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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2018

    there was a pointer to the generalization of “faithful functor” to 2-categories. I have added below that pointer to the corresponding version for (,1)(\infty,1)-categories.

    diff, v11, current

    • CommentRowNumber2.
    • CommentAuthorGrant_Bradley
    • CommentTimeJul 2nd 2021

    added properties of faithful functor in terms of hom-sets

    diff, v13, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2021

    If it wasn’t clear the way it was, then I suggest to expand on it while sticking to the notation that the paragraph starts with. I have made it show this (here):

    𝒞(x,y) F x,y 𝒟(F(x),F(y)) (xϕy) (F(x)F(ϕ)F(y)). \array{ \mathcal{C}(x,y) &\xhookrightarrow{\;\; F_{x,y} \;\;}& \mathcal{D}(F(x), F(y)) \\ (x \overset{\phi}{\to} y) &\mapsto& \big( F(x)\overset{F(\phi)}{\to} F(y) \big) \,. }

    Also I have removed the side-remark you added that faithful functors are called “embeddings”, because that terminology is fraught with issues. Instead I have added under “Related concepts” a pointer to embedding of categories, where this is discussed in more detail.

    diff, v14, current

    • CommentRowNumber4.
    • CommentAuthorHurkyl
    • CommentTimeJul 2nd 2021
    • (edited Jul 2nd 2021)

    I’m not sure that 0-truncated morphisms are the right thing to list under generalizations of the notion of faithful functor.

    It seems a reasonable characterization for functors between infinity groupoids, since XYX \to Y being 0-truncated means X(x,x)=Ω xXΩ yY=Y(y,y)X(x,x) = \Omega_x X \to \Omega_y Y = Y(y,y) is a monomorphism.

    But it doesn’t seem reasonable for functors between infinity categories. For example, if CC is a 1-category with no nontrivial isomorphisms, then when viewing CC as an infinity category, C*C \to * is a 0-truncated morphism of Cat\infty Cat (since the nerve of CC is a complete segal space whose levels are all sets). But C*C \to * need not be a faithful functor in the sence of 1-categories.

    I would expect “locally monic” would be the right generalization to infinity categories, and dually “locally an effective epimorphism” would be the right generalization of “full”.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJul 2nd 2021

    Maybe just say that 0-truncated morphisms in an (,1)(\infty,1)-category are a generalization of faithful functors between groupoids?

    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeJul 2nd 2021

    I understand the motivation of the existing text, and have crafted a suitable warning.

    Aside: is there any reason we shouldn’t define a faithful functor between (∞,1)-categories to be one that is locally monic?

    diff, v15, current

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