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