# [ALGTOP-L] "Cubical sets in nonabelian algebraic topology"

RONALD BROWN ronnie.profbrown at btinternet.com
Wed Sep 10 12:31:02 EDT 2014

```I gave a talk with this title on June 5, at the IHP, Paris, as part of a workshop on "Constructive mathematics and models of type theory". The invitation arose  from some interest in cubical models in this area (by Thierry Coquand).

Full and handout versions of a slightly revised version of the slides are available from my preprint page
http://pages.bangor.ac.uk/~mas010/brownpr.html

The talk  explained a little some successful aspects of cubical methods (when enhanced by the notion of connections)  in obtaining some nonabelian algebraic colimit results in algebraic topology, but I did not have time to discuss the many areas where the cubical theory has not been worked over, in contrast to the simplicial case.

If anyone is interested, I can say more on this.

Regards

Ronnie Brown
www.bangor.ac.uk/r.brown

Abstract: The talk will start from the 1-dimensional Seifert-van Kampen Theorem for the fundamental group, then groupoid, and so to a use of strict double groupoids for higher versions. These allow for some precise nonabelian calculations of some homotopy types, obtained by a gluing process. Cubical methods are involved because of the ease of writing multiple compositions, leading to "algebraic inverses to subdivision", relevant to higher dimensional local-to-global problems. Also the proofs involve some ideas of 2-dimensional formulae and rewriting. The use of strict multiple groupoids is essential to obtain precise descriptions as colimits and hence precise calculations. Another idea is to use both a "broad" and a "narrow" model of a particular kind of homotopy types, where the broad model is used for conjectures and proofs, while the narrow model is used for calculations and relation to classical methods. The algebraic proof of the equivalence of the
two models then gives a powerful tool.
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<html><body><div style="color:#000; background-color:#fff; font-family:times new roman, new york, times, serif;font-size:12pt"><div>I gave a talk with this title on June 5, at the IHP, Paris, as part of a workshop on "Constructive mathematics and models of type theory". The invitatio<span style="font-size: 12pt;">n arose  from some interest in cubical models in this area (by Thierry Coquand). </span></div><div style="color: rgb(0, 0, 0); font-size: 12pt; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;"><span style="font-size: 12pt;"><br></span></div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;"><span style="font-size: 12pt;">Full and handout versions of a slightly revised version of the slides are available from my preprint
page</span></div><div>http://pages.bangor.ac.uk/~mas010/brownpr.html</div><div><br></div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;">The talk  explained a little some successful aspects of cubical methods (when enhanced by the notion of connections)  in obtaining some nonabelian algebraic colimit results in algebraic topology, but I did not have time to discuss the many areas where the cubical theory has not been worked over, in contrast to the simplicial case. </div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;"><br></div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;">If anyone is interested, I can say
more on this.</div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;"><br></div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;">Regards</div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;"><br></div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;">Ronnie Brown</div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;">www.bangor.ac.uk/r.brown</div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family:
'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;"><br></div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;"><br></div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;"><span style="font-family: 'Times New Roman'; font-size: 16px; background-color: rgb(255, 255, 128);">Abstract: The talk will start from the 1-dimensional Seifert-van Kampen Theorem for the fundamental group, then groupoid, and so to a use of strict double groupoids for higher versions. These allow for some precise nonabelian calculations of some homotopy types, obtained by a gluing process. Cubical methods are involved because of the ease of writing multiple compositions, leading to "algebraic inverses to subdivision",
relevant to higher dimensional local-to-global problems. Also the proofs involve some ideas of 2-dimensional formulae and rewriting. The use of strict multiple groupoids is essential to obtain precise descriptions as colimits and hence precise calculations. Another idea is to use both a "broad" and a "narrow" model of a particular kind of homotopy types, where the broad model is used for conjectures and proofs, while the narrow model is used for calculations and relation to classical methods. The algebraic proof of the equivalence of the two models then gives a powerful tool.</span><br></div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;"><br></div><div style="color: rgb(0, 0, 0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;"><br></div><div style="color: rgb(0, 0,
0); font-size: 16px; font-family: 'times new roman', 'new york', times, serif; font-style: normal; background-color: transparent;"><br></div></div></body></html>
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