homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
The homotopy 2-category of an (∞,n)-category is the 2-category with the same objects and 1-morphisms as and with the 2-morphisms being the equivalence classes of 2-morphisms of .
In other words, for every pair of objects in , the hom-category is the ordinary homotopy category of the -category .
On the Toda bracket understood as homotopy-coherent pasting diagrams in a pointed homotopy 2-category:
Keith Hardie, Klaus Heiner Kamps, Rudger Kieboom, Higher homotopy groupoids and Toda brackets, Homology Homotopy Appl. Volume 1, Number 1 (1999), 117-134 (euclid:hha/1139840198)
Keith Hardie, Howard Marcum, Nobuyuki Oda, Bracket operations in the homotopy theory of a 2-category, Rend. Ist. Mat. Univ. Trieste 33, 19–70 (2001) (rendiconti:33/02)
Keith Hardie, Klaus Heiner Kamps, Howard Marcum, The Toda bracket in the homotopy category of a track bicategory, Journal of Pure and Applied Algebra Volume 175, Issues 1–3, 8 November 2002, Pages 109-133 (doi:10.1016/S0022-4049(02)00131-7)
Howard Marcum, Nobuyuki Oda, Long Box Bracket Operations in Homotopy Theory, Appl Categor Struct 19, 137–173 (doi:10.1007/s10485-009-9186-3)
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