×

The little bundles operad. (English) Zbl 1459.55006

Let \(E_n\) be the little \(n\)-disks operad. Recall that an operation in arity \(r\) of \(E_n\) is an embedding \(f\colon \coprod_r D^n\to D^n\), that on each copy of \(D^n\) amounts to rescaling and translation. Here \(D^n\) is the open unit disk in \(\mathbb R^n\).
Let \(T\) be a topological space. The authors construct a new operad \(E^T_n\). An operation in arity \(r\) of \(E^T_n\) consists of an operation \(f\) in arity \(r\) of \(E_n\), together with a map from the complement of the image of \(f\) into \(T\). Composition in \(E^T_n\) is defined using composition in \(E_n\), and gluing complements of embeddings in a natural way. In fact, \(E^T_n\) is a topological colored operad, whose colors are given by the mapping space \(\operatorname{Map}({\mathbb{S}}^{n-1}, T)\). The space of operations in arity \(r\) of \(E^T_n\), which the authors denote by \(W_n^T(r)\), fits in a homotopy fibration sequence \[ \operatorname{Map}(\vee_{r} {\mathbb{S}}^{n-1}, T)\to W_n^T(r) \to E_n(r). \]
For most of the paper the authors focus on the case when \(n=2\) and \(T=BG\) is the classifying space of a discrete group \(G\). The resulting operad is denoted by \(E_2^G\) and is named the operad of little \(G\)-bundles.
Recall that \(E_2(r)\) is the classifying space of the pure braid group on \(r\) strings. The space \(\operatorname{Map}(\vee_{r} {\mathbb{S}}^{1}, BG)\) is homotopy equivalent to the homotopy quotient \(G^r//G\), where \(G\) is acting on each coordinate of \(G^r\) by conjugation. It follows in particular that the spaces \(W_n^T(r)\) that constitute the operad \(E_2^G\) are aspherical. Therefore \(E_2^G\) can be realized as the simplicial nerve of an operad in groupoids. The authors give an explicit presentation of this operad in groupoids in terms of generators and relations, using so called parenthesized \(G\)-braids. The main result of the paper describes the algebras over this operad in groupoids in the category of small categories. The algebras are identified as braided \(G\)-crossed categories. The last section discusses applications to topological field theories.

MSC:

55P48 Loop space machines and operads in algebraic topology
18M75 Topological and simplicial operads
57R56 Topological quantum field theories (aspects of differential topology)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 10.1090/S0894-0347-10-00669-7 · Zbl 1202.14015 · doi:10.1090/S0894-0347-10-00669-7
[2] 10.1090/conm/431/08265 · doi:10.1090/conm/431/08265
[3] 10.1090/S0002-9904-1968-12070-1 · Zbl 0165.26204 · doi:10.1090/S0002-9904-1968-12070-1
[4] 10.1007/BFb0068547 · doi:10.1007/BFb0068547
[5] 10.1007/978-1-4757-6848-0 · doi:10.1007/978-1-4757-6848-0
[6] 10.2140/agt.2019.19.533 · Zbl 1420.18012 · doi:10.2140/agt.2019.19.533
[7] 10.1007/BF01443193 · JFM 05.0227.01 · doi:10.1007/BF01443193
[8] 10.1007/BF02684599 · Zbl 0181.48803 · doi:10.1007/BF02684599
[9] 10.4007/annals.2016.183.3.1 · Zbl 1342.14055 · doi:10.4007/annals.2016.183.3.1
[10] 10.4171/QT/6 · Zbl 1214.18007 · doi:10.4171/QT/6
[11] ; Fresse, Homotopy of operads and Grothendieck-Teichmüller groups, I : The algebraic theory and its topological background. Mathematical Surveys and Monographs, 217 (2017) · Zbl 1373.55014
[12] 10.1007/s11511-009-0036-9 · Zbl 1221.57039 · doi:10.1007/s11511-009-0036-9
[13] 10.1016/j.jalgebra.2017.05.027 · Zbl 1401.18021 · doi:10.1016/j.jalgebra.2017.05.027
[14] 10.2140/ant.2009.3.959 · Zbl 1201.18006 · doi:10.2140/ant.2009.3.959
[15] 10.1007/BF02102639 · Zbl 0807.17026 · doi:10.1007/BF02102639
[16] 10.1007/s11856-008-0006-5 · Zbl 1143.14003 · doi:10.1007/s11856-008-0006-5
[17] 10.1007/BF01199469 · JFM 23.0429.01 · doi:10.1007/BF01199469
[18] 10.1016/0022-4049(94)90097-3 · Zbl 0791.18010 · doi:10.1016/0022-4049(94)90097-3
[19] 10.1090/conm/310/05402 · doi:10.1090/conm/310/05402
[20] 10.4310/ATMP.2012.v16.n1.a7 · Zbl 1273.81198 · doi:10.4310/ATMP.2012.v16.n1.a7
[21] 10.1016/j.jalgebra.2004.02.026 · Zbl 1052.18004 · doi:10.1016/j.jalgebra.2004.02.026
[22] 10.4310/HHA.2020.v22.n1.a3 · Zbl 1453.57025 · doi:10.4310/HHA.2020.v22.n1.a3
[23] 10.1007/s40062-019-00242-3 · Zbl 1444.57020 · doi:10.1007/s40062-019-00242-3
[24] 10.1017/CBO9781107261457 · Zbl 1317.18001 · doi:10.1017/CBO9781107261457
[25] 10.1016/j.jpaa.2019.106213 · Zbl 1429.57031 · doi:10.1016/j.jpaa.2019.106213
[26] 10.1093/acprof:oso/9780199605880.001.0001 · doi:10.1093/acprof:oso/9780199605880.001.0001
[27] 10.1017/S0305004100055535 · Zbl 0392.18001 · doi:10.1017/S0305004100055535
[28] 10.4171/086 · Zbl 1243.81016 · doi:10.4171/086
[29] 10.1142/S0129167X12500942 · Zbl 1254.57012 · doi:10.1142/S0129167X12500942
[30] 10.1142/S0129167X1450027X · Zbl 1296.57016 · doi:10.1142/S0129167X1450027X
[31] ; Yau, Colored operads. Graduate Studies in Mathematics, 170 (2016) · Zbl 1348.18014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.