Homotopy theory of Moore flows (II)
DOI:
https://doi.org/10.17398/2605-5686.36.2.157Keywords:
enriched semicategory, semimonoidal structure, combinatorial model category, Quillen equivalence, locally presentable category, topologically enriched categoryAbstract
This paper proves that the q-model structures of Moore flows and of multipointed d-spaces are Quillen equivalent. The main step is the proof that the counit and unit maps of the Quillen adjunction are isomorphisms on the q-cofibrant objects (all objects are q-fibrant). As an application, we provide a new proof of the fact that the categorization functor from multipointed d-spaces to flows has a total left derived functor which induces a category equivalence between the homotopy categories. The new proof sheds light on the internal structure of the categorization functor which is neither a left adjoint nor a right adjoint. It is even possible to write an inverse up to homotopy of this functor using Moore flows.
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J. Adámek and J. Rosický. Locally presentable and accessible categories. Cambridge University Press, Cambridge, 1994. https://doi.org/10.1017/cbo9780511600579.004. 4, 19
F. Borceux. Handbook of categorical algebra. 2. Cambridge University Press, Cambridge, 1994. Categories and structures. https://doi.org/10.1017/cbo9780511525865. 4
D. Christensen, G. Sinnamon, and E. Wu. The D-topology for diffeological spaces. Pacific Journal of Mathematics, 272(1):87–110, 2014. https://doi.org/10.2140/pjm.2014.272.87. 5
W. G. Dwyer, P. S. Hirschhorn, D. M. Kan, and J. H. Smith. Homotopy limit functors on model categories and homotopical categories, volume 113 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2004. https://doi.org/10.1090/surv/113. 1, 52
U. Fahrenberg and M. Raussen. Reparametrizations of continuous paths. Journal of Homotopy and Related Structures, 2(2):93–117, 2007. 25
R. Garner, M. Kȩdziorek, and E. Riehl. Lifting accessible model structures. Journal of Topology, 13(1):59–76, 2020. https://doi.org/10.1112/topo.12123. 59
P. Gaucher. A model category for the homotopy theory of concurrency. Homology, Homotopy and Applications, 5(1):p.549–599, 2003. https://doi.org/10.4310/hha.2003.v5.n1.a20. 1, 3, 48, 50, 56, 57
P. Gaucher. Comparing globular complex and flow. New York Journal of Mathematics, 11:97–150, 2005. 2, 47
P. Gaucher. T-homotopy and refinement of observation (IV) : Invariance of the underlying homotopy type. New York Journal of Mathematics, 12:63–95, 2006. 21, 53
P. Gaucher. Towards a homotopy theory of process algebra. Homology, Homotopy and Applications, 10(1):353–388, 2008. https://doi.org/10.4310/HHA.2008.v10.n1.a16. 2
P. Gaucher. Homotopical interpretation of globular complex by multipointed d-space. Theory and Applications of Categories, 22(22):588–621, 2009. 1, 2, 8, 10, 47, 49, 52, 54
P. Gaucher. Enriched diagrams of topological spaces over locally contractible enriched categories. New York Journal of Mathematics, 25:1485–1510, 2019. 15, 59, 60
P. Gaucher. Flows revisited: the model category structure and its left determinedness. Cahiers de Topologie et Géométrie Différentielle Catégoriques, LXI-2:208–226, 2020. 59
P. Gaucher. Homotopy theory of Moore flows (I). Compositionality, 3, 2021. https://doi.org/10.32408/compositionality-3-3. 1, 2, 6, 10, 15, 16, 17, 19, 20, 31, 38, 47, 49, 50, 53, 54, 59, 60
P. Gaucher. Left properness of flows. Theory and Applications of Categories, 37(19):562–612, 2021. 2, 5, 10, 54, 55, 56, 59
P. Gaucher. Six model categories for directed homotopy. Categories and General Algebraic Structures with Applications, 15(1):145–181, 2021. https://doi.org/10.52547/cgasa.15.1.145. 8, 10, 48, 59
M. Grandis. Directed homotopy theory. I. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 44(4):281–316, 2003. 2
M. Grandis. Inequilogical spaces, directed homology and noncommutative geometry. Homology, Homotopy and Applications, 6(1):413–437, 2004. https://doi.org/10.4310/hha.2004.v6.n1.a21. 2
M. Grandis. Normed combinatorial homology and noncommutative tori. Theory and Applications of Categories, 13(7):114–128, 2004. 2
M. Grandis. Directed combinatorial homology and noncommutative tori (the breaking of symmetries in algebraic topology). Mathematical Proceedings of the Cambridge Philosophical Society, 138(2):233–262, 2005. https://doi.org/10.1017/S0305004104008217. 2
A. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. 6, 22
S. Henry. Minimal model structures. 2020. https://arxiv.org/abs/2011.13408. 3
K. Hess, M. Kȩdziorek, E. Riehl, and B. Shipley. A necessary and sufficient condition for induced model structures. Journal of Topology, 10(2):324–369, 2017. https://doi.org/10.1112/topo.12011. 59
P. S. Hirschhorn. Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2003. https://doi.org/10.1090/surv/099. 4, 59
M. Hovey. Model categories. American Mathematical Society, Providence, RI, 1999. https://doi.org/10.1090/surv/063. 2, 4
V. Isaev. On fibrant objects in model categories. Theory and Applications of Categories, 33(3):43–66, 2018. 59
G. M. Kelly. Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, (10):vi+137 pp., 2005. Reprint of the 1982 original [Cambridge Univ. Press, Cambridge; MR0651714]. 4
S. Krishnan. A convenient category of locally preordered spaces. Applied Categorical Structures, 17(5):445–466, 2009. https://doi.org/10.1007/s10485-008-9140-9. 2
S. Mac Lane. Categories for the working mathematician. Springer-Verlag, New York, second edition, 1998. https://doi.org/10.1007/978-1-4757-4721-8. 51, 58
J. P. May and J. Sigurdsson. Parametrized homotopy theory, volume 132 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2006. https://doi.org/10.1090/surv/132. 5
R. J. Piacenza. Homotopy theory of diagrams and CW-complexes over a category. Canadian Journal of Mathematics, 43(4):814–824, 1991. https://doi.org/10.4153/CJM-1991-046-3. 59
D. Quillen. Homotopical algebra. Springer-Verlag, Berlin, 1967. https://doi.org/10.1007/BFb0097438. 59
J. Rosický. On combinatorial model categories. Applied Categorical Structures, 17(3):303–316, 2009. https://doi.org/10.1007/s10485-008-9171-2. 4
