Beck's monadicity theorem

<h2 id="examples">Examples</h2>
<ul><li>The forgetful functor from topological spaces to sets is not monadic as it does not reflect isomorphisms: continuous bijections between (non-compact or non-Hausdorff) topological spaces need not be homeomorphisms.</li>
<li>Negrepontis (1971, §1) shows that the functor from commutative <a href="/facts/C*-algebras/B3tnGHIo">C*-algebras</a> to sets sending such an algebra A to the <a href="/facts/Unit_ball/5UeoTcug">unit ball</a>, i.e., the set 
 
 
 
 {
 a
 ∈
 A
 ,
 ‖
 a
 ‖
 ≤
 1
 }
 
 
 {\displaystyle \{a\in A,\|a\|\leq 1\}}
 
, is monadic. Negrepontis also deduces <a href="/facts/Gelfand_duality/QV54f7Jm">Gelfand duality</a>, i.e., the equivalence of categories between the opposite category of compact Hausdorff spaces and commutative C*-algebras can be deduced from this.</li>
<li>The powerset functor from Setop to Set is monadic, where Set is the category of sets. More generally Beck's theorem can be used to show that the powerset functor from Top to T is monadic for any topos T, which in turn is used to show that the topos T has finite colimits.</li>
<li>The forgetful functor from <a href="/facts/Semigroup/rgSdk2kt">semigroups</a> to sets is monadic. This functor does not preserve arbitrary coequalizers, showing that some restriction on the coequalizers in Beck's theorem is necessary if one wants to have conditions that are necessary and sufficient.</li>
<li>If B is a faithfully flat commutative ring over the commutative ring A, then the functor T from A modules to B modules taking M to B⊗AM is comonadic. This follows from the dual of Becks theorem, as the condition that B is flat implies that T preserves finite limits, while the condition that B is faithfully flat implies that T reflects isomorphisms. A coalgebra over T turns out to be essentially a B-module with descent data, so the fact that T is comonadic is equivalent to the main theorem of faithfully flat descent, saying that B-modules with descent are equivalent to A-modules.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://ncatlab.org/nlab/show/monadicity+theorem">monadicity theorem</a> at the <a href="/facts/NLab/lpPvRmgo">nLab</a></li>
<li><a href="https://ncatlab.org/nlab/show/monadic+descent">monadic descent</a> at the <a href="/facts/NLab/lpPvRmgo">nLab</a></li></ul>

<ul><li><a href="/facts/Paul_Balmer/5WN3zAMe">Balmer, Paul</a> (2012), "Descent in triangulated categories", <a href="/facts/Mathematische_Annalen/GPhGaonc">Mathematische Annalen</a>, 353 (1): 109–125, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs00208-011-0674-z">10.1007/s00208-011-0674-z</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2910783">2910783</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121964355">121964355</a></li>
<li>Barr, M.; Wells, C. (2013) [1985], Triples, toposes, and theories, Grundlehren der mathematischen Wissenschaften, vol. 278, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781489900234 <a href="http://www.math.mcgill.ca/barr/papers/tttall.pdf">pdf</a></li>
<li><a href="/facts/Jonathan_Mock_Beck/SQU5Pavt">Beck, Jonathan Mock</a> (2003) [1967], <a href="https://www.emis.de/journals/TAC/reprints/articles/2/tr2.pdf">"Triples, algebras and cohomology"</a> (PDF), Reprints in Theory and Applications of Categories, Columbia University PhD thesis, 2: 1–59, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1987896">1987896</a></li>
<li><a href="/facts/Jean_B%25C3%25A9nabou/mQlt5gFm">Bénabou, Jean</a>; <a href="/facts/Jacques_Roubaud/MWgc3z4J">Roubaud, Jacques</a> (1970-01-12), "Monades et descente", C. R. Acad. Sci. Paris, 270 (A): 96–98</li>
<li>Leinster, Tom (2013), "Codensity and the ultrafilter monad", Theory and Applications of Categories, 28: 332–370, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1209.3606">1209.3606</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2012arXiv1209.3606L">2012arXiv1209.3606L</a></li>
<li>Negrepontis, Joan W. (1971), "Duality in analysis from the point of view of triples", Journal of Algebra, 19 (2): 228–253, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0021-8693%2871%2990105-0">10.1016/0021-8693(71)90105-0</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0021-8693">0021-8693</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0280571">0280571</a></li>
<li>Pavlović, Duško (1991), "Categorical interpolation: descent and the Beck-Chevalley condition without direct images", in Carboni, A.; Pedicchio, M.C.; Rosolini, G. (eds.), Category theory, Lecture Notes in Mathematics, vol. 1488, Springer, pp. 306–325, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0084229">10.1007/BFb0084229</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-54706-8</li>
<li>Deligne, Pierre (1990), Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Progress in Mathematics, vol. 87, Birkhäuser, pp. 111–195</li>
<li>Grothendieck, A. (1962), "Fondements de la géométrie algébrique", [Extraits du Séminaire Bourbaki, 1957—1962], Paris: Secrétariat Math., <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0146040">0146040</a></li>
<li>Grothendieck, A.; Raynaud, M. (1971), Revêtements Etales et Groupe Fondamental, Lecture Notes in Mathematics, vol. 224, Springer, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.AG/0206203">math.AG/0206203</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0058656">10.1007/BFb0058656</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-36910-3</li>
<li>Borceux, Francis (1994), <a href="https://books.google.com/books?id=YfzImoopB-IC&pg=PP1">Basic Category Theory</a>, Handbook of Categorical Algebra, vol. 1, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-44178-0 (3 volumes).</li>
<li>Fantechi, Barbara; Göttsche, Lothar; Illusie, Luc; Kleiman, Steven L.; Nitsure, Nitin; Vistoli, Angelo (2005), <a href="https://books.google.com/books?id=KxH0BwAAQBAJ">Fundamental Algebraic Geometry: Grothendieck's FGA Explained</a>, Mathematical Surveys and Monographs, vol. 123, American Mathematical Society, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-4245-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2222646">2222646</a></li>
<li>Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004), Categorical foundations. Special topics in order, topology, algebra, and sheaf theory, Encyclopedia of Mathematics and Its Applications, vol. 97, Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-83414-7, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1034.18001">1034.18001</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Pedicchio & Tholen 2004, p. 228 - Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004), Categorical foundations. Special topics in order, topology, algebra, and sheaf theory, Encyclopedia of Mathematics and Its Applications, vol. 97, Cambridge: Cambridge University Press, ISBN 0-521-83414-7, Zbl 1034.18001 <a href="https://zbmath.org/?format=complete&q=an:1034.18001" target="_blank">https://zbmath.org/?format=complete&q=an:1034.18001</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Deligne 1990, §4.2 - Deligne, Pierre (1990), Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Progress in Mathematics, vol. 87, Birkhäuser, pp. 111–195 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Beck's monadicity theorem open-in-new

Beck's monadicity theorem