ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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Derived functor

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In <a href="Mathematics" title="Mathematics">mathematics</a>, certain <a href="Functor" title="Functor">functors</a> may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.

<div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2></div>

It was noted in various quite different settings that a <a href="Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations.

Suppose we are given a covariant <a href="Left_exact_functor" class="mw-redirect" title="Left exact functor">left exact functor</a> F : A → B between two <a href="Abelian_category" title="Abelian category">abelian categories</a> A and B. If 0 → A → B → C → 0 is a short exact sequence in A, then applying F yields the exact sequence 0 → F(A) → F(B) → F(C) and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if A is "nice" enough) there is one <a href="Canonical_form" title="Canonical form">canonical</a> way of doing so, given by the right derived functors of F. For every i≥1, there is a functor RiF: A → B, and the above sequence continues like so: 0 → F(A) → F(B) → F(C) → R1F(A) → R1F(B) → R1F(C) → R2F(A) → R2F(B) → ... . From this we see that F is an exact functor if and only if R1F = 0; so in a sense the right derived functors of F measure "how far" F is from being exact.

If the object A in the above short exact sequence is <a href="Injective_object" title="Injective object">injective</a>, then the sequence <a href="Splitting_lemma" title="Splitting lemma">splits</a>. Applying any additive functor to a split sequence results in a split sequence, so in particular R1F(A) = 0. Right derived functors (for i>0) are zero on injectives: this is the motivation for the construction given below.

<div class="mw-heading mw-heading2"><h2 id="Construction_and_first_properties">Construction and first properties</h2></div>

The crucial assumption we need to make about our abelian category A is that it has enough injectives, meaning that for every object A in A there exists a <a href="Monomorphism" title="Monomorphism">monomorphism</a> A → I where I is an <a href="Injective_object" title="Injective object">injective object</a> in A.

The right derived functors of the covariant left-exact functor F : A → B are then defined as follows. Start with an object X of A. Because there are enough injectives, we can construct a long exact sequence of the form

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<annotation encoding="application/x-tex">{\displaystyle 0\to X\to I^{0}\to I^{1}\to I^{2}\to \cdots }</annotation>

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</math><img src="./5649c3960ab6b38e091ffa63a88fd1c117581912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:30.748ex; height:2.676ex;" alt="{\displaystyle 0\to X\to I^{0}\to I^{1}\to I^{2}\to \cdots }" loading="lazy"></dd></dl>

where the I i are all injective (this is known as an <a href="Injective_resolution" class="mw-redirect" title="Injective resolution">injective resolution</a> of X). Applying the functor F to this sequence, and chopping off the first term, we obtain the <a href="Cochain_complex" class="mw-redirect" title="Cochain complex">cochain complex</a>

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<annotation encoding="application/x-tex">{\displaystyle 0\to F(I^{0})\to F(I^{1})\to F(I^{2})\to \cdots }</annotation>

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</math><img src="./a36b3bf7a8f4ae1ec0f1c5001c84fc6fd904f949.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.804ex; height:3.176ex;" alt="{\displaystyle 0\to F(I^{0})\to F(I^{1})\to F(I^{2})\to \cdots }" loading="lazy"></dd></dl>

Note: this is in general not an exact sequence anymore. But we can compute its <a href="Cohomology" title="Cohomology">cohomology</a> at the i-th spot (the kernel of the map from F(Ii) modulo the image of the map to F(Ii)); we call the result RiF(X). Of course, various things have to be checked: the result does not depend on the given injective resolution of X, and any morphism X → Y naturally yields a morphism RiF(X) → RiF(Y), so that we indeed obtain a functor. Note that left exactness means that

0 → F(X) → F(I0) → F(I1)

is exact, so R0F(X) = F(X), so we only get something interesting for i>0.

(Technically, to produce well-defined derivatives of F, we would have to fix an injective resolution for every object of A. This choice of injective resolutions then yields functors RiF. Different choices of resolutions yield <a href="Naturally_isomorphic" class="mw-redirect" title="Naturally isomorphic">naturally isomorphic</a> functors, so in the end the choice doesn't really matter.)

The above-mentioned property of turning short exact sequences into long exact sequences is a consequence of the <a href="Snake_lemma" title="Snake lemma">snake lemma</a>. This tells us that the collection of derived functors is a <a href="Delta-functor" title="Delta-functor">δ-functor</a>.

If X is itself injective, then we can choose the injective resolution 0 → X → X → 0, and we obtain that RiF(X) = 0 for all i ≥ 1. In practice, this fact, together with the long exact sequence property, is often used to compute the values of right derived functors.

An equivalent way to compute RiF(X) is the following: take an injective resolution of X as above, and let Ki be the image of the map Ii-1→Ii (for i=0, define Ii-1=0), which is the same as the kernel of Ii→Ii+1. Let φi : Ii-1→Ki be the corresponding surjective map. Then RiF(X) is the cokernel of F(φi).

<div class="mw-heading mw-heading2"><h2 id="Variations">Variations</h2></div>

If one starts with a covariant right-exact functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, and the category A has enough projectives (i.e. for every object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> of A there exists an epimorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow A}">

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<annotation encoding="application/x-tex">{\displaystyle P\rightarrow A}</annotation>

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</math><img src="./d0700ff39d1404e621633f859e007b5549dc6106.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.103ex; height:2.176ex;" alt="{\displaystyle P\rightarrow A}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> is a <a href="Projective_module" title="Projective module">projective object</a>), then one can define analogously the left-derived functors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{i}G}">

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<annotation encoding="application/x-tex">{\displaystyle L_{i}G}</annotation>

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</math><img src="./10655d4f7046bc90dcdbc0c8f74df431db1c39c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.209ex; height:2.509ex;" alt="{\displaystyle L_{i}G}" loading="lazy">. For an object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> of A we first construct a projective resolution of the form

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<annotation encoding="application/x-tex">{\displaystyle \cdots \to P_{2}\to P_{1}\to P_{0}\to X\to 0}</annotation>

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</math><img src="./9b76ce1da471cea35239d24b94acb6b79f5d3665.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.576ex; height:2.509ex;" alt="{\displaystyle \cdots \to P_{2}\to P_{1}\to P_{0}\to X\to 0}" loading="lazy"></dd></dl>

where the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{i}}</annotation>

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</math><img src="./3ba1396129f7be3c7f828a571b6649e6807d10d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}" loading="lazy"> are projective. We apply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> to this sequence, chop off the last term, and compute homology to get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{i}G(X)}">

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<annotation encoding="application/x-tex">{\displaystyle L_{i}G(X)}</annotation>

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</math><img src="./5577c759e42699a3a236ee777f4bbf7133131554.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.999ex; height:2.843ex;" alt="{\displaystyle L_{i}G(X)}" loading="lazy">. As before, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{0}G(X)=G(X)}">

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<annotation encoding="application/x-tex">{\displaystyle L_{0}G(X)=G(X)}</annotation>

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</math><img src="./fe553717b784fe3766d861816063563a304fbd8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.968ex; height:2.843ex;" alt="{\displaystyle L_{0}G(X)=G(X)}" loading="lazy">.

In this case, the long exact sequence will grow "to the left" rather than to the right:

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<annotation encoding="application/x-tex">{\displaystyle 0\to A\to B\to C\to 0}</annotation>

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</math><img src="./183430a88117f1e77462cf34d4f84b2a353e81e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.055ex; height:2.176ex;" alt="{\displaystyle 0\to A\to B\to C\to 0}" loading="lazy"></dd></dl>

is turned into

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<annotation encoding="application/x-tex">{\displaystyle \cdots \to L_{2}G(C)\to L_{1}G(A)\to L_{1}G(B)\to L_{1}G(C)\to G(A)\to G(B)\to G(C)\to 0}</annotation>

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</math><img src="./4a5fee9407ea1c6dfb1fff18d8bfc1f553b17167.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:81.112ex; height:2.843ex;" alt="{\displaystyle \cdots \to L_{2}G(C)\to L_{1}G(A)\to L_{1}G(B)\to L_{1}G(C)\to G(A)\to G(B)\to G(C)\to 0}" loading="lazy">.</dd></dl>

Left derived functors are zero on all projective objects.

One may also start with a contravariant left-exact functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">; the resulting right-derived functors are then also contravariant. The short exact sequence

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<annotation encoding="application/x-tex">{\displaystyle 0\to A\to B\to C\to 0}</annotation>

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</math><img src="./183430a88117f1e77462cf34d4f84b2a353e81e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.055ex; height:2.176ex;" alt="{\displaystyle 0\to A\to B\to C\to 0}" loading="lazy"></dd></dl>

is turned into the long exact sequence

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<annotation encoding="application/x-tex">{\displaystyle 0\to F(C)\to F(B)\to F(A)\to R^{1}F(C)\to R^{1}F(B)\to R^{1}F(A)\to R^{2}F(C)\to \cdots }</annotation>

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</math><img src="./69974026d34ef7a7f21cd0a3da97631514b3c075.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:81.235ex; height:3.176ex;" alt="{\displaystyle 0\to F(C)\to F(B)\to F(A)\to R^{1}F(C)\to R^{1}F(B)\to R^{1}F(A)\to R^{2}F(C)\to \cdots }" loading="lazy"></dd></dl>

These left derived functors are zero on projectives and are therefore computed via projective resolutions.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is an abelian category, then its category of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\{\ast \to \ast \}}}">

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<annotation encoding="application/x-tex">{\displaystyle A^{\{\ast \to \ast \}}}</annotation>

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</math><img src="./c3712dc58b11fc0709722fe355e9a50216efb615.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.906ex; height:2.843ex;" alt="{\displaystyle A^{\{\ast \to \ast \}}}" loading="lazy"> is also abelian. The functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker :A^{\{\ast \to \ast \}}\to A}">

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<annotation encoding="application/x-tex">{\displaystyle \ker :A^{\{\ast \to \ast \}}\to A}</annotation>

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</math><img src="./d2f982a9e0e23a0758733fc171f4456052d587e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.372ex; height:2.843ex;" alt="{\displaystyle \ker :A^{\{\ast \to \ast \}}\to A}" loading="lazy"> which maps each morphism to its kernel is left exact. Its right derived functors are</li></ul>

<dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{i}(\ker )(f)={\begin{cases}\ker(f)&i=0\\\operatorname {coker} (f)&i=1\\0&i>1\end{cases}}}">

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<mi>coker</mi>

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<annotation encoding="application/x-tex">{\displaystyle R^{i}(\ker )(f)={\begin{cases}\ker(f)&i=0\\\operatorname {coker} (f)&i=1\\0&i>1\end{cases}}}</annotation>

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</math><img src="./ec1a65b21f43d9654124dc3f46f76e3bc7b0bb79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:32.388ex; height:8.509ex;" alt="{\displaystyle R^{i}(\ker )(f)={\begin{cases}\ker(f)&i=0\\\operatorname {coker} (f)&i=1\\0&i>1\end{cases}}}" loading="lazy"></dd></dl></dd>

<dd>Dually the functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {coker} }">

<mi>coker</mi>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {coker} }</annotation>

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</math><img src="./a710773531cc6bad1147a2401db79b24f0ea99dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.366ex; height:2.176ex;" alt="{\displaystyle \operatorname {coker} }" loading="lazy"> is right exact and its left derived functors are

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{i}(\operatorname {coker} )(f)={\begin{cases}\operatorname {coker} (f)&i=0\\\ker(f)&i=1\\0&i>1\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle L_{i}(\operatorname {coker} )(f)={\begin{cases}\operatorname {coker} (f)&i=0\\\ker(f)&i=1\\0&i>1\end{cases}}}</annotation>

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</math><img src="./47811e3b1bb3ce3607ecdd53b7bad57370590918.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:34.402ex; height:8.509ex;" alt="{\displaystyle L_{i}(\operatorname {coker} )(f)={\begin{cases}\operatorname {coker} (f)&i=0\\\ker(f)&i=1\\0&i>1\end{cases}}}" loading="lazy"></dd></dl></dd>

<dd>This is a manifestation of the <a href="Snake_lemma" title="Snake lemma">snake lemma</a>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Homology_and_cohomology">Homology and cohomology</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Sheaf_cohomology"><a href="Sheaf_cohomology" title="Sheaf cohomology">Sheaf cohomology</a></h4></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a <a href="Topological_space" title="Topological space">topological space</a>, then the category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Sh(X)}">

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<annotation encoding="application/x-tex">{\displaystyle Sh(X)}</annotation>

</semantics>

</math><img src="./8d0ce678eeea773d68148f9fc8f1db290640d9c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.628ex; height:2.843ex;" alt="{\displaystyle Sh(X)}" loading="lazy"> of all <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaves</a> of <a href="Abelian_group" title="Abelian group">abelian groups</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is an abelian category with enough injectives. The functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma :Sh(X)\to Ab}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma :Sh(X)\to Ab}</annotation>

</semantics>

</math><img src="./7c742831991954b8c4701cb5f187226e633166ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.372ex; height:2.843ex;" alt="{\displaystyle \Gamma :Sh(X)\to Ab}" loading="lazy"> which assigns to each such sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> the group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma ({\mathcal {F}}):={\mathcal {F}}(X)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma ({\mathcal {F}}):={\mathcal {F}}(X)}</annotation>

</semantics>

</math><img src="./09d9a830c0b815b9e84cef4113d44db0667a321d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.65ex; height:2.843ex;" alt="{\displaystyle \Gamma ({\mathcal {F}}):={\mathcal {F}}(X)}" loading="lazy"> of global sections is left exact, and the right derived functors are the <a href="Sheaf_cohomology" title="Sheaf cohomology">sheaf cohomology</a> functors, usually written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{i}(X,{\mathcal {F}})}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{i}(X,{\mathcal {F}})}</annotation>

</semantics>

</math><img src="./677bacb8895f30ce6eb63c0a72f47be368ac730f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.653ex; height:3.176ex;" alt="{\displaystyle H^{i}(X,{\mathcal {F}})}" loading="lazy">. Slightly more generally: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,{\mathcal {O}}_{X})}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {O}}_{X})}</annotation>

</semantics>

</math><img src="./46716c1fec068ffe9981107a5a215fa1fc9e7d5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.306ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {O}}_{X})}" loading="lazy"> is a <a href="Ringed_space" title="Ringed space">ringed space</a>, then the category of all sheaves of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>

</semantics>

</math><img src="./9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy">-modules is an abelian category with enough injectives, and we can again construct sheaf cohomology as the right derived functors of the global section functor.

There are various notions of cohomology which are a special case of this:

<ul><li><a href="De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a> is the sheaf cohomology of the sheaf of <a href="Locally_constant_function" title="Locally constant function">locally constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">-valued functions on a <a href="Manifold" title="Manifold">manifold</a>. The De Rham complex is a resolution of this sheaf not by injective sheaves, but by <a href="Fine_sheaf" class="mw-redirect" title="Fine sheaf">fine sheaves</a>.</li>

<li><a href="%C3%89tale_cohomology" title="Étale cohomology">Étale cohomology</a> is another cohomology theory for sheaves over a scheme. It is the right derived functor of the global sections of abelian sheaves on the <a href="%C3%89tale_topology" title="Étale topology">étale site</a>.</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Ext_functors"><a href="Ext_functor" title="Ext functor">Ext functors</a></h4></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>, then the category of all left <a href="Module_(mathematics)" title="Module (mathematics)"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-modules</a> is an abelian category with enough injectives. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a fixed left <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-module, then the functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (A,-):R{\text{-Mod}}\to {\mathfrak {Ab}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (A,-):R{\text{-Mod}}\to {\mathfrak {Ab}}}</annotation>

</semantics>

</math><img src="./f3e0abecd06a90a98f9d1bef01397424d6e0fadb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.773ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (A,-):R{\text{-Mod}}\to {\mathfrak {Ab}}}" loading="lazy"> is left exact, and its right derived functors are the <a href="Ext_functor" title="Ext functor">Ext functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ext} _{R}^{i}(A,-)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ext} _{R}^{i}(A,-)}</annotation>

</semantics>

</math><img src="./ab572b912135a1c1bc5c3ba0c4f744349a75ef01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.589ex; height:3.176ex;" alt="{\displaystyle \operatorname {Ext} _{R}^{i}(A,-)}" loading="lazy">. Alternatively <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ext} _{R}^{i}(-,B)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ext} _{R}^{i}(-,B)}</annotation>

</semantics>

</math><img src="./e67ce67c47dbb5adaf6f8d81517431ee4625275a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.61ex; height:3.176ex;" alt="{\displaystyle \operatorname {Ext} _{R}^{i}(-,B)}" loading="lazy"> can also be obtained as the left derived functor of the right exact functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{R}(-,B):R{\text{-Mod}}\to {\mathfrak {Ab}}^{op}}">

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{R}(-,B):R{\text{-Mod}}\to {\mathfrak {Ab}}^{op}}</annotation>

</semantics>

</math><img src="./a945883a0aec3ba114fa4551826024b61b83bccf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.13ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} _{R}(-,B):R{\text{-Mod}}\to {\mathfrak {Ab}}^{op}}" loading="lazy">.

Various notions of cohomology are special cases of Ext functors and therefore also derived functors.

<ul><li><a href="Group_cohomology" title="Group cohomology">Group cohomology</a> is the right derived functor of the invariants functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-)^{G}:k[G]{\text{-Mod}}\to k[G]{\text{-Mod}}}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-)^{G}:k[G]{\text{-Mod}}\to k[G]{\text{-Mod}}}</annotation>

</semantics>

</math><img src="./2123ca8addbb14c40824a2700a9fb9e009519b57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.077ex; height:3.176ex;" alt="{\displaystyle (-)^{G}:k[G]{\text{-Mod}}\to k[G]{\text{-Mod}}}" loading="lazy"> which is the same as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{k[G]}(k,-)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{k[G]}(k,-)}</annotation>

</semantics>

</math><img src="./835b77773e277b7aff12dd194e7af55773113cf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.999ex; height:3.176ex;" alt="{\displaystyle \operatorname {Hom} _{k[G]}(k,-)}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k[G]}</annotation>

</semantics>

</math><img src="./4a13dacf8d6ff682a6b5d59e84f93e589cba3fa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.332ex; height:2.843ex;" alt="{\displaystyle k[G]}" loading="lazy">-module) and therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{i}(G,M)=\operatorname {Ext} _{k[G]}^{i}(k,M)}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{i}(G,M)=\operatorname {Ext} _{k[G]}^{i}(k,M)}</annotation>

</semantics>

</math><img src="./d3c155c5c3b3fb42dc8f288de90c3770ac1c3f73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:26.621ex; height:3.676ex;" alt="{\displaystyle H^{i}(G,M)=\operatorname {Ext} _{k[G]}^{i}(k,M)}" loading="lazy">.</li>

<li><a href="Lie_algebra_cohomology" title="Lie algebra cohomology">Lie algebra cohomology</a> of a <a href="Lie_algebra" title="Lie algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation>

</semantics>

</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> over some commutative ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is the right derived functor of the invariants functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-)^{\mathfrak {g}}:{\mathfrak {g}}{\text{-Mod}}\to k{\text{-Mod}}}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-)^{\mathfrak {g}}:{\mathfrak {g}}{\text{-Mod}}\to k{\text{-Mod}}}</annotation>

</semantics>

</math><img src="./0dd1f93f8e180929805fb15ad24f58ae1db97d1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.334ex; height:2.843ex;" alt="{\displaystyle (-)^{\mathfrak {g}}:{\mathfrak {g}}{\text{-Mod}}\to k{\text{-Mod}}}" loading="lazy"> which is the same as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{U({\mathfrak {g}})}(k,-)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{U({\mathfrak {g}})}(k,-)}</annotation>

</semantics>

</math><img src="./f40ad8ce46d08c9ee878f8a16848c05801b36a2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.305ex; height:3.176ex;" alt="{\displaystyle \operatorname {Hom} _{U({\mathfrak {g}})}(k,-)}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is again the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation>

</semantics>

</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">-module and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U({\mathfrak {g}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U({\mathfrak {g}})}</annotation>

</semantics>

</math><img src="./d259de5e97b5d78daf491cddb6b914ad2ef6fffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.764ex; height:2.843ex;" alt="{\displaystyle U({\mathfrak {g}})}" loading="lazy"> is the <a href="Universal_enveloping_algebra" title="Universal enveloping algebra">universal enveloping algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation>

</semantics>

</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">). Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{i}({\mathfrak {g}},M)=\operatorname {Ext} _{U({\mathfrak {g}})}^{i}(k,M)}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{i}({\mathfrak {g}},M)=\operatorname {Ext} _{U({\mathfrak {g}})}^{i}(k,M)}</annotation>

</semantics>

</math><img src="./c6d61a7f39605008d13da230cc4d54289041e191.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:26.271ex; height:3.676ex;" alt="{\displaystyle H^{i}({\mathfrak {g}},M)=\operatorname {Ext} _{U({\mathfrak {g}})}^{i}(k,M)}" loading="lazy">.</li>

<li><a href="Hochschild_cohomology" class="mw-redirect" title="Hochschild cohomology">Hochschild cohomology</a> of some <a href="Associative_algebra" title="Associative algebra"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is the right derived functor of invariants <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-)^{A}:(A,A){\text{-Bimod}}\to k{\text{-Mod}}}">

<msup>

</mrow>

</msup>

<mtext>-Bimod</mtext>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-)^{A}:(A,A){\text{-Bimod}}\to k{\text{-Mod}}}</annotation>

</semantics>

</math><img src="./ba876948c3ad68ca8b196d36285f18be49a1d0f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.993ex; height:3.176ex;" alt="{\displaystyle (-)^{A}:(A,A){\text{-Bimod}}\to k{\text{-Mod}}}" loading="lazy"> mapping a <a href="Bimodule" title="Bimodule">bimodule</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> to its center, also called its set of invariants <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{A}:=Z(M):=\{m\in M\mid \forall a\in A:am=ma\}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M^{A}:=Z(M):=\{m\in M\mid \forall a\in A:am=ma\}}</annotation>

</semantics>

</math><img src="./d0987083e5bd97b04bb046f8447783cf771f37ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.654ex; height:3.176ex;" alt="{\displaystyle M^{A}:=Z(M):=\{m\in M\mid \forall a\in A:am=ma\}}" loading="lazy"> which is the same as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{A^{e}}(A,M)}">

<msub>

<msup>

</mrow>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{A^{e}}(A,M)}</annotation>

</semantics>

</math><img src="./c14aeb83c1fadedcda5c0dcab3981e7cbcbe8c87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.121ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} _{A^{e}}(A,M)}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{e}:=A\otimes _{k}A^{op}}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A^{e}:=A\otimes _{k}A^{op}}</annotation>

</semantics>

</math><img src="./ef373c109738e8f11b98b0d029a741981d3bf884.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.759ex; height:2.676ex;" alt="{\displaystyle A^{e}:=A\otimes _{k}A^{op}}" loading="lazy"> is the enveloping algebra of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is considered an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,A)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A,A)}</annotation>

</semantics>

</math><img src="./4cf5882c77887d53c393039604d105a6086a5966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (A,A)}" loading="lazy">-bimodule via the usual left and right multiplication). Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle HH^{i}(A,M)=\operatorname {Ext} _{A^{e}}^{i}(A,M)}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle HH^{i}(A,M)=\operatorname {Ext} _{A^{e}}^{i}(A,M)}</annotation>

</semantics>

</math><img src="./9a9938ce0da1d040c32a7b1e4834763c91a70d25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.088ex; height:3.343ex;" alt="{\displaystyle HH^{i}(A,M)=\operatorname {Ext} _{A^{e}}^{i}(A,M)}" loading="lazy">:</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Tor_functors"><a href="Tor_functor" title="Tor functor">Tor functors</a></h4></div>

The category of left <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-modules also has enough projectives. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a fixed right <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-module, then the <a href="Tensor_product" title="Tensor product">tensor product</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> gives a right exact covariant functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\otimes _{R}-:R{\text{-Mod}}\to Ab}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\otimes _{R}-:R{\text{-Mod}}\to Ab}</annotation>

</semantics>

</math><img src="./33558fe5872bc244629934595fc52692aad18c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.288ex; height:2.509ex;" alt="{\displaystyle A\otimes _{R}-:R{\text{-Mod}}\to Ab}" loading="lazy">; The category of modules has enough projectives so that left derived functors always exists. The left derived functors of the tensor functor are the <a href="Tor_functor" title="Tor functor">Tor functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Tor} _{i}^{R}(A,-)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tor} _{i}^{R}(A,-)}</annotation>

</semantics>

</math><img src="./6cf61f9f5b97ff8d0e7deaa1d4fbccd6c8b5a6d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.626ex; height:3.176ex;" alt="{\displaystyle \operatorname {Tor} _{i}^{R}(A,-)}" loading="lazy">. Equivalently <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Tor} _{i}^{R}(-,B)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tor} _{i}^{R}(-,B)}</annotation>

</semantics>

</math><img src="./d4069cb209b5b36c2025668a54c493db7103ac0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.647ex; height:3.176ex;" alt="{\displaystyle \operatorname {Tor} _{i}^{R}(-,B)}" loading="lazy"> can be defined symmetrically as the left derived functors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\otimes B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\otimes B}</annotation>

</semantics>

</math><img src="./79019b9d3aedc5339502b682cbb9b92f2ee35b6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.413ex; height:2.343ex;" alt="{\displaystyle -\otimes B}" loading="lazy">. In fact one can combine both definitions and define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Tor} _{i}^{R}(-,-)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tor} _{i}^{R}(-,-)}</annotation>

</semantics>

</math><img src="./0407322fbbf37c07b3521b4280dbbc79afb58d8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.691ex; height:3.176ex;" alt="{\displaystyle \operatorname {Tor} _{i}^{R}(-,-)}" loading="lazy"> as the left derived of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\otimes -:{\text{Mod-}}R\times R{\text{-Mod}}\to Ab}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\otimes -:{\text{Mod-}}R\times R{\text{-Mod}}\to Ab}</annotation>

</semantics>

</math><img src="./6acb56fd338032c9d3921ae06b6fd0d9415bfbe0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:31.838ex; height:2.343ex;" alt="{\displaystyle -\otimes -:{\text{Mod-}}R\times R{\text{-Mod}}\to Ab}" loading="lazy">.

This includes several notions of homology as special cases. This often mirrors the situation with Ext functors and cohomology.

<ul><li><a href="Group_homology" class="mw-redirect" title="Group homology">Group homology</a> is the left derived functor of taking coinvariants <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-)_{G}:k[G]{\text{-Mod}}\to k{\text{-Mod}}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-)_{G}:k[G]{\text{-Mod}}\to k{\text{-Mod}}}</annotation>

</semantics>

</math><img src="./f04b5aea7ab319ed1f34c1bc1f64111a40702209.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.957ex; height:2.843ex;" alt="{\displaystyle (-)_{G}:k[G]{\text{-Mod}}\to k{\text{-Mod}}}" loading="lazy"> which is the same as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\otimes _{k[G]}-}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\otimes _{k[G]}-}</annotation>

</semantics>

</math><img src="./c2347a6994bc87a0be4404a8791aeffe30173d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.155ex; height:3.009ex;" alt="{\displaystyle k\otimes _{k[G]}-}" loading="lazy">.</li>

<li><a href="Lie_algebra_homology" class="mw-redirect" title="Lie algebra homology">Lie algebra homology</a> is the left derived functor of taking coinvariants <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}{\text{-Mod}}\to k{\text{-Mod}},M\mapsto M/[{\mathfrak {g}},M]}">

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}{\text{-Mod}}\to k{\text{-Mod}},M\mapsto M/[{\mathfrak {g}},M]}</annotation>

</semantics>

</math><img src="./10ea68aaf398f81f3affacd1efebbd90d894a499.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.355ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {g}}{\text{-Mod}}\to k{\text{-Mod}},M\mapsto M/[{\mathfrak {g}},M]}" loading="lazy"> which is the same as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\otimes _{U({\mathfrak {g}})}-}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\otimes _{U({\mathfrak {g}})}-}</annotation>

</semantics>

</math><img src="./574f90bc6b3930183ab53be3b92d66786d457958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.46ex; height:3.009ex;" alt="{\displaystyle k\otimes _{U({\mathfrak {g}})}-}" loading="lazy">.</li>

<li><a href="Hochschild_homology" title="Hochschild homology">Hochschild homology</a> is the left derived functor of taking coinvariants <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,A){\text{-Bimod}}\to k{\text{-Mod}},M\mapsto M/[A,M]}">

<mtext>-Bimod</mtext>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A,A){\text{-Bimod}}\to k{\text{-Mod}},M\mapsto M/[A,M]}</annotation>

</semantics>

</math><img src="./83c18b07234f5195adb185affe7d3dfd8e466f8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.181ex; height:2.843ex;" alt="{\displaystyle (A,A){\text{-Bimod}}\to k{\text{-Mod}},M\mapsto M/[A,M]}" loading="lazy"> which is the same as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\otimes _{A^{e}}-}">

<msub>

<msup>

</mrow>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\otimes _{A^{e}}-}</annotation>

</semantics>

</math><img src="./cf2725d1f6cbe14b698043e90bb800bc4c95ea10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.643ex; height:2.509ex;" alt="{\displaystyle A\otimes _{A^{e}}-}" loading="lazy">.</li></ul>

Instead of taking individual left derived functors one can also take the total derived functor of the tensor functor. This gives rise to the <a href="Derived_tensor_product" title="Derived tensor product">derived tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\otimes ^{L}-:D({\text{Mod-}}R)\times D(R{\text{-Mod}})\to D(Ab)}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\otimes ^{L}-:D({\text{Mod-}}R)\times D(R{\text{-Mod}})\to D(Ab)}</annotation>

</semantics>

</math><img src="./d515e12a7c17e3faaae1d062153a0f62e6153453.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.39ex; height:3.176ex;" alt="{\displaystyle -\otimes ^{L}-:D({\text{Mod-}}R)\times D(R{\text{-Mod}})\to D(Ab)}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> is the <a href="Derived_category" title="Derived category">derived category</a>.

<div class="mw-heading mw-heading2"><h2 id="Naturality">Naturality</h2></div>

Derived functors and the long exact sequences are "natural" in several technical senses.

First, given a <a href="Commutative_diagram" title="Commutative diagram">commutative diagram</a> of the form

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mover>

<msub>

</mrow>

</msub>

</mpadded>

</mover>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mover>

<msub>

</mrow>

</msub>

</mpadded>

</mover>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mover>

<msub>

</mrow>

</msub>

</mpadded>

</mover>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mover>

<msub>

</mrow>

</msub>

</mpadded>

</mover>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ccccccccc}0&\to &A_{1}&{\xrightarrow {f_{1}}}&B_{1}&{\xrightarrow {g_{1}}}&C_{1}&\to &0\\&&\alpha \downarrow \quad &&\beta \downarrow \quad &&\gamma \downarrow \quad &&\\0&\to &A_{2}&{\xrightarrow {f_{2}}}&B_{2}&{\xrightarrow {g_{2}}}&C_{2}&\to &0\end{array}}}</annotation>

</semantics>

</math><img src="./82876ecee7b15062aa4f6527d6bede7561773f2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; margin-top: -0.437ex; width:48.677ex; height:12.343ex;" alt="{\displaystyle {\begin{array}{ccccccccc}0&\to &A_{1}&{\xrightarrow {f_{1}}}&B_{1}&{\xrightarrow {g_{1}}}&C_{1}&\to &0\\&&\alpha \downarrow \quad &&\beta \downarrow \quad &&\gamma \downarrow \quad &&\\0&\to &A_{2}&{\xrightarrow {f_{2}}}&B_{2}&{\xrightarrow {g_{2}}}&C_{2}&\to &0\end{array}}}" loading="lazy"></dd></dl>

(where the rows are exact), the two resulting long exact sequences are related by commuting squares:

Second, suppose η : F → G is a <a href="Natural_transformation" title="Natural transformation">natural transformation</a> from the left exact functor F to the left exact functor G. Then natural transformations Riη : RiF → RiG are induced, and indeed Ri becomes a functor from the <a href="Functor_category" title="Functor category">functor category</a> of all left exact functors from A to B to the full functor category of all functors from A to B. Furthermore, this functor is compatible with the long exact sequences in the following sense: if

<mover>

</mpadded>

</mover>

</mrow>

<mover>

</mpadded>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\to A{\xrightarrow {f}}B{\xrightarrow {g}}C\to 0}</annotation>

</semantics>

</math><img src="./f9d536024e8eedc43873c9bc607d3fb41c33f4b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.327ex; width:19.474ex; height:3.843ex;" alt="{\displaystyle 0\to A{\xrightarrow {f}}B{\xrightarrow {g}}C\to 0}" loading="lazy"></dd></dl>

is a short exact sequence, then a commutative diagram

is induced.

Both of these naturalities follow from the naturality of the sequence provided by the <a href="Snake_lemma" title="Snake lemma">snake lemma</a>.

Conversely, the following characterization of derived functors holds: given a family of functors Ri: A → B, satisfying the above, i.e. mapping short exact sequences to long exact sequences, such that for every injective object I of A, Ri(I)=0 for every positive i, then these functors are the right derived functors of R0.

<div class="mw-heading mw-heading2"><h2 id="Generalization">Generalization</h2></div>

The more modern (and more general) approach to derived functors uses the language of <a href="Derived_category" title="Derived category">derived categories</a>.

In 1968 <a href="Daniel_Quillen" title="Daniel Quillen">Quillen</a> developed the theory of <a href="Model_category" title="Model category">model categories</a>, which give an abstract category-theoretic system of fibrations, cofibrations and weak equivalences. Typically one is interested in the underlying <a href="Homotopy_category" title="Homotopy category">homotopy category</a> obtained by localizing against the weak equivalences. A <a href="Quillen_adjunction" title="Quillen adjunction">Quillen adjunction</a> is an adjunction between model categories that descends to an adjunction between the homotopy categories. For example, the category of topological spaces and the category of simplicial sets both admit Quillen model structures whose <a href="Simplicial_set" title="Simplicial set">nerve and realization</a> adjunction gives a Quillen adjunction that is in fact an equivalence of homotopy categories. Particular objects in a model structure have “nice properties” (concerning the existence of lifts against particular morphisms), the “fibrant” and “cofibrant” objects, and every object is weakly equivalent to a fibrant-cofibrant “resolution.”

Although originally developed to handle the category of topological spaces Quillen model structures appear in numerous places in mathematics; in particular the category of chain complexes from any Abelian category (modules, sheaves of modules on a topological space or <a href="Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a>, etc.) admit a model structure whose weak equivalences are those morphisms between chain complexes preserving homology. Often we have a functor between two such model categories (e.g. the global sections functor sending a complex of Abelian sheaves to the obvious complex of Abelian groups) that preserves weak equivalences within the subcategory of “good” (fibrant or cofibrant) objects. By first taking a fibrant or cofibrant resolution of an object and then applying that functor, we have successfully extended it to the whole category in such a way that weak equivalences are always preserved (and hence it descends to a functor from the homotopy category). This is the “derived functor.” The “derived functors” of sheaf cohomology, for example, are the homologies of the output of this derived functor. Applying these to a sheaf of Abelian groups interpreted in the obvious way as a complex concentrated in homology, they measure the failure of the global sections functor to preserve weak equivalences of such, its failure of “exactness.” General theory of model structures shows the uniqueness of this construction (that it does not depend of choice of fibrant or cofibrant resolution, etc.)

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFManinGelfand2003" class="citation cs2"><a href="Yuri_Ivanovich_Manin" class="mw-redirect" title="Yuri Ivanovich Manin">Manin, Yuri Ivanovich</a>; Gelfand, Sergei I. (2003), Methods of Homological Algebra, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-43583-9</bdi></cite></li>

<li><cite id="CITEREFWeibel1994" class="citation book cs1"><a href="Charles_Weibel" title="Charles Weibel">Weibel, Charles A.</a> (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-55987-4</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1269324">1269324</a>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/36131259">36131259</a>.</cite></li></ul>

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</style><div id="Functor_types17" style="font-size:114%;margin:0 4em"><a href="Functor" title="Functor">Functor</a> types</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Additive_functor" class="mw-redirect" title="Additive functor">Additive</a></li>

<li><a href="Adjoint_functors" title="Adjoint functors">Adjoint</a></li>

<li><a href="Conservative_functor" title="Conservative functor">Conservative</a></li>

<li><a href="Diagonal_functor" title="Diagonal functor">Diagonal</a></li>

<li><a href="Enriched_functor" class="mw-redirect" title="Enriched functor">Enriched</a></li>

<li><a href="Essentially_surjective_functor" title="Essentially surjective functor">Essentially surjective</a></li>

<li><a href="Exact_functor" title="Exact functor">Exact</a></li>

<li><a href="Forgetful_functor" title="Forgetful functor">Forgetful</a></li>

<li><a href="Full_and_faithful_functors" title="Full and faithful functors">Full and faithful</a></li>

<li><a href="Logical_functor" class="mw-redirect" title="Logical functor">Logical</a></li>

<li><a href="Monoidal_functor" title="Monoidal functor">Monoidal</a></li>

<li><a href="Representable_functor" title="Representable functor">Representable</a></li>

<li><a href="Smooth_functor" title="Smooth functor">Smooth</a></li></ul>

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