In the present work, we extract pairs of topological spaces from maps between coloured operads. We prove that those pairs are weakly equivalent to explicit algebras over the one dimensional Swiss-Cheese operad . Thereafter, we show that the pair formed by the space of long embeddings and the manifold calculus limit of -immersions from to is an -algebra.
A partir d’un morphisme d’opérades colorées, on introduit un couple d’espaces topologiques que l’on identifie explicitement à une algèbre sous l’opérade Swiss-Cheese de dimension . Nous sommes alors en mesure d’identifier le couple formé des plongements longs et de l’approximation polynomiale des -immersions de vers à une algèbre sous l’opérade Swiss-Cheese de dimension .
Revised:
Accepted:
Published online:
Keywords: coloured operads, loop spaces, space of knots, model category
Mot clés : opérades colorées, espaces de lacets, espaces de plongements, catégorie modèle
@article{AIF_2018__68_2_661_0, author = {Ducoulombier, Julien}, title = {From maps between coloured operads to {Swiss-Cheese} algebras}, journal = {Annales de l'Institut Fourier}, pages = {661--724}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {2}, year = {2018}, doi = {10.5802/aif.3175}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.3175/} }
TY - JOUR AU - Ducoulombier, Julien TI - From maps between coloured operads to Swiss-Cheese algebras JO - Annales de l'Institut Fourier PY - 2018 SP - 661 EP - 724 VL - 68 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3175/ DO - 10.5802/aif.3175 LA - en ID - AIF_2018__68_2_661_0 ER -
%0 Journal Article %A Ducoulombier, Julien %T From maps between coloured operads to Swiss-Cheese algebras %J Annales de l'Institut Fourier %D 2018 %P 661-724 %V 68 %N 2 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3175/ %R 10.5802/aif.3175 %G en %F AIF_2018__68_2_661_0
Ducoulombier, Julien. From maps between coloured operads to Swiss-Cheese algebras. Annales de l'Institut Fourier, Volume 68 (2018) no. 2, pp. 661-724. doi : 10.5802/aif.3175. http://www.numdam.org/articles/10.5802/aif.3175/
[1] On the rational homology of high dimensional analogues of spaces of long knots, Geom. Topol., Volume 18 (2014) no. 3, pp. 1261-1322 | DOI | Zbl
[2] Axiomatic homotopy theory for operads, Comment. Math. Helv., Volume 78 (2003) no. 4, pp. 805-831 | DOI | Zbl
[3] The Boardman-Vogt resolution of operads in monoidal model categories, Topology, Volume 45 (2006) no. 5, pp. 807-849 | DOI | Zbl
[4] Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, 347, Springer, 1973, x+257 pages | Zbl
[5] Spaces of smooth embeddings and configuration categories (2015) (https://arxiv.org/abs/1502.01640)
[6] Homology of non-k-overlapping discs, Homology Homotopy Appl., Volume 17 (2015) no. 2, pp. 261-290 | DOI | Zbl
[7] Swiss-cheese action on the totalization of action-operads, Algebr. Geom. Topol., Volume 16 (2016) no. 3, 1683.1726 pages | DOI | Zbl
[8] Delooping derived mapping spaces of bimodules over an operad (2017) (https://arxiv.org/abs/1704.07062)
[9] Delooping the manifold calculus tower for closed discs (2017) (https://arxiv.org/abs/1708.02203)
[10] Long knots and maps between operads, Geom. Topol., Volume 16 (2012) no. 2, pp. 919-955 | DOI | Zbl
[11] Modules over operads and functors, Lecture Notes in Mathematics, 1967, Springer, 2009, ix+308 pages | Zbl
[12] Calculus. I. The first derivative of pseudoisotopy theory, -Theory, Volume 4 (1990) no. 1, pp. 1-27 | DOI | MR | Zbl
[13] Calculus. II. Analytic functors, -Theory, Volume 5 (1991/92) no. 4, pp. 295-332 | DOI | MR | Zbl
[14] Calculus. III. Taylor series, Geom. Topol., Volume 7 (2003), pp. 645-711 | DOI | MR | Zbl
[15] Multiple disjunction for spaces of smooth embeddings, J. Topol., Volume 8 (2015) no. 3, pp. 651-674 | DOI | Zbl
[16] Embeddings from the point of view of immersion theory. II, Geom. Topol., Volume 3 (1999), pp. 103-118 | DOI | MR | Zbl
[17] -actions and Recognition of Relative Loop Spaces, Topology Appl., Volume 206 (2016), pp. 126-147 | DOI | Zbl
[18] Model categories, Mathematical Surveys and Monographs, 63, American Mathematical Society, 1999, xii+209 pages | Zbl
[19] Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series, 64, Cambridge University Press, 1982 | Zbl
[20] Operads and motives in deformation quantization, Lett. Math. Phys., Volume 48 (1999) no. 1, pp. 35-72 | DOI | Zbl
[21] The geometry of iterated loop spaces, Lecture Notes in Mathematics, 271, Springer, 1972, ix+175 pages | Zbl
[22] Operads and cosimplicial objects: an introduction, Axiomatic, enriched and motivic homotopy theory (NATO Science Series II: Mathematics, Physics and Chemistry), Volume 131, Kluwer Academic Publishers, 2004, pp. 133-171 | DOI | Zbl
[23] Operads and knot spaces, J. Am. Math. Soc., Volume 19 (2006) no. 2, pp. 461-486 | DOI | Zbl
[24] Context-free manifold functor calculus and the Fulton-MacPherson operad, Algebr. Geom. Topol., Volume 13 (2013) no. 3, pp. 1243-1271 | DOI | Zbl
[25] Delooping totalization of a multiplicative operad, J. Homotopy Relat. Struct., Volume 9 (2014) no. 2, pp. 349-418 | DOI | MR | Zbl
[26] Cofibrant operads and universal -operads, Topology Appl., Volume 133 (2003) no. 1, pp. 69-87 | DOI | Zbl
[27] The Swiss-cheese operad, Homotopy invariant algebraic structure (Baltimore, MD, 1998) (Contemporary Mathematics), Volume 239, American Mathematical Society, 1999, pp. 365-373 | Zbl
[28] Embeddings from the point of view of immersion theory. I, Geom. Topol., Volume 3 (1999), pp. 67-101 erratum in ibid. 15 (2011), no. 1, p. 407-409 | DOI | MR | Zbl
Cited by Sources: