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  1. I corrected a couple of wrong claims and added the link to a counter-example


    diff, v9, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2019

    made the reference-link work, by adding the missing anchor


    diff, v10, current

  2. exibits -> exhibits

    Ian Coley

    diff, v11, current

  3. Added the fact that (weak) localizations are initial and final, together with the fact that the initial vertex map from the category of simplices is a weak localization (and hence initial and final). Added references to Shah and Cisinski about this and noted that this can be used to prove a Bousfield-Kan formula for homotopy (co)limits in infinity categories. Hopefully didn’t screw up the whole page- I’m horrible at html/computers in general…


    diff, v13, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 3rd 2020
    • (edited Jul 3rd 2020)

    I have added an Example-section “Example: Cofiber products in coslice categories” (here). Currently it reads as follows:

    Consider the inclusion of the walking span-category, into the result of adjoining an initial object tt:

    (1){x b y}AAAA{ t x b y} \mathllap{ (1) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; } \Big\{ \array{ x &\longleftarrow& b &\longrightarrow& y } \Big\} \;\; \overset{\phantom{AAAA}}{\hookrightarrow} \;\; \left\{ \array{ && t \\ & \swarrow & \downarrow & \searrow \\ x &\longleftarrow& b &\longrightarrow& y } \right\}

    One readily sees that for each object on the right, its comma category over this inclusion has contractible nerve, whence Theorem \ref{Recognition} implies that this inclusion is a final \infty-functor.

    As an application of the finality of (1), observe that for 𝒞\mathcal{C} an (∞,1)-category and T𝒞T \in \mathcal{C} an object, (∞,1)-colimits in the under-(∞,1)-category

    𝒞 T/U𝒞 \mathcal{C}^{T/} \overset{\;\;U\;\;}{\longrightarrow} \mathcal{C}

    are given by the \infty-colimit in 𝒞\mathcal{C} itself of the given cone of the original diagram, with tip XX (by this Prop.): For

    F:𝒞 T/ F \;\colon\; \mathcal{I} \longrightarrow \mathcal{C}^{T/}

    a small diagram, we have

    U(limF)lim(T/U(F)) U \big( \underset{\longrightarrow}{\lim}\, F \big) \;\simeq\; \underset{\longrightarrow}{\lim}\, \big( T/U(F) \big)

    (when either \infty-colimit exists).

    Now for \mathcal{I} the walking span diagram on the left of (1), this means that homotopy cofiber products in 𝒞 T/\mathcal{C}^{T/} are computed as \infty-colimits in 𝒞\mathcal{C} of diagrams of the shape on the right of (1). But since the inclusion in (1) is final, these are just homotopy cofiber products in 𝒞\mathcal{C}.

    Explicitly: Given

    T = T = T ϕ X ϕ B ϕ Y X f B g Y \array{ T &=& T &=& T \\ {}^{\mathllap{ \phi_X }} \big\downarrow && {}^{\mathllap{ \phi_B }} \big\downarrow && {}^{\mathllap{ \phi_Y }} \big\downarrow \\ X & \underset{ f }{\longleftarrow} & B & \underset{ g }{ \longrightarrow } & Y }

    regarded as a span in 𝒞 T\mathcal{C}^T, hence with underlying objects

    U((X,ϕ X))=X,U((B,ϕ B))=B,U((Y,ϕ Y))=Y, U\big( (X,\phi_X) \big) \;=\; X \,, \;\;\;\;\;\; U\big( (B,\phi_B) \big) \;=\; B \,, \;\;\;\;\;\; U\big( (Y,\phi_Y) \big) \;=\; Y \,,

    we have:

    U((X,ϕ X)(B,ϕ B)(Y,ϕ Y))XBY. U \Big( \; (X,\phi_X) \underset{ (B,\phi_B) }{\coprod} (Y,\phi_Y) \; \Big) \;\;\;\simeq\;\;\; X \underset{B}{\coprod} Y \,.

    In particular, if (B,ϕ B)(T,id T)(B,\phi_B) \;\coloneqq\; (T,id_T) is the initial object in 𝒞 T/\mathcal{C}^{T/}, in which case the cofiber product is just the coproduct

    (X,ϕ X)(Y,ϕ Y)=(X,ϕ X)(T,id T)(Y,ϕ Y) (X,\phi_X) \coprod (Y,\phi_Y) \;\;=\;\; (X,\phi_X) \underset{ (T,id_T) }{\coprod} (Y,\phi_Y)

    we find that the coproduct in the co-slice category is the co-fiber product under the given tip object in the underlying category

    U((X,ϕ X)(Y,ϕ Y))XTY. U \Big( \; (X,\phi_X) \coprod (Y,\phi_Y) \; \Big) \;\;\;\simeq\;\;\; X \underset{T}{\coprod} Y \,.

    diff, v14, current

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 19th 2021



    The term “cofinal (∞,1)-functor” can mean either a functor for which the precomposition functor preserves colimits (in Lurie’s Higher Topos Theory) or limits (in Cisinski’s Higher Categories and Homotopical Algebra.

    Given that the two main sources for quasicategories assign opposite meanings to this term, it is best to avoid its usage altogether.

    Further adding to the confusion is that some sources, like Borceux’s Handbook of Categorical Algebra use the term “final functor” for a functor for which the precomposition functor preserves limits, in contrast to the majority of the literature. Such usage, fortunately, is marginal.

    diff, v17, current

    • CommentRowNumber7.
    • CommentAuthorHurkyl
    • CommentTimeDec 29th 2021

    Remarked that a final functor between categories need not be final (∞,1)-functor. (but the converse is true)

    diff, v19, current

    • CommentRowNumber8.
    • CommentAuthorHurkyl
    • CommentTimeDec 29th 2021

    Remarked that a final functor between categories need not be final (∞,1)-functor. (but the converse is true)

    diff, v19, current

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 7th 2022

    Added the following paragraph about the new terminology introduced by Kerodon (as if the previous mess wasn’t enough):

    In Kerodon, final (∞,1)-functors are referred to as right cofinal functors. Likewise, initial (∞,1)-functors are known as left cofinal functors.

    diff, v20, current

    • CommentRowNumber10.
    • CommentAuthorRuneHaugseng
    • CommentTimeMay 7th 2022

    Re #6: I don’t think it’s great for the nlab to say that the term “cofinal” should be avoided when it’s by far the most common terminology in the literature on \infty-categories. What should definitely be avoided is using both “final” and “cofinal” as dual notions, but (with the unfortunate exception of Cisinski’s book, apparently) I believe the term “cofinal” unambiguously refers to colimits, while “final” has been used for both variants often enough that I think there’s a stronger argument for avoiding that word…

    • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 7th 2022

    Added a new section (also in response to #10 above):


    In most major references, “final map” is used to refer to (∞,1)-functors such that restriction along them preserves colimit cones and “initial map” is used for limit cones. A mnemonic rule for this terminology is that final objects are picked out by final functors and likewise for initial objects.

    Lurie’s work is a notable exception to this rule.

    diff, v21, current

    • CommentRowNumber12.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 7th 2022

    Re #10: Is there any source other than Borceux that uses “final” to refer to the other notion?

    • CommentRowNumber13.
    • CommentAuthorHurkyl
    • CommentTimeAug 6th 2022

    Changed “generalization” to “vertical categorification” in the idea section, to reflect the fact that final functors between (1,1)-categories need not be final (∞,1)-functors.

    diff, v22, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2022
    • (edited Aug 6th 2022)

    The section “Warning” overlaps with the material in the section “Terminology”. Since a warning should not be a section anyway, I have changed it from a section to a remark (and used the occasion to point to this remark from the first line of the idea section). But the overlap of material remains. Maybe somebody could streamline the section “Terminology” a little more.

    diff, v23, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2022

    I have now adjusted a little in the (merged) section Terminology (here).

    Also added direct cross-links with the corresponding remark at final functor.

    diff, v24, current

    • CommentRowNumber16.
    • CommentAuthorMarc Hoyois
    • CommentTimeMar 16th 2023

    Lurie’s Higher Algebra uses the opposite convention as Kerodon for final/initial functors. Added this to the list.

    diff, v26, current

    • CommentRowNumber17.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 13th 2024

    Added a reference to Hirschhorn.

    diff, v28, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMar 14th 2024

    completed this and other bibitems and brought them into chronological order

    diff, v29, current