Homotopic distance between functors
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Homotopic distance between functors E. Macías-Virgós1 · D. Mosquera-Lois1 Received: 9 December 2019 / Accepted: 1 October 2020 © Tbilisi Centre for Mathematical Sciences 2020
Abstract We introduce a notion of categorical homotopic distance between functors by adapting the notion of homotopic distance in topological spaces, recently defined by the authors, to the context of small categories. Moreover, this notion generalizes the work on categorical LS-category of small categories by Tanaka. Keywords Homotopic distance · Lusternik-Schnirelmann category · Small categories Mathematics Subject Classification 55U10 · 55M30
1 Introduction Recently, some topological concepts were extended to small categories. This is the case of the Euler characteristic by Leinster [2,12] and both a notion of LusternikSchnirelman category [25] and a theory of Euler Calculus in the context of small categories by Tanaka [23,24]. Moreover, the authors have generalized both the LScategory and the Topological Complexity by means of a new notion of homotopic distance between continuous maps [13]. The purpose of this work is to adapt the notion of homotopic distance to the context of functors between small categories. Furthermore, this homotopic distance between functors generalizes the categorical LS-category introduced by Tanaka [25] and allows us to define a notion of Topological
Communicated by Emily Riehl. Partially supported by Xunta de Galicia ED431C 2019/10 with FEDER funds. The first author was partially supported by MINECO-FEDER research project MTM2016–78647–P. The second author was partly supported by Ministerio de Ciencia, Innovación y Universidades, Grant FPU17/03443.
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D. Mosquera-Lois [email protected] E. Macías-Virgós [email protected]
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Institute of Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain
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E. Macías-Virgós, D. Mosquera-Lois
Complexity for categories, which may be thought as an adaptation of the Topological Complexity introduced by Farber [4] to the context of categories. The organization of the paper is as follows: In Sect. 2 we recall the well known definitions of homotopy and weak homotopy for functors between categories and then we introduce two corresponding definitions of categorical distance, which we call categorical homotopic distance and weak categorical homotopic distance between functors, respectively. Section 3 is devoted to present particular cases of categorical homotopic distance such as the categorical LS-category introduced by Tanaka [25] and a new notion of Topological Complexity for categories. In Sect. 4 we prove several properties of the categorical homotopic distance such as its behavior under compositions and products and its homotopical invariance. Moreover, we prove that the homotopic distance between functors is bounded above by the category of the domain. Afterwards, we relate the two notions of homotopic distance between the functors F, G to the homotopic distance of the continuous maps BF, BG associated by the classifying space functo
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