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? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> Showing changes from revision #45 to #46: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='homological_algebra'>Homological algebra</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/homological+algebra'>homological algebra</a> (also <a class='existingWikiWord' href='/nlab/show/diff/nonabelian+homological+algebra'>nonabelian homological algebra</a>) <a class='existingWikiWord' href='/schreiber/show/diff/Introduction+to+Homological+Algebra' title='schreiber'>Introduction</a> Context <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/additive+and+abelian+categories'>additive and abelian categories</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Ab-enriched+category'>Ab-enriched category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/pre-additive+category'>pre-additive category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/additive+category'>additive category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/pre-abelian+category'>pre-abelian category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/abelian+category'>abelian category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+category'>Grothendieck category</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf'>abelian sheaves</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/semi-abelian+category'>semi-abelian category</a> </li> </ul> Basic definitions <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/kernel'>kernel</a>, <a class='existingWikiWord' href='/nlab/show/diff/cokernel'>cokernel</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/complex'>complex</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential'>differential</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homology'>homology</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/category+of+chain+complexes'>category of chain complexes</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/chain+map'>chain map</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/chain+homotopy'>chain homotopy</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/chain+homology+and+cohomology'>chain homology and cohomology</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homological+resolution'>homological resolution</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cochain+on+a+simplicial+set'>simplicial homology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/generalized+homology'>generalized homology</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>exact sequence</a>, <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>short exact sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>long exact sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/split+exact+sequence'>split exact sequence</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/injective+object'>injective object</a>, <a class='existingWikiWord' href='/nlab/show/diff/projective+object'>projective object</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/projective+resolution'>injective resolution</a>, <a class='existingWikiWord' href='/nlab/show/diff/projective+resolution'>projective resolution</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/flat+resolution'>flat resolution</a> </li> </ul> </li> </ul> Stable homotopy theory notions <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/derived+category'>derived category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/triangulated+category'>triangulated category</a>, <a class='existingWikiWord' href='/nlab/show/diff/enhanced+triangulated+category'>enhanced triangulated category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/stable+model+category'>stable model category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/pretriangulated+dg-category'>pretriangulated dg-category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/A-infinity-category'>A-∞-category</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+chain+complexes'>(∞,1)-category of chain complexes</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>derived functor</a>, <a class='existingWikiWord' href='/nlab/show/diff/derived+functor+in+homological+algebra'>derived functor in homological algebra</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Tor'>Tor</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/lim%5E1+and+Milnor+sequences'>lim^1 and Milnor sequences</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fr-code'>fr-code</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a> </li> </ul> Constructions <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/double+complex'>double complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Koszul-Tate+resolution'>Koszul-Tate resolution</a>, <a class='existingWikiWord' href='/nlab/show/diff/BV-BRST+formalism'>BRST-BV complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence'>spectral sequence</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence+of+a+filtered+complex'>spectral sequence of a filtered complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence+of+a+double+complex'>spectral sequence of a double complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+spectral+sequence'>Grothendieck spectral sequence</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Leray+spectral+sequence'>Leray spectral sequence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Serre+spectral+sequence'>Serre spectral sequence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hochschild-Serre+spectral+sequence'>Hochschild-Serre spectral sequence</a> </li> </ul> </li> </ul> </li> </ul> Lemmas <a class='existingWikiWord' href='/nlab/show/diff/diagram+chasing'>diagram chasing</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/3x3+lemma'>3x3 lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/four+lemma'>four lemma</a>, <a class='existingWikiWord' href='/nlab/show/diff/five+lemma'>five lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/snake+lemma'>snake lemma</a>, <a class='existingWikiWord' href='/nlab/show/diff/connecting+homomorphism'>connecting homomorphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/horseshoe+lemma'>horseshoe lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Baer%27s+criterion'>Baer's criterion</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/Schanuel%27s+lemma'>Schanuel's lemma</a> Homology theories <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/singular+homology'>singular homology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cyclic+homology'>cyclic homology</a> </li> </ul> Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Dold-Kan+correspondence'>Dold-Kan correspondence</a> / <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal</a>, <a class='existingWikiWord' href='/nlab/show/diff/operadic+Dold-Kan+correspondence'>operadic</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Moore+complex'>Moore complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/Alexander-Whitney+map'>Alexander-Whitney map</a>, <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Zilber+map'>Eilenberg-Zilber map</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Zilber+theorem'>Eilenberg-Zilber theorem</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cochain+on+a+simplicial+set'>cochain on a simplicial set</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/universal+coefficient+theorem'>universal coefficient theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/K%C3%BCnneth+theorem'>Künneth theorem</a> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a><ul><li><a href='#ToolboxInStableHomotopyTheory'>As a toolbox in stable homotopy theory</a></li><li><a href='#nonabelian_variants'>Non-abelian variants</a></li></ul></li><li><a href='#entries_in_homological_algebra'>Entries in homological algebra</a></li><li><a href='#References'>References</a><ul><li><a href='#ReferencesGeneral'>General</a></li><li><a href='#LectureNotes'>Lecture notes and course notes</a></li><li><a href='#in_constructive_mathematics'>In constructive mathematics</a></li><li><a href='#ReferencesHistory'>History</a></li></ul></li></ul></div> <h2 id='idea'>Idea</h2> In an <a class='existingWikiWord' href='/nlab/show/diff/abelian+category'>abelian category</a> <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math>, homological algebra is the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a> of <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complexes</a> in <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> up to <a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphism of chain complexes</a>. Hence it is the study of the <a class='existingWikiWord' href='/nlab/show/diff/simplicial+localization'>(infinity,1)-categorical localization</a> of the <a class='existingWikiWord' href='/nlab/show/diff/category+of+chain+complexes'>category of chain complexes</a> at the class of <a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphisms</a>, or in other words the <a class='existingWikiWord' href='/nlab/show/diff/derived+category'>derived (infinity,1)-category</a> of <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math>. When considering nonnegatively graded chain complexes, homological algebra may be viewed as a linearized version of the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a> of <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy types</a> or <a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>infinity-groupoids</a>, by the <a class='existingWikiWord' href='/nlab/show/diff/Dold-Kan+correspondence'>Dold-Kan correspondence</a>. When considering unbounded chain complexes, it may be viewed as a linearized and stabilized version, by the <a class='existingWikiWord' href='/nlab/show/diff/stable+Dold-Kan+correspondence'>stable Dold-Kan correspondence</a>. Conversely, we may view <a class='existingWikiWord' href='/nlab/show/diff/homotopical+algebra'>homotopical algebra</a> as a nonabelian generalization of homological algebra. Hence homological algebra is <ul> <li> The study of a particularly simple sort of <a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-categories</a>, namely those derived from categories of chain complexes. See <a href='#ToolboxInStableHomotopyTheory'>As a toolbox in stable homotopy theory</a> below and the discussion at <a class='existingWikiWord' href='/nlab/show/diff/cosmic+cube'>cosmic cube</a>. </li> <li> The study of properties of <a class='existingWikiWord' href='/nlab/show/diff/module'>modules</a> over <a class='existingWikiWord' href='/nlab/show/diff/ring'>rings</a> of various types, by the use of methods adapted from <a class='existingWikiWord' href='/nlab/show/diff/topology'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/generalized+homology'>homology theory</a>. </li> <li> A simple fragment of, and toolbox for, <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable homotopy theory</a> — and hence, by extension, unstable <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>. From this point of view, an archetypical motivating example is the <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complex</a> <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C_\bullet(X)</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/singular+cohomology'>singular chains</a> in a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, whose <a class='existingWikiWord' href='/nlab/show/diff/chain+homology+and+cohomology'>chain homology</a> is the <a class='existingWikiWord' href='/nlab/show/diff/singular+homology'>singular homology</a> <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H_\bullet(X)</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, which is a linear approximation to the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy groups</a> of <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. Accordingly, <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C_\bullet(X)</annotation></semantics></math> itself serves as a linearized approximation to the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a> of <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. </li> </ul> <h3 id='ToolboxInStableHomotopyTheory'>As a toolbox in stable homotopy theory</h3> With homological algebra being a topic in stabilized <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>, it is really the study of <a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-categories</a> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+chain+complexes'>of chain complexes</a> – and thus, by the stable <a href='module%20spectrum#StableDoldKanCorrespondence'>Dold-Kan correspondence</a>, of Eilenberg-MacLane <a class='existingWikiWord' href='/nlab/show/diff/module+spectrum'>module spectra</a>. Historically this modern perspective has developed only in stages out of more “concrete” (more <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>1-categorical</a>) notions, which now form the body of homological algebra, in the form of a box of tools for computing linearized problems in homotopy theory. The following list indicates how these traditional notions serve to present constructions in stable homotopy theory. <ol> <li> The notion of <a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphism</a> between chain complexes – <a class='existingWikiWord' href='/nlab/show/diff/chain+map'>chain maps</a> which induce <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphisms</a> on <a class='existingWikiWord' href='/nlab/show/diff/homology'>homology groups</a> – is the stable version of <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalences</a> of topological spaces. The <a class='existingWikiWord' href='/nlab/show/diff/derived+category'>derived category</a> of chain complexes <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>D(\mathcal{A})</annotation></semantics></math> obtained by <a class='existingWikiWord' href='/nlab/show/diff/localization'>localizing</a> <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch_\bullet(\mathcal{A})</annotation></semantics></math> at these <a class='existingWikiWord' href='/nlab/show/diff/weak+equivalence'>weak equivalences</a> is the corresponding <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a>, the context where all <a class='existingWikiWord' href='/nlab/show/diff/chain+map'>chain maps</a> are identified up to <a class='existingWikiWord' href='/nlab/show/diff/chain+homotopy'>chain homotopy</a> between good representatives of these objects. (On the other hand, in more general situations this correspondence is less immediate, and the notion of quasi-isomorphism may not be the best choice; see at <a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead theorem</a>.) </li> <li> By the discussion at <a class='existingWikiWord' href='/nlab/show/diff/localization'>localization</a> the morphisms in <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>D(\mathcal{A})</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/diff/zigzag'>zig-zags</a> of <a class='existingWikiWord' href='/nlab/show/diff/chain+map'>chain maps</a> that involve <a class='existingWikiWord' href='/nlab/show/diff/resolution'>resolutions</a> by non-isomorphic but <a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphic</a> chain complexes. By the various <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+chain+complexes'>model structures on chain complexes</a> these resolutions can concretely be constructed as <a class='existingWikiWord' href='/nlab/show/diff/projective+resolution'>injective resolutions</a>, <a class='existingWikiWord' href='/nlab/show/diff/projective+resolution'>projective resolutions</a> and/or more general sorts of resolutions (such as <a class='existingWikiWord' href='/nlab/show/diff/flat+resolution'>flat resolutions</a>, soft, flabby, etc.) of chain complexes, and much of the theory revolves around handling these. </li> <li> Notably, <a class='existingWikiWord' href='/nlab/show/diff/functor'>functors</a> between categories of chain complexes may extend to functors on these derived categories by evaluating them on suitable resolutions – accordingly called <a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>derived functors</a>. (In homological algebra, the phrase “derived functor” is traditionally applied to the homology groups of what abstract homotopy theory calls the “derived functor”, these being the invariants that one can compute.) Much of the theory revolves around computing and characterizing derived functors, for instance in the definition of <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a> and hence there are powerful tools for these computations, notably <a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence'>spectral sequences</a>. </li> <li> However, the <a class='existingWikiWord' href='/nlab/show/diff/derived+category'>derived category</a> <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>D(\mathcal{A})</annotation></semantics></math> is still a rather coarse approximation to the full <a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+chain+complexes'>of chain complexes</a> in <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math>. There is a series of extra property and structures added to it which gives better approximations, and large parts of modern homological algebra study these: First of all the derived category is automatically a <a class='existingWikiWord' href='/nlab/show/diff/triangulated+category'>triangulated category</a>, which is a means of remembering the operation of <a class='existingWikiWord' href='/nlab/show/diff/suspension'>suspension</a> and de-suspension (<a class='existingWikiWord' href='/nlab/show/diff/looping'>looping</a>) of chain complexes. Further structure added to these goes by names such as <a class='existingWikiWord' href='/nlab/show/diff/enhanced+triangulated+category'>enhanced triangulated category</a>. A <a class='existingWikiWord' href='/nlab/show/diff/stable+derivator'>stable derivator</a> is a strong enhancement which encodes basically all the requisite structure for internal computations. Finally, the further promotion of these to <a class='existingWikiWord' href='/nlab/show/diff/stable+model+category'>stable model categories</a> or <a class='existingWikiWord' href='/nlab/show/diff/pretriangulated+dg-category'>pretriangulated dg-categories</a>/linear <a class='existingWikiWord' href='/nlab/show/diff/A-infinity-category'>A-∞ categories</a> of chain complexes makes them capture the full information present in the <a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a>. </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/algebra'>Algebra</a> in <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable homotopy theory</a> is <a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a> over <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ rings</a>, and homological algebra provides approximations to that: by the <a class='existingWikiWord' href='/nlab/show/diff/stable+Dold-Kan+correspondence'>stable Dold-Kan correspondence</a> <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complexes</a> of <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/module'>modules</a> are a presentation for <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Mac+Lane+spectrum'>HR</a>-<a class='existingWikiWord' href='/nlab/show/diff/module+spectrum'>module spectra</a>. Moreover, <a class='existingWikiWord' href='/nlab/show/diff/A-infinity-algebra'>A-infinity algebras</a> in <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>HR</mi></mrow><annotation encoding='application/x-tex'>HR</annotation></semantics></math>-module spectra <a href='http://www.ncatlab.org/nlab/show/model+structure+on+dg-algebras#RelationToAInfinityAlgebras'>are presented by</a> <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebras</a>, hence by ordinary <a class='existingWikiWord' href='/nlab/show/diff/associative+unital+algebra'>associative algebras</a> in <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complexes</a>. Similarly <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+algebra'>E-infinity algebras</a> <a href='http://www.ncatlab.org/nlab/show/model+structure+on+dg-algebras#RelationToEInfinityAlgebras'>are presented by</a> <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>commutative dg-algebras</a>, hence by <a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid'>commutative algebras</a> internal to chain complexes. By variation of this theme a multitude of notions in <a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a> finds their representation in homological algebra, for instance <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebras</a> in terms of <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+Lie+algebra'>dg-Lie algebras</a>: <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra'>Lie algebras</a> internal to chain complexes. </li> </ol> <h3 id='nonabelian_variants'>Non-abelian variants</h3> There are variants of the tools of homological algebra that can also be applied to more non-linear phenomena, see for instance at <a class='existingWikiWord' href='/nlab/show/diff/Dold-Kan+correspondence'>Dold-Kan correspondence</a> the section <a href='Dold-Kan%20correspondence#StatementGeneral'>non-abelian case</a>. These include non-Abelian (co)homology and crossed and quadratic versions that use a small degree of non-linearity in the models. These latter theories make extensive use of techniques from <a class='existingWikiWord' href='/nlab/show/diff/homotopical+algebra'>homotopical algebra</a> in the wide sense of that term and <a class='existingWikiWord' href='/nlab/show/diff/simplicial+homotopy+theory'>simplicial homotopy theory</a> to avoid the crushing of homotopical information that can occur when passing to chain complexes. <h2 id='entries_in_homological_algebra'>Entries in homological algebra</h2> <a class='existingWikiWord' href='/nlab/show/diff/homological+algebra'>homological algebra</a> (also <a class='existingWikiWord' href='/nlab/show/diff/nonabelian+homological+algebra'>nonabelian homological algebra</a>) <a class='existingWikiWord' href='/schreiber/show/diff/Introduction+to+Homological+Algebra' title='schreiber'>Introduction</a> Context <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/additive+and+abelian+categories'>additive and abelian categories</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Ab-enriched+category'>Ab-enriched category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/pre-additive+category'>pre-additive category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/additive+category'>additive category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/pre-abelian+category'>pre-abelian category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/abelian+category'>abelian category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+category'>Grothendieck category</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf'>abelian sheaves</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/semi-abelian+category'>semi-abelian category</a> </li> </ul> Basic definitions <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/kernel'>kernel</a>, <a class='existingWikiWord' href='/nlab/show/diff/cokernel'>cokernel</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/complex'>complex</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential'>differential</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homology'>homology</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/category+of+chain+complexes'>category of chain complexes</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/chain+map'>chain map</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/chain+homotopy'>chain homotopy</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/chain+homology+and+cohomology'>chain homology and cohomology</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homological+resolution'>homological resolution</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cochain+on+a+simplicial+set'>simplicial homology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/generalized+homology'>generalized homology</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>exact sequence</a>, <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>short exact sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>long exact sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/split+exact+sequence'>split exact sequence</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/injective+object'>injective object</a>, <a class='existingWikiWord' href='/nlab/show/diff/projective+object'>projective object</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/projective+resolution'>injective resolution</a>, <a class='existingWikiWord' href='/nlab/show/diff/projective+resolution'>projective resolution</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/flat+resolution'>flat resolution</a> </li> </ul> </li> </ul> Stable homotopy theory notions <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/derived+category'>derived category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/triangulated+category'>triangulated category</a>, <a class='existingWikiWord' href='/nlab/show/diff/enhanced+triangulated+category'>enhanced triangulated category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/stable+model+category'>stable model category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/pretriangulated+dg-category'>pretriangulated dg-category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/A-infinity-category'>A-∞-category</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+chain+complexes'>(∞,1)-category of chain complexes</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>derived functor</a>, <a class='existingWikiWord' href='/nlab/show/diff/derived+functor+in+homological+algebra'>derived functor in homological algebra</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Tor'>Tor</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/lim%5E1+and+Milnor+sequences'>lim^1 and Milnor sequences</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fr-code'>fr-code</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a> </li> </ul> Constructions <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/double+complex'>double complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Koszul-Tate+resolution'>Koszul-Tate resolution</a>, <a class='existingWikiWord' href='/nlab/show/diff/BV-BRST+formalism'>BRST-BV complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence'>spectral sequence</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence+of+a+filtered+complex'>spectral sequence of a filtered complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence+of+a+double+complex'>spectral sequence of a double complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+spectral+sequence'>Grothendieck spectral sequence</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Leray+spectral+sequence'>Leray spectral sequence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Serre+spectral+sequence'>Serre spectral sequence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hochschild-Serre+spectral+sequence'>Hochschild-Serre spectral sequence</a> </li> </ul> </li> </ul> </li> </ul> Lemmas <a class='existingWikiWord' href='/nlab/show/diff/diagram+chasing'>diagram chasing</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/3x3+lemma'>3x3 lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/four+lemma'>four lemma</a>, <a class='existingWikiWord' href='/nlab/show/diff/five+lemma'>five lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/snake+lemma'>snake lemma</a>, <a class='existingWikiWord' href='/nlab/show/diff/connecting+homomorphism'>connecting homomorphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/horseshoe+lemma'>horseshoe lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Baer%27s+criterion'>Baer's criterion</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/Schanuel%27s+lemma'>Schanuel's lemma</a> Homology theories <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/singular+homology'>singular homology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cyclic+homology'>cyclic homology</a> </li> </ul> Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Dold-Kan+correspondence'>Dold-Kan correspondence</a> / <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal</a>, <a class='existingWikiWord' href='/nlab/show/diff/operadic+Dold-Kan+correspondence'>operadic</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Moore+complex'>Moore complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/Alexander-Whitney+map'>Alexander-Whitney map</a>, <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Zilber+map'>Eilenberg-Zilber map</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Zilber+theorem'>Eilenberg-Zilber theorem</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cochain+on+a+simplicial+set'>cochain on a simplicial set</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/universal+coefficient+theorem'>universal coefficient theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/K%C3%BCnneth+theorem'>Künneth theorem</a> </li> </ul> <h2 id='References'>References</h2> The following lists references on homological algebra: <ul> <li> <a href='#ReferencesGeneral'>General</a> </li> <li> <a href='#LectureNotes'>Lecture notes and course notes</a> </li> <li> <a href='#ReferencesHistory'>History</a> </li> </ul> <h3 id='ReferencesGeneral'>General</h3> Classical accounts: <ul> <ins class='diffins'><li id='EilenbergMacLane42'> <a class='existingWikiWord' href='/nlab/show/diff/Samuel+Eilenberg'>Samuel Eilenberg</a>, <a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders MacLane</a>, Group Extensions and Homology, Annals of Mathematics 43 4 (1942) 757-831 [[doi:10.2307/1968966](https://doi.org/10.2307/1968966), <a href='https://www.jstor.org/stable/1968966'>jstor:1968966</a>] </li></ins><ins class='diffins'> </ins><li> D. A. Buchsbaum, Exact categories and duality, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 (<a href='http://www.jstor.org/stable/1993003'>JSTOR</a>) </li> <li id='CartanEilenberg'> <a class='existingWikiWord' href='/nlab/show/diff/Henri+Cartan'>Henri Cartan</a>, <a class='existingWikiWord' href='/nlab/show/diff/Samuel+Eilenberg'>Samuel Eilenberg</a>, <del class='diffmod'>Homological algebra</del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Homological+Algebra'>Homological Algebra</a></ins> , Princeton Univ. Press<del class='diffmod'> (1956)</del><ins class='diffmod'> (1956),</ins><del class='diffmod'> (</del><ins class='diffmod'> Princeton</ins><ins class='diffins'> Mathematical</ins><ins class='diffins'> Series</ins><del class='diffmod'><a href='https://press.princeton.edu/books/paperback/9780691049915/homological-algebra-pms-19-volume-19'>ISBN:9780691049915</a></del><ins class='diffmod'>19</ins><del class='diffmod'> ,</del><ins class='diffmod'> </ins><ins class='diffins'> (1999)</ins><ins class='diffins'> [[ISBN:9780691049915](https://press.princeton.edu/books/paperback/9780691049915/homological-algebra-pms-19-volume-19),</ins><ins class='diffins'><a href='https://doi.org/10.1515/9781400883844'>doi:10.1515/9781400883844</a></ins><ins class='diffins'>, </ins><a href='http://www.math.stonybrook.edu/~mmovshev/BOOKS/homologicalalgeb033541mbp.pdf'>pdf</a><del class='diffmod'> )</del><ins class='diffmod'> ]</ins> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Alexander+Grothendieck'>A. Grothendieck</a>, <a class='existingWikiWord' href='/nlab/show/diff/T%C3%B4hoku'>Sur quelques points d'algèbre homologique</a> (1957) (<a href='http://projecteuclid.org/euclid.tmj/1178244774'>part1</a>, <a href='http://projecteuclid.org/euclid.tmj/1178244774'>part2</a>) </li> <li id='Godement58'> <a class='existingWikiWord' href='/nlab/show/diff/Roger+Godement'>Roger Godement</a>, Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), <a class='existingWikiWord' href='/nlab/files/Godement-TopologieAlgebrique.pdf' title='pdf'>pdf</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders MacLane</a>, Homology, Grundlehren der Mathematischen Wissenschaften 114, Springer (1963, 1974), reprinted as: Classics in Mathematics, Springer (1995) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/maclane_homology.pdf), <a href='https://doi.org/10.1007/978-3-642-62029-4'>doi:10.1007/978-3-642-62029-4</a>] </li> <li id='HiltonStammbach71'> <a class='existingWikiWord' href='/nlab/show/diff/Peter+Hilton'>Peter Hilton</a>, <a class='existingWikiWord' href='/nlab/show/diff/Urs+Stammbach'>Urs Stammbach</a>, A course in homological algebra, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 4 (<a href='https://link.springer.com/book/10.1007/978-1-4419-8566-8'>doi:10.1007/978-1-4419-8566-8</a>, <a href='https://web.math.rochester.edu/people/faculty/doug/otherpapers/hilton-stammbach.pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders Mac Lane</a>, Homology (1975) reprinted as Classics in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp. ISBN 3-540-58662-8 (<a href='https://link.springer.com/book/10.1007/978-3-642-62029-4'>doi:10.1007/978-3-642-62029-4</a>) </li> </ul> A standard modern textbook introduction is <ul> <li id='Weibel94'><a class='existingWikiWord' href='/nlab/show/diff/Charles+Weibel'>Charles Weibel</a>, <a class='existingWikiWord' href='/nlab/show/diff/An+Introduction+to+Homological+Algebra'>An introduction to homological algebra</a>, Cambridge Studies in Adv. Math. 38, Cambridge University Press (1994) [[doi:10.1017/CBO9781139644136](https://doi.org/10.1017/CBO9781139644136), <a href='https://web.math.rochester.edu/people/faculty/doug//otherpapers/weibel-hom.pdf'>pdf</a>]</li> </ul> and a more systematic modern development of the theory is in sections 8 and 12-18 of <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Masaki+Kashiwara'>Masaki Kashiwara</a>, <a class='existingWikiWord' href='/nlab/show/diff/Pierre+Schapira'>Pierre Schapira</a>, <a class='existingWikiWord' href='/nlab/show/diff/Categories+and+Sheaves'>Categories and Sheaves</a>, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006)</li> </ul> Non-abelian variants of homological algebra are disussed for instance in <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Francis+Borceux'>Francis Borceux</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dominique+Bourn'>Dominique Bourn</a>, <a class='existingWikiWord' href='/nlab/show/diff/Malcev%2C+protomodular%2C+homological+and+semi-abelian+categories'>Mal'cev, protomodular, homological and semi-abelian categories</a>, Mathematics and Its Applications 566, Kluwer (2004)</li> </ul> The foundations of the formulation in the broader context of <a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a> theory is in <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, <a class='existingWikiWord' href='/nlab/show/diff/Stable+Infinity-Categories'>Stable Infinity-Categories</a>, (<a href='http://www.arXiv.org/abs/math.CT/0608228'>math.CT/0608228</a>)</li> </ul> Other textbooks include <ul> <li> I. Bucur, A. Deleanu, Introduction to the theory of categories and functors, 1968 </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Sergei+Gelfand'>Sergei Gelfand</a>, <a class='existingWikiWord' href='/nlab/show/diff/Yuri+Manin'>Yuri Manin</a>, <a class='existingWikiWord' href='/nlab/show/diff/Methods+of+homological+algebra'>Methods of homological algebra</a>, transl. from the 1988 Russian (Nauka Publ.) original, Springer (1996, 2002) [[doi:10.1007/978-3-662-12492-5](https://doi.org/10.1007/978-3-662-12492-5)] </li> </ul> See also <ul> <li> Wikipedia, <a href='http://en.wikipedia.org/wiki/Homological_algebra'>Homological algebra</a> </li> <li> Springer Online Encyclopeadia of Mathematics: <a href='http://eom.springer.de/H/h047710.htm'>homological algebra</a> </li> </ul> <h3 id='LectureNotes'>Lecture notes and course notes</h3> <ul id='Collins'> <li> <a class='existingWikiWord' href='/nlab/show/diff/Alexander+Beilinson'>Alexander Beilinson</a>, Introduction to homological algebra (handwritten notes, summer 2007, pdf) <a href='http://www.math.uchicago.edu/~mitya/beilinson/hom-alg-1.pdf'>lec1</a>, <a href='http://www.math.uchicago.edu/~mitya/beilinson/hom-alg-2.pdf'>lec2</a>, <a href='http://www.math.uchicago.edu/~mitya/beilinson/hom-alg-3.pdf'>lec3</a>, <a href='http://www.math.uchicago.edu/~mitya/beilinson/hom-alg-4.pdf'>lec4</a> </li> <li> Julia Collins, Homological algebra (2006)<del class='diffmod'> (</del><ins class='diffmod'> [[pdf](http://www.maths.ed.ac.uk/~s0681349/HomologicalAlgebra.pdf)]</ins><del class='diffdel'><a href='http://www.maths.ed.ac.uk/~s0681349/HomologicalAlgebra.pdf'>pdf</a></del><del class='diffdel'>)</del> </li><ins class='diffins'> </ins><ins class='diffins'><li> <a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>Rick Jardine</a>, Homological algebra, course notes, 2009 (<a href='http://www.math.uwo.ca/~jardine/papers/HomAlg/index.shtml'>index</a>) </li></ins><ins class='diffins'> </ins><ins class='diffins'><li> <a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, Notes on Tor and Ext (<a href='http://www.math.uchicago.edu/~may/MISC/TorExt.pdf'>pdf</a>) </li></ins> </ul> <del class='diffdel'><ul id='May'> <li> <a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>Rick Jardine</a>, Homological algebra, course notes, 2009 (<a href='http://www.math.uwo.ca/~jardine/papers/HomAlg/index.shtml'>index</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, Notes on Tor and Ext (<a href='http://www.math.uchicago.edu/~may/MISC/TorExt.pdf'>pdf</a>) </li> </ul></del><del class='diffdel'> </del><ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Pierre+Schapira'>Pierre Schapira</a>, Categories and homological algebra, lecture notes (2011) (<a href='http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Urs+Schreiber'>Urs Schreiber</a>, <a class='existingWikiWord' href='/schreiber/show/diff/Introduction+to+Homological+Algebra' title='schreiber'>Introduction to Homological Algebra</a> </li> </ul> <h3 id='in_constructive_mathematics'>In constructive mathematics</h3> Discussion of homological algebra in <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive mathematics</a>: <ul> <li id='CoquandSpiwack'> <a class='existingWikiWord' href='/nlab/show/diff/Thierry+Coquand'>Thierry Coquand</a>, <a class='existingWikiWord' href='/nlab/show/diff/Arnaud+Spiwack'>Arnaud Spiwack</a>, Towards constructive homological algebra in type theory, in: Towards Mechanized Mathematical Assistants. MKM Calculemus 2007, Lecture Notes in Computer Science 4573 Springer (2007) [[doi:10.1007/978-3-540-73086-6_4](https://doi.org/10.1007/978-3-540-73086-6_4), <a href='https://hal.inria.fr/inria-00432525/document'>pdf</a>] <blockquote> (via <a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a>) </blockquote> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Julio+Rubio'>Julio Rubio</a>, <a class='existingWikiWord' href='/nlab/show/diff/Francis+Sergeraert'>Francis Sergeraert</a>, Constructive homological algebra and applications [[arXiv:1208.3816](http://arxiv.org/abs/1208.3816)] </li> </ul> <h3 id='ReferencesHistory'>History</h3> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Charles+Weibel'>Charles Weibel</a>, A history of homological algebra, <a href='http://www.math.rutgers.edu/~weibel/HA-history.dvi'>dvi</a></li> </ul> From the introduction of<del class='diffdel'> (</del><a href='#Collins'> Collins<ins class='diffins'> 2006</ins></a><del class='diffmod'> ):</del><ins class='diffmod'> :</ins> <blockquote> The word “<a class='existingWikiWord' href='/nlab/show/diff/homology'>homology</a>” was first used in a <a class='existingWikiWord' href='/nlab/show/diff/topology'>topological</a> context by <a class='existingWikiWord' href='/nlab/show/diff/Henri+Poincar%C3%A9'>Poincaré</a> in 1895, who used it to think about <a class='existingWikiWord' href='/nlab/show/diff/manifold'>manifolds</a> which were the <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundaries</a> of higher-dimensional manifolds. It was <a class='existingWikiWord' href='/nlab/show/diff/Emmy+Noether'>Emmy Noether</a> in the 1920s who began thinking of homology in terms of <a class='existingWikiWord' href='/nlab/show/diff/group'>groups</a>, and who developed algebraic techniques such as the idea of <a class='existingWikiWord' href='/nlab/show/diff/module'>modules</a> over a <a class='existingWikiWord' href='/nlab/show/diff/ring'>ring</a>. These are both absolutely crucial ingredients in the modern theory of homological algebra, yet for the next twenty years homology theory was to remain confined to the realm of topology. In 1942 came the first move forward towards homological algebra as we know it today, with the arrival of a<del class='diffdel'> paper</del><del class='diffdel'> by</del><del class='diffdel'> Samuel</del><del class='diffdel'> Eilenberg</del><del class='diffdel'> and</del><del class='diffdel'> Saunders</del><del class='diffdel'> MacLane.</del><del class='diffdel'> In</del><del class='diffdel'> it</del><del class='diffdel'> we</del><del class='diffdel'> find</del><ins class='diffins'><a href='#EilenbergMacLane42'>paper by Samuel Eilenberg and Saunders MacLane</a></ins><ins class='diffins'>. In it we find </ins><a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>Hom</a> and <a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext</a> defined for the very first time, and along with it the notions of a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> and <a class='existingWikiWord' href='/nlab/show/diff/natural+isomorphism'>natural isomorphism</a>. These were needed to provide a precise language for talking about the properties of <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Hom</mi><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Hom(A,B)</annotation></semantics></math>; in particular the fact that it varies naturally, contravariantly in <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and covariantly in <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>. Only three years later this language<del class='diffdel'> was</del><del class='diffdel'> expanded</del><del class='diffdel'> to</del><del class='diffdel'> include</del><ins class='diffins'><a href='category#EilenbergMacLane45'>was expanded</a></ins><ins class='diffins'> to include </ins><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> and <a class='existingWikiWord' href='/nlab/show/diff/natural+equivalence'>natural equivalence</a>. However, this terminology was not widely accepted by the mathematical community until the appearance of <a href='#CartanEilenberg'>Cartan and Eilenberg’s book</a><del class='diffmod'> </del><ins class='diffmod'> .</ins><del class='diffdel'> in</del><del class='diffdel'> 1956.</del> Cartan and Eilenberg’s book was truly a revolution in the subject, and in fact it was here that the term “Homological Algebra” was first coined. The book used<a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>derived functors</a> in a systematic way which united all the previous <a class='existingWikiWord' href='/nlab/show/diff/generalized+homology'>homology theories</a>, which in the past ten years had arisen in <a class='existingWikiWord' href='/nlab/show/diff/group+theory'>group theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra'>Lie algebras</a> and <a class='existingWikiWord' href='/nlab/show/diff/algebraic+geometry'>algebraic geometry</a>. The sheer list of terms that were first defined in this book may give the reader an idea of how much of this project is due to the existence of that one book! They defined what it means for an object to be <a class='existingWikiWord' href='/nlab/show/diff/projective+object'>projective</a> or <a class='existingWikiWord' href='/nlab/show/diff/injective+object'>injective</a>, and defined the notions of <a class='existingWikiWord' href='/nlab/show/diff/projective+resolution'>projective</a> and <a class='existingWikiWord' href='/nlab/show/diff/projective+resolution'>injective resolutions</a>. It is here that we find the first mention of <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Hom</mi></mrow><annotation encoding='application/x-tex'>Hom</annotation></semantics></math> being <a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>left exact</a> and the first occurrence of <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>Ext^n</annotation></semantics></math> as the <a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>right derived functors</a> of <math class='maruku-mathml' display='inline' id='mathml_97af40568526acca9f9869c31dc96fb123e7493b_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Hom</mi></mrow><annotation encoding='application/x-tex'>Hom</annotation></semantics></math>. </blockquote> </div> <div class="revisedby"> Last revised on August 23, 2023 at 14:17:45. 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