Abstract
We study here a notion of simplicial satellites, as a first step towards a characterisation of simplicial derived functors, a problem unsolved since the latter were introduced.
The problem comes from the fact that, in contrast with the abelian case, simplicial derived functors do not produce by themselves an exact sequence. Our solution consists in extending them to commutative k-cubes, for all k, forming thus an “exact system” of functors universal within the “connected” ones; or, in other words, a system of simplicial satellites. The tool we develop here for this extension is the homotopy kernel of a commutative k-dimensional cubic diagram, generalising the homotopy kernel of a map; its 2-dimensional version has already been proved essential in other homotopical topics.
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Grandis, M. Simplicial Homotopical Algebra and Satellites. Applied Categorical Structures 5, 75–97 (1997). https://doi.org/10.1023/A:1008670229771
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DOI: https://doi.org/10.1023/A:1008670229771