I_F^\bullet) \to H^{n+1}(Cone(f)^\bullet) \to \cdots
  \,.
$$

Observe that with $f^\bullet$ a quasi-isomorphism $Cone(f^\bullet)$ is quasi-isomorphic to 0. Therefore
the second item above implies with lemma \ref{FPreservesNullnessOfFInjectiveComplexes} that also $F(Cone(f))$ is quasi-isomorphic to 0. This finally means that the above homology exact sequences consists of exact pieces of the form

$$
  0 \to (R^n F(A)\coloneqq H^n(I^\bullet) \stackrel{\simeq}{\to} H^n(I_F^\bullet) \to 0
  \,.
$$

=--

### Derived adjoint functors

+-- {: .num_note}
###### Observation

If

$$
  (F \dashv G) : \mathcal{A}
  \stackrel{\overset{G}{\leftarrow}}{\underset{F}{\to}}
   \mathcal{B}
$$

is a pair of [[additive functor|additive]] [[adjoint functors]], then

* the [[left adjoint]] $F$ is [[right exact functor|right exact]];

* the [[right adjoint]] $G$ is [[left exact functor|left exact]];

=--

(...)

### Preservation of further limits and colimits

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

Let $F \colon \mathcal{A} \to \mathcal{B}$ be an [[additive functor|additive]] [[right exact functor]] with codomain an [[additive and abelian categories|AB5-category]].

Then the [[left derived functors]] $L_n F$ preserves [[filtered colimits]] precisely if filtered colimits of [[projective objects]] in $\mathcal{A}$ are $F$-[[acyclic objects]].

=--

See [here](http://mathoverflow.net/questions/97658/left-derived-functors-commute-with-filtered-colimits).

## Examples

(...)

## Related concepts

* [[delta-functor]]

## References

A standard textbook introduction is chapter 2 of

* {#Weibel} [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_.

A systematic discussion from the point of view of [[homotopy theory]] and [[derived categories]] is in chapter 7 of

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

* [[Masaki Kashiwara]], [[Pierre Schapira]], section 13 of _[[Categories and Sheaves]]_, Grundlehren der Mathematischen Wissenschaften __332__, Springer (2006)

The above text is taken from

* [[Urs Schreiber]], _[[schreiber:HAI|Introduction to Homological Algebra]]_

[[!redirects derived functors in homological algebra]]


derived functor on a derived category

#Idea#

A special case of the general notion of a [[derived functor]] on the [[homotopy category]] of a [[homotopical category]] is that of a derived functor on the [[category of chain complexes]] of an [[abelian category]]. This is the original case in which derived functors were considered in [[homological algebra]]. This entry discusses special aspects of this special situation.

+--{: .query}
[[Mike Shulman|Mike]]: Somewhere, we should talk about derived functors in the very traditional sense of "extending the image of a short exact sequence to a long exact sequence."

[[Zoran Skoda]] Not only universal $\delta$ and $\delta^*$ functors, but we should also have satelites.

Title of this entry misleading, and nonstandard (I can say I never heard of exact phrase "derived functor on a derived category"). It seems that Urs wanted to do all kinds of derived functors in the setting of chain complexes, not only inducing functors between derived categories from functors between abelian. The example below are classical derived functors between abelian categories, not between derived (I see downstairs Ext and Tor).
=--

#Derived functors in homological algebra#

Here are some peculiarities of the concept of derived functors in [[homological algebra]], mostly due to historical reasons:

1. Every functor $F : C \to C'$ of [[abelian category|abelian categories]] canonically induces a functor $K(F) : K(C) \to K(C')$ of [[category of chain complexes|categories of chain complexes]]. The (right, say) _derived functor_ $R F$ of $F$ is the derived functor of $K(F)$ and usually only functors of chain complexes of the form $K(F)$ are considered.

1. Similarly the output of the derived functor is usually taken to be in $C'$ by postcomposing with the [[homology]] functor. One writes $R^k F := H^k \circ R F$. The derived functor $R F$ in its totality as a functor with values in $K(C)$ is then sometimes denoted $R^\bullet F$.

#Definition#

#Examples#

The most famous derived functors are the derived version of the hom-functor and the tensor product functor, whose derived functors are traditionally denoted $Ext$ and $Tor$.

## Ext ##

Consider

$$
  Hom^\bullet : K(C)^{op} \times K(C) \to K(Ab)
$$
$$
  (X', Y') \mapsto tot Hom^{\bullet, \bullet}(X', Y')
$$

Then

$$
  Ext^k(X,Y) \simeq H^k(R Hom(X, Y))
$$

(...)

## Tor ##

(...)