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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Michael Barr (mathematician)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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.infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox biography vcard"><tbody><tr><th colspan="2" class="infobox-above" style="font-size:125%;"><div class="fn">Michael Barr</div></th></tr><tr><th scope="row" class="infobox-label">Born</th><td class="infobox-data"><span style="display:none"> (<span class="bday">1937-01-22</span>) </span>January 22, 1937<span class="noprint ForceAgeToShow"> (age 87)</span><br /><div style="display:inline" class="birthplace"><a href="/wiki/Philadelphia" title="Philadelphia">Philadelphia</a>, <a href="/wiki/Pennsylvania" title="Pennsylvania">Pennsylvania</a>, U.S.</div></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"></td></tr><tr><th colspan="2" class="infobox-header">Academic background</th></tr><tr><th scope="row" class="infobox-label">Education</th><td class="infobox-data"><a href="/wiki/University_of_Pennsylvania" title="University of Pennsylvania">University of Pennsylvania</a> (<a href="/wiki/Bachelor_of_Science" title="Bachelor of Science">BS</a>, <a href="/wiki/PhD" class="mw-redirect" title="PhD">PhD</a>)</td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"></tr><tr><th colspan="2" class="infobox-header">Academic work</th></tr><tr><th scope="row" class="infobox-label">Discipline</th><td class="infobox-data"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></td></tr><tr><th scope="row" class="infobox-label">Sub-discipline</th><td class="infobox-data"><a href="/wiki/Homological_algebra" title="Homological algebra">Homological algebra</a><br /><a href="/wiki/Category_theory" title="Category theory">Category theory</a><br /><a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">Theoretical computer science</a></td></tr><tr><th scope="row" class="infobox-label">Institutions</th><td class="infobox-data"><a href="/wiki/Columbia_University" title="Columbia University">Columbia University</a><br /><a href="/wiki/University_of_Illinois_Urbana-Champaign" title="University of Illinois Urbana-Champaign">University of Illinois Urbana-Champaign</a><br /><a href="/wiki/McGill_University" title="McGill University">McGill University</a></td></tr><tr style="display:none"><td colspan="2"> </td></tr></tbody></table> <p><b>Michael Barr</b> (born January 22, 1937) is an American mathematician who is the <a href="/wiki/Peter_Redpath" title="Peter Redpath">Peter Redpath</a> Emeritus Professor of Pure Mathematics at <a href="/wiki/McGill_University" title="McGill University">McGill University</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Early_life_and_education">Early life and education</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Michael_Barr_(mathematician)&action=edit&section=1" title="Edit section: Early life and education"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>He was born in <a href="/wiki/Philadelphia" title="Philadelphia">Philadelphia</a>, <a href="/wiki/Pennsylvania" title="Pennsylvania">Pennsylvania</a>, and graduated from the 202nd class of <a href="/wiki/Central_High_School_(Philadelphia)" title="Central High School (Philadelphia)">Central High School</a> in June 1954. He graduated from the <a href="/wiki/University_of_Pennsylvania" title="University of Pennsylvania">University of Pennsylvania</a> in February 1959 and received a PhD from the same school in June 1962. </p> <div class="mw-heading mw-heading2"><h2 id="Career">Career</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Michael_Barr_(mathematician)&action=edit&section=2" title="Edit section: Career"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Barr studied mathematics at the University of Pennsylvania, graduating with a bachelor's degree in 1959 and a doctorate in 1962 under <a href="/wiki/David_Kent_Harrison" title="David Kent Harrison">David Kent Harrison</a> <i>(Cohomology of Commutative Algebras</i>). He was then an instructor at <a href="/wiki/Columbia_University" title="Columbia University">Columbia University</a> and from 1964 Assistant Professor and later Associate Professor at the <a href="/wiki/University_of_Illinois_Urbana-Champaign" title="University of Illinois Urbana-Champaign">University of Illinois Urbana-Champaign</a>. In 1968 he became Associate Professor and in 1972 Professor at McGill University. </p><p>In 1967 and 1975/76 he was a visiting scientist at <a href="/wiki/ETH_Zurich" title="ETH Zurich">ETH Zurich</a> and in 1970/71 at the <a href="/wiki/University_of_Fribourg" title="University of Fribourg">University of Fribourg</a> and in 1989/90 a visiting professor at the University of Pennsylvania. </p><p>In 1970 he was an invited speaker at the <a href="/wiki/International_Congress_of_Mathematicians" title="International Congress of Mathematicians">International Congress of Mathematicians</a> in <a href="/wiki/Nice" title="Nice">Nice</a> <i>(Non-abelian full embedding: outline</i>). </p><p>His earlier work was in <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>, but his principal research area for a number of years has been <a href="/wiki/Category_theory" title="Category theory">category theory</a>. He is well known to <a href="/wiki/Theoretical_computer_scientist" class="mw-redirect" title="Theoretical computer scientist">theoretical computer scientists</a> for his book <i>Category Theory for Computing Science</i> (1990) with <a href="/wiki/Charles_Wells_(mathematician)" title="Charles Wells (mathematician)">Charles Wells</a>, as well as for the development of <a href="/wiki/*-autonomous_category" title="*-autonomous category">*-autonomous categories</a> and <a href="/wiki/Chu_space" title="Chu space">Chu spaces</a> which have found various applications in computer science. His monograph <i>*-autonomous categories</i> (1979), and his books <i>Toposes, Triples, and Theories</i> (1985),<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> also coauthored with Wells, and <i>Acyclic Models</i> (2002), are aimed at more specialized audiences. In 2011 Michael Barr and his wife Marcia published an English translation of <a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck</a>'s fundamental <a href="/wiki/Grothendieck%27s_T%C3%B4hoku_paper" title="Grothendieck's Tôhoku paper"><i>Tôhoku</i> paper</a>. </p><p>Barr is on the editorial boards of <i>Mathematical Structures in Computer Science</i> and the electronic journal <i><a href="/wiki/Homology,_Homotopy_and_Applications" title="Homology, Homotopy and Applications">Homology, Homotopy and Applications</a></i>, and is editor of the electronic journal <i>Theory and Applications of Categories</i>. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Michael_Barr_(mathematician)&action=edit&section=3" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mcgill.ca/study/2010-2011/faculties/science/undergraduate/ug_science_math">"Mathematics and Statistics"</a>. McGill University<span class="reference-accessdate">. Retrieved <span class="nowrap">11 August</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Mathematics+and+Statistics&rft.pub=McGill+University&rft_id=https%3A%2F%2Fwww.mcgill.ca%2Fstudy%2F2010-2011%2Ffaculties%2Fscience%2Fundergraduate%2Fug_science_math&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMichael+Barr+%28mathematician%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPitts1991" class="citation cs2">Pitts, Andrew (March 1991), "Review of Toposes, Triples and Theories by Barr, M., & Wells, C.", <i><a href="/wiki/Journal_of_Symbolic_Logic" title="Journal of Symbolic Logic">Journal of Symbolic Logic</a></i>, <b>56</b> (1): 340–341, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2274934">10.2307/2274934</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2274934">2274934</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Logic&rft.atitle=Review+of+Toposes%2C+Triples+and+Theories+by+Barr%2C+M.%2C+%26+Wells%2C+C.&rft.volume=56&rft.issue=1&rft.pages=340-341&rft.date=1991-03&rft_id=info%3Adoi%2F10.2307%2F2274934&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2274934%23id-name%3DJSTOR&rft.aulast=Pitts&rft.aufirst=Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMichael+Barr+%28mathematician%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRota1986" class="citation cs2"><a href="/wiki/Gian-Carlo_Rota" title="Gian-Carlo Rota">Rota, Gian-Carlo</a> (August 1986), "Toposes, triples and theories: M. Barr and C. Wells, Springer, 1985, 345 pp.", <i><a href="/wiki/Advances_in_Mathematics" title="Advances in Mathematics">Advances in Mathematics</a></i>, <b>61</b> (2): 184, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0001-8708%2886%2990076-9">10.1016/0001-8708(86)90076-9</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Mathematics&rft.atitle=Toposes%2C+triples+and+theories%3A+M.+Barr+and+C.+Wells%2C+Springer%2C+1985%2C+345+pp.&rft.volume=61&rft.issue=2&rft.pages=184&rft.date=1986-08&rft_id=info%3Adoi%2F10.1016%2F0001-8708%2886%2990076-9&rft.aulast=Rota&rft.aufirst=Gian-Carlo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMichael+Barr+%28mathematician%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Michael_Barr_(mathematician)&action=edit&section=4" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf"><i>Toposes, Triples and Theories</i></a>, updated edition of text first published in 1985.</li> <li><a rel="nofollow" class="external text" href="http://www.tac.mta.ca/tac/reprints/articles/22/tr22.pdf"><i>Category Theory for Computing Science</i></a>, updated 3rd edition of the book.</li> <li><a rel="nofollow" class="external text" href="https://www.math.mcgill.ca/barr/papers/gk.pdf">Some aspects of homological Algebra</a>, translation of Grothendieck's <i>Tôhoku</i> paper</li> <li><a rel="nofollow" class="external free" href="http://www.tac.mta.ca/tac">http://www.tac.mta.ca/tac</a> (Theory and Applications of Categories)</li> <li><a rel="nofollow" class="external free" href="https://web.archive.org/web/20080704125156/http://www.math.rutgers.edu/hha/geninfo.html">https://web.archive.org/web/20080704125156/http://www.math.rutgers.edu/hha/geninfo.html</a> (Homology, Homotopy and Applications)</li> <li><a rel="nofollow" class="external text" href="https://mathgenealogy.org/id.php?id=5652">Michael Barr</a> at the <a href="/wiki/Mathematics_Genealogy_Project" title="Mathematics Genealogy Project">Mathematics Genealogy Project</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline 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