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Higher-dimensional algebra
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<h2 id="higher-dimensional-categories">Higher-dimensional categories</h2> <p class="note">Main article: <a href="/facts/Category_theory/6zS3DXPa">Category theory § Higher-dimensional categories</a></p> <p>A first step towards defining higher dimensional algebras is the concept of <a href="/facts/2-category/YeMVfoAc">2-category</a> of <a href="/facts/Higher_category_theory/7Pp6a0VN">higher category theory</a>, followed by the more 'geometric' concept of double category.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>A higher level concept is thus defined as a <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> of categories, or super-category, which generalises to higher dimensions the notion of <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> – regarded as any structure which is an interpretation of <a href="/facts/Lawvere/x0j3IlCj">Lawvere</a>'s axioms of the elementary theory of abstract categories (ETAC).<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> Thus, a supercategory and also a <a href="/facts/Functor_category/r7sWdb2h">super-category</a>, can be regarded as natural extensions of the concepts of <a href="/facts/Multicategory/T0fRWmMV">meta-category</a>,<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> <a href="/facts/Multicategory/T0fRWmMV">multicategory</a>, and <a href="/facts/Tur%C3%A1n_graph/y0hzJqQQ">multi-graph, <i>k</i>-partite graph</a>, or <a href="/facts/Colored_graph/J6HoBfP0">colored graph</a> (see a color figure, and also its definition in <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>). </p><p>Supercategories were first introduced in 1970,<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> and were subsequently developed for applications in <a href="/facts/Theoretical_physics/Cs2Y12rQ">theoretical physics</a> (especially <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a> and <a href="/facts/Topological_quantum_field_theory/zHjrp3b8">topological quantum field theory</a>) and <a href="/facts/Mathematical_biology/pV9KCBSD">mathematical biology</a> or <a href="/facts/Mathematical_biology/pV9KCBSD">mathematical biophysics</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p><p>Other pathways in higher-dimensional algebra involve: <a href="/facts/Bicategory/vaYsrIbg">bicategories</a>, homomorphisms of bicategories, variable categories (also known as indexed or parametrized categories), <a href="/facts/Topoi/zzttCIwC">topoi</a>, effective descent, and <a href="/facts/Enriched_category/VgphwkBu">enriched</a> and <a href="/facts/Internal_category/Bttn1sI5">internal categories</a>. </p> <h2 id="double-groupoids">Double groupoids</h2> <p class="note">Main article: <a href="/facts/Double_groupoid/EaNqz8E2">double groupoid</a></p> <p>In higher-dimensional algebra (HDA), a <i>double groupoid</i> is a generalisation of a one-dimensional <a href="/facts/Groupoid/hvj8wvWe">groupoid</a> to two dimensions,<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> and the latter groupoid can be considered as a special case of a category with all invertible arrows, or <a href="/facts/Morphism/Kdpuyz3S">morphisms</a>. </p><p><a href="/facts/Double_groupoid/EaNqz8E2">Double groupoids</a> are often used to capture information about <a href="/facts/Geometrical/wTQf5ffQ">geometrical</a> objects such as <a href="/facts/N-dimensional_space/0ktmUyac">higher-dimensional manifolds</a> (or <a href="/facts/List_of_manifolds/fL1Eor4V"><i>n</i>-dimensional manifolds</a>).<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> In general, an <a href="/facts/List_of_manifolds/fL1Eor4V"><i>n</i>-dimensional manifold</a> is a space that locally looks like an <a href="/facts/N-dimensional_Euclidean_space/R2UbzmzM"><i>n</i>-dimensional Euclidean space</a>, but whose global structure may be <a href="/facts/Non-Euclidean/efKa7De2">non-Euclidean</a>. </p><p>Double groupoids were first introduced by <a href="/facts/Ronald_Brown_(mathematician)/mT6ArHj0">Ronald Brown</a> in <i>Double groupoids and crossed modules</i> (1976),<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> and were further developed towards applications in <a href="/facts/Non-abelian_group/KQYHrwJl">nonabelian</a> <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> A related, 'dual' concept is that of a double <a href="/facts/Lie_algebroid/Go9mKaX5">algebroid</a>, and the more general concept of <a href="/facts/R-algebroid/ojcWRjHv">R-algebroid</a>. </p> <h2 id="nonabelian-algebraic-topology">Nonabelian algebraic topology</h2> <p>See <a href="/facts/Nonabelian_algebraic_topology/gS8XxoW5">Nonabelian algebraic topology</a> </p> <h2 id="applications">Applications</h2> <h3>Theoretical physics</h3> <p>In <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a>, there exist quantum categories.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a><a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> and quantum double groupoids.<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> One can consider quantum double groupoids to be <a href="/facts/Fundamental_group/VyYtLqSP">fundamental groupoids</a> defined via a <a href="/facts/2-functor/pMtYoHqx">2-functor</a>, which allows one to think about the physically interesting case of quantum fundamental groupoids (QFGs) in terms of the <a href="/facts/Bicategory/vaYsrIbg">bicategory</a> Span(Groupoids), and then constructing 2-<a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a> and 2-<a href="/facts/Linear_map/Q1PBKNVV">linear maps</a> for manifolds and <a href="/facts/Cobordism/ctN2FSTB">cobordisms</a>. At the next step, one obtains <a href="/facts/Cobordism/ctN2FSTB">cobordisms</a> with corners via <a href="/facts/Natural_transformation/xwPumTHt">natural transformations</a> of such 2-functors. A claim was then made that, with the <a href="/facts/Gauge_group/fSW3gggv">gauge group</a> <a href="/facts/SU(2)/Qgy2orcu">SU(2)</a>, "the extended <a href="/facts/TQFT/zHjrp3b8">TQFT</a>, or ETQFT, gives a theory equivalent to the Ponzano–Regge model of <a href="/facts/Quantum_gravity/t7whBMNE">quantum gravity</a>";<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> similarly, the Turaev–Viro model would be then obtained with <a href="/facts/Representation_(mathematics)/mpsX51I9">representations</a> of SU<i>q</i>(2). Therefore, one can describe the <a href="/facts/State_space/H1vjb3LL">state space</a> of a gauge theory – or many kinds of <a href="/facts/Quantum_field_theories/VoewsNJV">quantum field theories</a> (QFTs) and local quantum physics, in terms of the <a href="/facts/Transformation_groupoid/hvj8wvWe">transformation groupoids</a> given by symmetries, as for example in the case of a gauge theory, by the <a href="/facts/Gauge_transformation/fSW3gggv">gauge transformations</a> acting on states that are, in this case, connections. In the case of symmetries related to <a href="/facts/Quantum_group/huNTuaXs">quantum groups</a>, one would obtain structures that are representation categories of <a href="/facts/Quantum_groupoid/5Fsa82QE">quantum groupoids</a>,<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> instead of the 2-<a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> that are representation categories of groupoids. </p> <h3>Quantum physics</h3> <ul><li><a href="/facts/Topological_order/EfEnKRSw">Quantum Algebraic Topology</a></li> <li><a href="/facts/Quantum_geometry/u2kx8MJY">Quantum geometry</a></li> <li><a href="/facts/Quantum_gravity/t7whBMNE">Quantum gravity</a></li> <li><a href="/facts/Quantum_group/huNTuaXs">Quantum group</a></li> <li><a href="/facts/Topological_quantum_field_theory/zHjrp3b8">Topological quantum field theory</a></li> <li><a href="/facts/Local_quantum_field_theory/XqBgUQva">Local quantum field theory</a></li></ul> <h2 id="see-also">See also</h2> <ul> <li>Mathematics portal</li></ul> <ul><li><a href="/facts/Timeline_of_category_theory_and_related_mathematics/hJ0Lfckq">Timeline of category theory and related mathematics</a></li> <li><a href="/facts/Higher_category_theory/7Pp6a0VN">Higher category theory</a></li> <li><a href="/facts/Ronald_Brown_(mathematician)/mT6ArHj0">Ronald Brown</a></li> <li><a href="/facts/Lie_algebroid/Go9mKaX5">Lie algebroid</a></li> <li><a href="/facts/Double_groupoid/EaNqz8E2">Double groupoid</a></li> <li><a href="/facts/Anabelian_geometry/zqvz8gqf">Anabelian geometry</a></li> <li><a href="/facts/Noncommutative_geometry/WSwggL1p">Noncommutative geometry</a></li> <li><a href="/facts/Categorical_algebra/vyMHqRp4">Categorical algebra</a></li> <li><a href="/facts/Grothendieck%27s_Galois_theory/NggfvWW9">Grothendieck's Galois theory</a></li> <li><a href="/facts/Grothendieck_topology/Przxx5sH">Grothendieck topology</a></li> <li><a href="/facts/Topological_dynamics/QYCym7RF">Topological dynamics</a></li> <li><a href="/facts/Symbolic_dynamics/kApuvAj8">Categorical dynamics</a></li> <li><a href="/facts/Crossed_module/IXMVWce9">Crossed module</a></li> <li><a href="/facts/Pseudoalgebra/1cVPBvdp">Pseudoalgebra</a></li> <li><a href="/facts/2-ring/N5g2NPCQ">2-ring</a></li> <li><a href="/facts/Lie_n-algebra/lfBzYQ0P">Lie n-algebra</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="further-reading">Further reading</h2> <ul><li>Brown, R.; Higgins, P.J.; Sivera, R. (2011). <a href="http://groupoids.org.uk/nonab-a-t.html"><i>Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids</i></a>. Vol. Tracts Vol 15. European Mathematical Society. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0407275">math/0407275</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4171%2F083">10.4171/083</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-03719-083-8. (<a href="http://groupoids.org.uk/nonab-a-t.html">Downloadable PDF available</a>)</li> <li>Brown, R.; Mosa, G.H. (1999). <a href="https://www.emis.de/journals/TAC/volumes/1999/n7/5-07abs.html">"Double categories, thin structures and connections"</a>. <i>Theory and Applications of Categories</i>. 5: 163–175. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.438.8991">10.1.1.438.8991</a>.</li> <li>Brown, R. (2002). <i>Categorical Structures for Descent and Galois Theory</i>. <a href="/facts/Fields_Institute/PeNa4cve">Fields Institute</a>.</li> <li>Brown, R. (1987). <a href="http://groupoids.org.uk/pdffiles/groupoidsurvey.pdf">"From groups to groupoids: a brief survey"</a> (PDF). <i><a href="/facts/Bulletin_of_the_London_Mathematical_Society/jdcf3vhX">Bulletin of the London Mathematical Society</a></i>. 19 (2): 113–134. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.363.1859">10.1.1.363.1859</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fblms%2F19.2.113">10.1112/blms/19.2.113</a>. <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/10338.dmlcz%2F140413">10338.dmlcz/140413</a>. This give some of the history of groupoids, namely the origins in work of <a href="/facts/Heinrich_Brandt/opNq9yFz">Heinrich Brandt</a> on quadratic forms, and an indication of later work up to 1987, with 160 references.</li> <li>Brown, Ronald (2018). <a href="http://groupoids.org.uk/hdaweb2.html">"Higher Dimensional Group Theory"</a>. <i>groupoids.org.uk</i>. Bangor University. A web article with many references explaining how the groupoid concept has led to notions of higher-dimensional groupoids, not available in group theory, with applications in homotopy theory and in group cohomology.</li> <li>Brown, R.; Higgins, P.J. (1981). "On the algebra of cubes". <i><a href="/facts/Journal_of_Pure_and_Applied_Algebra/Cnn9fRjc">Journal of Pure and Applied Algebra</a></i>. 21 (3): 233–260. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0022-4049%2881%2990018-9">10.1016/0022-4049(81)90018-9</a>.</li> <li>Mackenzie, K.C.H. (2005). <a href="https://web.archive.org/web/20050310034123/http://www.shef.ac.uk/~pm1kchm/gt.html"><i>General theory of Lie groupoids and Lie algebroids</i></a>. London Mathematical Society Lecture Note Series. Vol. 213. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-49928-6. Archived from <a href="http://www.shef.ac.uk/~pm1kchm/gt.html">the original</a> on 2005-03-10.</li> <li>Brown, R. (2006). <a href="http://groupoids.org.uk/topgpds.html"><i>Topology and Groupoids</i></a>. <a href="/facts/Booksurge/Imzk2wKk">Booksurge</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4196-2722-4. Revised and extended edition of a book previously published in 1968 and 1988. E-version available from website.</li> <li>Borceux, F.; Janelidze, G. (2001). <a href="https://archive.today/20121223050454/http://www.cup.cam.ac.uk/catalogue/catalogue.asp?isbn=9780521803090"><i>Galois theories</i></a>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-07041-6. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/1167627177">1167627177</a>. Archived from <a href="http://www.cup.cam.ac.uk/catalogue/catalogue.asp?isbn=9780521803090">the original</a> on 2012-12-23. Shows how generalisations of <a href="/facts/Galois_theory/54wscJfj">Galois theory</a> lead to Galois groupoids.</li> <li>Baez, J.; Dolan, J. (1998). "Higher-Dimensional Algebra III. <i>n</i>-Categories and the Algebra of Opetopes". <i><a href="/facts/Advances_in_Mathematics/kXtcl4pM">Advances in Mathematics</a></i>. 135 (2): 145–206. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/q-alg/9702014">q-alg/9702014</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1997q.alg.....2014B">1997q.alg.....2014B</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1006%2Faima.1997.1695">10.1006/aima.1997.1695</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:18857286">18857286</a>.</li> <li>Baianu, I.C. (1970). <a href="http://cogprints.org/7752/3/OSIImultist1970.pdf">"Organismic Supercategories: II. On Multistable Systems"</a> (PDF). <i>The Bulletin of Mathematical Biophysics</i>. 32 (4): 539–61. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02476770">10.1007/BF02476770</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/4327361">4327361</a>.</li> <li>Baianu, I.C.; Marinescu, M. (1974). "On A Functorial Construction of (<i>M</i>, <i>R</i>)-Systems". <i>Revue Roumaine de Mathématiques Pures et Appliquées</i>. 19: 388–391.</li> <li>Baianu, I.C. (1987). <a href="http://cogprints.org/3687/">"Computer Models and Automata Theory in Biology and Medicine"</a>. In M. Witten (ed.). <i>Mathematical Models in Medicine</i>. Vol. 7. <a href="/facts/Pergamon_Press/qe6ctSTB">Pergamon Press</a>. pp. 1513–77. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-08-034692-2. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/939260427">939260427</a>. CERN <i>Preprint</i> No. EXT-2004-072. ‹See Tfd›<a href="/facts/Amazon_Standard_Identification_Number/twP2MtLk">ASIN</a> <a href="https://www.amazon.ca/dp/0080346928">0080346928</a> ‹See Tfd›<a href="/facts/Amazon_Standard_Identification_Number/twP2MtLk">ASIN</a> <a href="https://www.amazon.ca/dp/0080346928">0080346928</a>.</li> <li><a href="https://web.archive.org/web/20090813044843/http://planetphysics.org/encyclopedia/HigherDimensionalHomotopy.html">"Higher dimensional Homotopy"</a>. PlanetPhysics. Archived from <a href="http://planetphysics.org/encyclopedia/HigherDimensionalHomotopy.html">the original</a> on 2009-08-13.</li> <li>Janelidze, George (1990). "Pure Galois theory in categories". <i>Journal of Algebra</i>. 132 (2): 270–286. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0021-8693%2890%2990130-G">10.1016/0021-8693(90)90130-G</a>.</li> <li>Janelidze, George (1993). "Galois theory in variable categories". <i>Applied Categorical Structures</i>. 1: 103–110. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF00872989">10.1007/BF00872989</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:22258886">22258886</a>..</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Double Categories and Pseudo Algebras" (PDF). Archived from the original (PDF) on 2010-06-10. <a href="https://web.archive.org/web/20100610024231/http://www.math.uchicago.edu/~fiore/1/fiorefolding.pdf" target="_blank">https://web.archive.org/web/20100610024231/http://www.math.uchicago.edu/~fiore/1/fiorefolding.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Brown, R.; Loday, J.-L. (1987). "Homotopical excision, and Hurewicz theorems, for n-cubes of spaces". Proceedings of the London Mathematical Society. 54 (1): 176–192. CiteSeerX 10.1.1.168.1325. doi:10.1112/plms/s3-54.1.176. <a href="/wiki/Proceedings_of_the_London_Mathematical_Society" target="_blank">/wiki/Proceedings_of_the_London_Mathematical_Society</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Batanin, M.A. (1998). "Monoidal Globular Categories As a Natural Environment for the Theory of Weak n-Categories". Advances in Mathematics. 136 (1): 39–103. doi:10.1006/aima.1998.1724. <a href="https://doi.org/10.1006%2Faima.1998.1724" target="_blank">https://doi.org/10.1006%2Faima.1998.1724</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Lawvere, F. W. (1964). "An Elementary Theory of the Category of Sets". Proceedings of the National Academy of Sciences of the United States of America. 52 (6): 1506–1511. Bibcode:1964PNAS...52.1506L. doi:10.1073/pnas.52.6.1506. PMC 300477. 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