Abstract
We have introduced, in a previous paper, the fundamental lax 2-category of a ‘directed space’ \(X\). Here we show that, when \( X \) has a \(T_1\)-topology, this structure can be embedded into a larger one, with the same objects (the points of \( X\)), the same arrows (the directed paths) and the same cells (based on directed homotopies of paths), but a larger system of comparison cells. The new comparison cells are absolute, in the sense that they only depend on the arrows themselves rather than on their syntactic expression, as in the usual settings of lax or weak structures. It follows that, in the original structure, all the diagrams of comparison cells commute, even if not constructed in a natural way and even if the composed cells need not stay within the old system.
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References
Bauer, A., Birkedal, L., Scott, D.S.: Equilogical spaces. Theor. Comp. Sci. 315, 35–59 (2004)
Bénabou, J.: Catégories relatives. C. R. Acad. Sci. Paris 260, 3824–3827 (1965)
Bénabou, J.: Introduction to bicategories. In: Reports of the Midwest Category Seminar, Lecture Notes in Mathematics, vol. 47, pp. 1–77. Springer, Berlin Heidelberg New York (1967)
Brown, R., Hardie, K.A., Kamps, K.H., Porter, T.: A homotopy double groupoid of a Hausdorff space. Theory Appl. Categ. 10, 71–93 (2002)
Brown, R., Kamps, K.H., Porter, T.: A homotopy double groupoid of a Hausdorff space II: A van Kampen theorem. Theory Appl. Categ. 14, 200–220 (2005)
Burroni, A.: T-catégories. Cah. Topol. Géom. Différ. 12, 215–321 (1971)
Ehresmann, C.: Catégories et Structures. Dunod, Paris (1965)
Fajstrup, L., Raussen, M., Goubault, E., Haucourt, E.: Components of the fundamental category. Appl. Categ. Struct. 12, 81–108 (2004)
Goubault, E.: Geometry and concurrency: A user’s guide. Geometry and concurrency. Math. Struct. Comput. Sci. 10(4), 411–425 (2000)
Grandis, M.: Directed homotopy theory. I. The fundamental category. Cah. Topol. Géom. Différ. Catég. 44, 281–316 (2003)
Grandis, M.: Inequilogical spaces, directed homology and noncommutative geometry. Homology Homotopy Appl. 6, 413–437 (2004)
Grandis, M.: The shape of a category up to directed homotopy. Theory Appl. Categ. 15(4), 95–146 (2005) (CT2004)
Grandis, M.: Lax 2-categories and directed homotopy. Cah. Topol. Géom. Diff. Catég., to appear. Dip. Mat. Univ. Genova, Preprint 530 (2005). http://www.dima.unige.it/~grandis/LCat.pdf
Kelly, G.M.: On Mac Lane’s conditions for coherence of natural associativities, commutativities, etc. J. Algebra 1, 397–402 (1964)
Leinster, T.: Higher Operads, Higher Categories. Cambridge University Press, Cambridge (2004)
Mac Lane, S.: Natural associativity and commutativity. Rice Univ. Stud. 49, 28–46 (1963)
Scott, D.: A new category? Domains, spaces and equivalence relations, Unpublished manuscript (1996). http://www.cs.cmu.edu/Groups/LTC/
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Grandis, M. Absolute Lax 2-categories. Appl Categor Struct 14, 191–214 (2006). https://doi.org/10.1007/s10485-006-9017-8
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DOI: https://doi.org/10.1007/s10485-006-9017-8
Key words
- 2-categories
- bicategories
- lax 2-categories
- homotopy theory
- directed algebraic topology
- fundamental categories