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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title></title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> </div> </div> <h1 id='section_table_of_contents'>Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#construction'>Construction</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> it turns out that a host of common (co)homological constructions (such as <a class="existingWikiWord" href="/nlab/show/group+homology">group homology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a>, etc.) may be cast in a unified way as <a class="existingWikiWord" href="/nlab/show/homotopy+limits">homotopy limits</a> of <a class="existingWikiWord" href="/nlab/show/functors">functors</a> on <a class="existingWikiWord" href="/nlab/show/categories">categories</a> of <a class="existingWikiWord" href="/nlab/show/presentations">presentations</a> of the given algebraic <a class="existingWikiWord" href="/nlab/show/structure">structure</a> (<a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a>, etc.) [<a href="#IvanovMikhailov15">Ivanov & Mikhailov 2015</a>], in fact all these functors may systematically be indexed by “fr-codes” [<a href="#IvanovMikhailov17">Ivanov & Mikhailov 2017</a>].</p> <h2 id="construction">Construction</h2> <p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>Pres</mi><mo>→</mo><mi>Ring</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}F\colon Pres\to Ring</annotation></semantics></math>, a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> of <a class="existingWikiWord" href="/nlab/show/rings">rings</a> on the category of all free <a class="existingWikiWord" href="/nlab/show/extensions">extensions</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>→</mo><mi>R</mi><mo>→</mo><mi>F</mi><mo>→</mo><mi>G</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1\to R\to F\to G\to 1</annotation></semantics></math>, which takes a free <a class="existingWikiWord" href="/nlab/show/extension">extension</a> (<a class="existingWikiWord" href="/nlab/show/group+presentation">free presentations</a>) and sends it to the <a class="existingWikiWord" href="/nlab/show/group+ring">group ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}F</annotation></semantics></math>. There are two functorial <a class="existingWikiWord" href="/nlab/show/ideals">ideals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>f</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{f}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>r</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{r}</annotation></semantics></math> in the (functorial) ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}F</annotation></semantics></math> that are defined as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>f</mi></mstyle><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>ker</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mi>F</mi><mo>→</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{f} \;\coloneqq\; ker(\mathbb{Z}F \to \mathbb{Z}) \,, </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>r</mi></mstyle><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>ker</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mi>F</mi><mo>→</mo><mi>ℤ</mi><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{r} \;\coloneqq\; ker(\mathbb{Z}F \to \mathbb{Z}G) \,. </annotation></semantics></math></div> <p>That is, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>f</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{f}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/augmentation+ideal">augmentation ideal</a> of the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>, and it is generated by expressions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(w - 1)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">w \in F</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>r</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{r}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/subobject">sub</a>-<a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>f</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{f}</annotation></semantics></math> generated by expressions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(r - 1)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math>.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>f</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>r</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{f},\mathbf{r}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/ideals">ideals</a> of the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> of <a class="existingWikiWord" href="/nlab/show/rings">rings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ZF</mi></mrow><annotation encoding="application/x-tex">ZF</annotation></semantics></math>, one may form sums and intersections of monomials: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>rf</mi></mstyle><mo>∩</mo><mstyle mathvariant="bold"><mi>fr</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mrow><mi>r</mi><mo>+</mo><mi>ff</mi></mrow></mstyle><mo>,</mo><msup><mstyle mathvariant="bold"><mi>r</mi></mstyle> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mstyle mathvariant="bold"><mi>f</mi></mstyle><msup><mstyle mathvariant="bold"><mi>r</mi></mstyle> <mi>k</mi></msup><mstyle mathvariant="bold"><mi>f</mi></mstyle><mo>,</mo><mi>…</mi></mrow><annotation encoding="application/x-tex">\mathbf{rf}\cap \mathbf{fr}, \mathbf{r+ff}, \mathbf{r}^{k+1}+\mathbf{f}\mathbf{r}^k\mathbf{f}, \dots</annotation></semantics></math></p> <p>These are <a class="existingWikiWord" href="/nlab/show/functors">functors</a> on the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/free+group">free</a> <a class="existingWikiWord" href="/nlab/show/extensions">extensions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pres</mi></mrow><annotation encoding="application/x-tex">Pres</annotation></semantics></math> with values in <a class="existingWikiWord" href="/nlab/show/Ab">abelian groups</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pres</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>Pres</mi></mrow><annotation encoding="application/x-tex">Pres(G)\subset Pres</annotation></semantics></math> denote a fiber of a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pres</mi><mo>→</mo><mi>Gr</mi></mrow><annotation encoding="application/x-tex">Pres\to Gr</annotation></semantics></math> sending a free <a class="existingWikiWord" href="/nlab/show/extension">extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>→</mo><mi>R</mi><mo>→</mo><mi>F</mi><mo>→</mo><mi>G</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1\to R\to F\to G\to 1</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. Given an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mrow><mi>f</mi><mo lspace="0em" rspace="thinmathspace">r</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{f\r}</annotation></semantics></math>-expression <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>f</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>r</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">w(\mathbf{f},\mathbf{r})</annotation></semantics></math> and a group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, one can define</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>f</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>r</mi></mstyle><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msup><mi>lim</mi> <mi>i</mi></msup><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>f</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>r</mi></mstyle><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><mi>Pres</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">{}^i[w(\mathbf{f},\mathbf{r})](G) := lim^i (w(\mathbf{f},\mathbf{r})|_{Pres(G)}).</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>lim</mi> <mi>i</mi></msup><mo stretchy="false">(</mo><mi>ℱ</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">lim^i (\mathcal{F}: C\to Ab)</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>i</mi> </mrow> <annotation encoding="application/x-tex">i</annotation> </semantics> </math>-th right derived</a> functor of the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mo>:</mo><mi>Fun</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>Ab</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Ab</mi></mrow><annotation encoding="application/x-tex">lim: Fun(C,Ab)\to Ab</annotation></semantics></math>.</p> <p>It turns out that by exploiting some features of the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pres</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pres(G)</annotation></semantics></math> this construction can be made functorial in <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. The first such feature is that it has all binary <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> (in particular, its <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> is <a class="existingWikiWord" href="/nlab/show/contractible+space">contractible</a>). That feature is used in by authors of [<a href="#IvanovMikhailov15">Ivanov & Mikhailov 2015</a>], [<a href="#IvanovMikhailov17">Ivanov & Mikhailov 2017</a>] extensively, since it ensures triviality of <a class="existingWikiWord" href="/nlab/show/derived+limit+functor">higher limits</a> of <a class="existingWikiWord" href="/nlab/show/constant+functors">constant functors</a>.</p> <p>Secondly, this category is strongly connected, in that the hom-set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo lspace="0em" rspace="thinmathspace">sf</mo><mi>hom</mi></mrow><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">{\sf hom}(c,c')</annotation></semantics></math> is not empty for any pair of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">c'</annotation></semantics></math>. Hence, with each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mrow><mi>f</mi><mo lspace="0em" rspace="thinmathspace">r</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{f\r}</annotation></semantics></math>-expression <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>f</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>r</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">w(\mathbf{f},\mathbf{r})</annotation></semantics></math> we associate a graded <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from the category <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a> to the category <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>f</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>r</mi></mstyle><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo lspace="verythinmathspace">:</mo><mi>Gr</mi><mo>→</mo><mi>Ab</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex">{}^i[w(\mathbf{f},\mathbf{r})] \colon Gr \to Ab\,.</annotation></semantics></math></p> <p>In [<a href="#GolubNikita24">Golub 2024</a>] author suggests a homotopy theoretic construction not using the homological algebra.</p> <h2 id="related_entries">Related entries</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/higher+limit+approach+to+homology">higher limit approach to homology</a></li> </ul> <h2 id="references">References</h2> <p>The original articles</p> <ul> <li id="IvanovMikhailov15"> <p><a class="existingWikiWord" href="/nlab/show/Sergei+O.+Ivanov">Sergei O. Ivanov</a>, <a class="existingWikiWord" href="/nlab/show/Roman+Mikhailov">Roman Mikhailov</a>: <em>A higher limit approach to homology theories</em>, Journal of Pure and Applied Algebra <strong>219</strong> 6 (2015) 1915-1939 [<a href="https://arxiv.org/abs/1309.4920">arXiv:1309.4920</a>, <a href="https://doi.org/10.1016/j.jpaa.2014.07.016">doi:10.1016/j.jpaa.2014.07.016</a>]</p> </li> <li id="IvanovMikhailov17"> <p><a class="existingWikiWord" href="/nlab/show/Sergei+O.+Ivanov">Sergei O. Ivanov</a>, <a class="existingWikiWord" href="/nlab/show/Roman+Mikhailov">Roman Mikhailov</a>: <em>Higher limits, homology theories and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>fr</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{fr}</annotation></semantics></math>-codes</em>, in: <em>Combinatorial and Toric Homotopy</em>, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore (2017) 229-261 [<a href="https://arxiv.org/abs/1510.09044">arXiv:1510.09044</a>, <a href="https://doi.org/10.1142/9789813226579_0004">doi:10.1142/9789813226579_0004</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sergei+O.+Ivanov">Sergei O. Ivanov</a>, <a class="existingWikiWord" href="/nlab/show/Roman+Mikhailov">Roman Mikhailov</a>, Fedor Pavutnitskiy: <em>Limits, standard complexes and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>fr</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{fr}</annotation></semantics></math>-codes</em>, Sb. Math. <strong>211</strong> (2020) 1568 [<a href="https://arxiv.org/abs/1906.08793">arXiv:1906.08793</a>, <a href="https://doi.org/10.1070/SM9348">doi:10.1070/SM9348</a>, <a href="https://doi.org/10.1070/SM9348">doi:10.1070/SM9348</a>]</p> </li> </ul> <p>Further discussion:</p> <ul> <li id="GolubNikita24"> <p>Nikita Golub: <em>Functorial languages in homological algebra and the lower central series</em> [<a href="https://arxiv.org/abs/2410.05708">arXiv:2410.05708</a>]</p> </li> <li> <p>Nikita Golub: <em>Functorial Languages in Homological Algebra</em>, <a href="CQTS#Golub2024">talk at</a> <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> @ NYU Abu Dhabi (Oct 2024) [slides:<a class="existingWikiWord" href="/nlab/files/Golub-CQTSOct2024.pdf" title="pdf">pdf</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 23, 2024 at 17:56:13. See the <a href="/nlab/history/fr-code" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/fr-code" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/18557/#Item_7">Discuss</a><span class="backintime"><a href="/nlab/revision/fr-code/9" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/fr-code" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/fr-code" accesskey="S" class="navlink" id="history" rel="nofollow">History (9 revisions)</a> <a href="/nlab/show/fr-code/cite" style="color: black">Cite</a> <a href="/nlab/print/fr-code" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/fr-code" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>