Abstract
We show that the quasicategory defined as the localization of the category of (simple) graphs at the class of A-homotopy equivalences does not admit colimits. In particular, we settle in the negative the question of whether the A-homotopy equivalences in the category of graphs are part of a model structure.
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References
Atkin, R.H.: An algebra for patterns on a complex. I. Int. J. Man Mach. Stud. 6, 285–307 (1974)
Atkin, R.H.: An algebra for patterns on a complex. II. Int. J. Man Mach. Stud. 8(5), 483–498 (1976)
Babson, E., Barcelo, H., de Longueville, M., Laubenbacher, R.: Homotopy theory of graphs. J. Algebr. Comb. 24(1), 31–44 (2006)
Barcelo, H., Kramer, X., Laubenbacher, R., Weaver, C.: Foundations of a connectivity theory for simplicial complexes. Adv. Appl. Math. 26(1), 97–128 (2001)
Barcelo, H., Laubenbacher, R.: Perspectives on \(A\)-homotopy theory and its applications. Discrete Math. 298(1–3), 39–61 (2005)
Carranza, D., Kapulkin, K.: Cubical setting for discrete homotopy theory, revisited (2022) preprint
Carranza, D., Kapulkin, K., Lindsey, Z.: Calculus of fractions for quasicategories (2023) preprint
Dochtermann, A.: Hom complexes and homotopy theory in the category of graphs. Eur. J. Comb. 30(2), 490–509 (2009)
Goyal, S., Santhanam, R.: (Lack of) model structures on the category of graphs. Appl. Categ. Struct. 29(4), 671–683 (2021)
Goyal, S., Santhanam, R.: Study of cofibration category structures on finite graphs (2023)
Kapulkin, K., Szumiło, K.: Quasicategories of frames of cofibration categories. Appl. Categ. Struct. 25(3), 323–347 (2017)
Szumiło, K.: Frames in cofibration categories. J. Homotopy Relat. Struct. 12(3), 577–616 (2017)
Funding
This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930 while the first two authors participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the 2022–23 academic year.
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All authors (DC, CK, and JK) contributed equally to the research and preparation of this work.
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Communicated by Tobias Dyckerhoff.
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Carranza, D., Kapulkin, K. & Kim, J. Nonexistence of Colimits in Naive Discrete Homotopy Theory. Appl Categor Struct 31, 41 (2023). https://doi.org/10.1007/s10485-023-09746-9
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DOI: https://doi.org/10.1007/s10485-023-09746-9