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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 8th 2018

    Consider a model category M and a bifibrant replacement functor R: M→M.

    Define a new category M’ with the same objects as M by setting hom_M’(X,Y) := hom_M(R(X),R(Y)), with the obvious notions of composition and identity.

    The category M’ behaves in an intuitively correct way: homotopies are well-behaved, homotopy classes of pointed maps S^n→X compute the homotopy groups of X, etc.

    What would be a good name for morphisms of M’?

    For instance, one could say for M=sSet:

    for any integer d there is a (⋯) map S^n→S^n of degree d, where S^n=Δ^n/∂Δ^n is the standard simplicial sphere.

    What would be a good substitute for (⋯)?

    I am currently considering “weak map” and “derived map”, and possibly “∞-map”, but perhaps there are better alternatives.

    Of course, one could say “a point in the derived mapping space”, but this is a rather awkward name for such a simple idea.

    I do not want to restrict to bifibrant objects in M (where one could talk about ordinary maps in M), since in the case M=sSet this would eliminate all finite nondiscrete simplicial sets, which are an important source of examples.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeAug 9th 2018

    I think ’weak’ and ’derived’ are reasonable adjectives. Stretching an analogy one might also call it an ’ana-map’.

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 9th 2018
    • (edited Aug 9th 2018)

    I just discovered that the page homotopical cohomology theory (a rather weird name) refers to a similar concept as ana-morphisms, using zigzags instead of bifibrant replacements.

    And Urs refers to these as ∞-anafunctors in DCCT, §1.2.5.2.1.

    Considering the centrality of this concept for everything that concerns homotopy theory, a standardized name would be highly beneficial.

    Perhaps a reasonable choice would be to refer to 0-simplices of homs of hammock localizations as anamaps. Anamaps organize into a category, there is a natural notion of homotopy of anamaps (namely, a zigzag of 1-simplices in the above homs), and if an anamap is a weak equivalence (i.e., becomes invertible in the homotopy category), then it is invertible up to a homotopy.