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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeNov 8th 2017
    • (edited Nov 8th 2017)

    In the case of strict ω-categories, there is an ostensible generalization of the comma construction, where, given maps f:ABC:gf:A \to B \leftarrow C:g we let

    f ngf\downarrow_n g denote the limit of the cospan

    A D n1× B D n1C D n1B D n1× B D n1B D n1B D nA^{D^{n-1}} \times_{B^{\partial D^{n-1}}} C^{D^{n-1}} \to B^{D^{n-1}} \times_{B^{\partial D^{n-1}}} B^{D^{n-1}} \leftarrow B^{D^n}.

    The objects of this strict ω-category are pairs of n1n-1-cells aAa\in A, cCc \in C whose images under ff resp. gg are parallel, together with an nn-cell γ:f(a)g(c)\gamma: f(a) \to g(c), and the arrows are obvious arrows forming commuting diagrams.

    I think this construction might be interesting since we have a natural equipment of pairs of projections π s,t:f ngf n1g\pi_{s,t}: f \downarrow_n g \to f \downarrow_{n-1} g for each nn.

    In particular, this should be interesting if we want to define cartesian fibrations of ω-categories globally, when, given an isofibration p:EBp:E\to B

    we ask for the existence of a left adjoint right inverse to the induced maps E D np nBE^{D^n} \to p\downarrow_n B for each nn\in \mathbb{N}, and moreover we require that these are compatible with the projections (some kind of Beck-Chevalley condition, I think, and what this should mean intuitively, is that every cartesian nn-cell lies between cartesian n1n-1-cells).

    Is this kind of idea covered anywhere?

    • CommentRowNumber2.
    • CommentAuthorHarry Gindi
    • CommentTimeNov 12th 2017
    • (edited Nov 12th 2017)

    I found something in Gray’s old papers through Mike’s old notes on 3-toposes, and the correct construction uses the (op)lax Gray cotensor but is related.