Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T17:25:12.864Z Has data issue: false hasContentIssue false
  • English
  • Français

Dwyer–Kan homotopy theory for cyclic operads

Part of: Applied homological algebra and category theory Categories with structure Homotopy theory

Published online by Cambridge University Press:  14 January 2021

Gabriel C. Drummond-Cole  and
Philip Hackney
Gabriel C. Drummond-Cole
Affiliation:
Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang37673, Republic of Korea (gabriel.c.drummond.cole@gmail.com)
Philip Hackney
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, USA (philip@phck.net)
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a general definition for coloured cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from coloured cyclic operads to coloured operads has both adjoints, each of which is relatively simple. Explicit formulae for these adjoints allow us to lift the Cisinski–Moerdijk model structure on the category of coloured operads enriched in simplicial sets to the category of coloured cyclic operads enriched in simplicial sets.

Keywords

cyclic operadQuillen model categorycoloured operadadjoint functorDwyer–Kan equivalence

MSC classification

Primary: 55P48: Loop space machines, operads
Secondary: 55U35: Abstract and axiomatic homotopy theory 18D50: Operads 18D20: Enriched categories (over closed or monoidal categories)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 1 , February 2021 , pp. 29 - 58
Check for updates
Copyright
Copyright © The Author(s) 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barr, M., *-autonomous categories, Lecture Notes in Mathematics, Volume 752 (Springer, Berlin, 1979). With an appendix by Po Hsiang Chu.CrossRefGoogle Scholar
Berger, C. and Moerdijk, I., On the homotopy theory of enriched categories, Q. J. Math. 64(3) (2013), 805–846.CrossRefGoogle Scholar
Bergner, J. E., A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359(5) (2007), 2043–2058.CrossRefGoogle Scholar
Bergner, J. E., A survey of $(\infty ,1)$-categories, in Towards higher categories, IMA Vol. Math. Appl., Volume 152, pp. 69–83 (Springer, New York, 2010).Google Scholar
Caviglia, G., Dagger categories, uncirculated manuscript, September 2014.Google Scholar
Caviglia, G., A model structure for enriched coloured operads, preprint arXiv:1401.6983.Google Scholar
Caviglia, G., The Dwyer-Kan model structure for enriched coloured PROPs, preprint arXiv:1510.01289.Google Scholar
Cheng, E., Gurski, N. and Riehl, E., Cyclic multicategories, multivariable adjunctions and mates, J. K-Theory 13(2) (2014), 337–396.CrossRefGoogle Scholar
Cisinski, D.-C. and Moerdijk, I., Dendroidal Segal spaces and $\infty$-operads, J. Topol. 6(3) (2013), 675–704.CrossRefGoogle Scholar
Cisinski, D.-C. and Moerdijk, I., Dendroidal sets and simplicial operads, J. Topol. 6(3) (2013), 705–756.CrossRefGoogle Scholar
Cockett, J. R. B. and Seely, R. A. G., Weakly distributive categories, in Applications of categories in computer science (Durham, 1991), London Math. Soc. Lecture Note Ser., Volume 177, pp. 45–65 (Cambridge Univ. Press, Cambridge, 1992).CrossRefGoogle Scholar
Cockett, J. R. B. and Seely, R. A. G., Weakly distributive categories, J. Pure Appl. Algebra 114(2) (1997), 133–173.CrossRefGoogle Scholar
Curien, P.-L. and Obradović, J., A formal language for cyclic operads, Higher Structures 1(1) (2017), 22–55.Google Scholar
Curien, P.-L. and Obradović, J., Categorified cyclic operads, Appl. Categ. Structures 28(1) (2020), 59–112.CrossRefGoogle Scholar
Drummond-Cole, G. C. and Hackney, P., A criterion for existence of right-induced model structures, Bull. Lond. Math. Soc. 51(2) (2019), 309–326.CrossRefGoogle Scholar
Drummond-Cole, G. C. and Hackney, P., Coextension of scalars in operad theory, preprint arXiv:1906.12275v1.Google Scholar
Dwyer, W. G. and Kan, D. M., Function complexes in homotopical algebra, Topology 19(4) (1980), 427–440.CrossRefGoogle Scholar
Gan, W. L., Koszul duality for dioperads, Math. Res. Lett. 10 (2003), 109–124.Google Scholar
Getzler, E. and Kapranov, M. M., Cyclic operads and cyclic homology, in Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, Volume IV, pp. 167–201 (Int. Press, Cambridge, MA, 1995).Google Scholar
Hackney, P. and Robertson, M., The homotopy theory of simplicial props, Israel J. Math. 219(2) (2017), 835–902.Google Scholar
Hackney, P., Robertson, M. and Yau, D., A simplicial model for infinity properads, High. Struct. 1(1) (2017), 1–21.Google Scholar
Hackney, P., Robertson, M. and Yau, D., Higher cyclic operads, Algebr. Geom. Topol. 19(2) (2019), 863–940.Google Scholar
Hackney, P., Robertson, M. and Yau, D., A graphical category for higher modular operads, Adv. Math. 365 (2020), 107044.CrossRefGoogle Scholar
Harpaz, Y. and Prasma, M., The Grothendieck construction for model categories, Adv. Math. 281 (2015), 1306–1363.CrossRefGoogle Scholar
Hinich, V. and Vaintrob, A., Cyclic operads and algebra of chord diagrams, Selecta Math. (N.S.) 8(2) (2002), 237–282.Google Scholar
Hyland, J. M. E., Proof theory in the abstract, Ann. Pure Appl. Logic 114(1–3) (2002), 43–78. Commemorative Symposium Dedicated to Anne S. Troelstra (Noordwijkerhout, 1999).Google Scholar
Intermont, M. and Johnson, M. W., Model structures on the category of ex-spaces, Topol. Appl. 119(3) (2002), 325–353.Google Scholar
Jacobs, B., Categorical logic and type theory, Studies in Logic and the Foundations of Mathematics, Volume 141 (North-Holland Publishing Co., Amsterdam, 1999).Google Scholar
Joyal, A. and Kock, J., Feynman graphs, and nerve theorem for compact symmetric multicategories (extended abstract), Electron. Notes Theor. Comput. Sci. 270(2) (2011), 105–113.CrossRefGoogle Scholar
Kaufmann, R. M. and Lucas, J., Decorated Feynman categories, J. Noncommut. Geom. 11 (2017), 1437–1464.CrossRefGoogle Scholar
Leinster, T., Higher operads, higher categories, London Mathematical Society Lecture Note Series, Volume 298 (Cambridge University Press, Cambridge, 2004).Google Scholar
Markl, M., Modular envelopes, OSFT and nonsymmetric (non-$\Sigma$) modular operads, J. Noncommut. Geom. 10(2) (2016), 775–809.Google Scholar
Markl, M., Shnider, S. and Stasheff, J., Operads in algebra, topology and physics, Mathematical Surveys and Monographs, Volume 96 (American Mathematical Society, Providence, RI, 2002).Google Scholar
Moerdijk, I. and Weiss, I., Dendroidal sets, Algebr. Geom. Topol. 7 (2007), 1441–1470.CrossRefGoogle Scholar
Muro, F., Dwyer-Kan homotopy theory of enriched categories, J. Topol. 8(2) (2015), 377–413.CrossRefGoogle Scholar
Obradović, J., Monoid-like definitions of cyclic operad, Theory Appl. Categ. 32(12) (2017), 396–436.Google Scholar
Riehl, E., Category theory in context, in Aurora Modern Math Originals (Dover, 2016).Google Scholar
Robertson, M., The homotopy theory of simplicially enriched multicategories, preprint arXiv:1111.4146.Google Scholar
Shulman, M., The 2-Chu-Dialectica construction and the polycategory of multivariable adjunctions, Theory Appl. Categ. 35 (2020), 89–136.Google Scholar
Templeton, J. J., Self-dualities, graphs and operads, Ph.D. thesis, University of Cambridge, 2003.Google Scholar
Walde, T., $2$-Segal spaces as invertible $\infty$-operads, preprint arXiv:1709.09935v1.Google Scholar
Ward, B. C., Six operations formalism for generalized operads, Theory Appl. Categ. 34(6) (2019), 121–169.Google Scholar
Yau, D., Dwyer–Kan homotopy theory of algebras over operadic collections, preprint arXiv:1608.01867.Google Scholar
Yau, D. and Johnson, M. W., A foundation for PROPs, algebras, and modules, Mathematical Surveys and Monographs, Volume 203 (American Mathematical Society, Providence, RI, 2015).CrossRefGoogle Scholar