Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundle bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched etcs fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorHurkyl
    • CommentTimeMay 28th 2020

    Characterized full faithfulness in terms of a pullback

    diff, v15, current

    • CommentRowNumber2.
    • CommentAuthorHurkyl
    • CommentTimeMay 28th 2020

    (I have a gap in my argument so I’ve retracted it for now)

    diff, v15, current

    • CommentRowNumber3.
    • CommentAuthorRichard Williamson
    • CommentTimeMay 28th 2020
    • (edited May 28th 2020)

    What was the argument you were trying to give? Maybe it can be repaired, and anyhow it would be interesting to know how you were thinking about this!

    • CommentRowNumber4.
    • CommentAuthorRichard Williamson
    • CommentTimeMay 28th 2020
    • (edited Jun 3rd 2020)

    One can probably for example say: a functor f:ABf: A \rightarrow B is fully faithful if, for any arrow gg of BB, viewed as a functor IBI \rightarrow B, where II is the interval category, the pullback of ff and gg can be taken to be II with the identity arrow as the projection to II if the pullback of ff and the restriction of gg to the inclusion of 111 \sqcup 1 into II is non-empty (it is necessarily empty otherwise). Is that the kind of thing you had in mind?

    • CommentRowNumber5.
    • CommentAuthorRichard Williamson
    • CommentTimeMay 28th 2020
    • (edited May 28th 2020)

    One can probably also say: the pullback of the functor B IB 11B^I \rightarrow B^{1 \sqcup 1} and the functor f 11:A 11B 11f^{1 \sqcup 1}: A^{1 \sqcup 1} \rightarrow B^{1 \sqcup 1} can be taken to be A IA^I with the obvious projections.

    • CommentRowNumber6.
    • CommentAuthorRichard Williamson
    • CommentTimeMay 28th 2020
    • (edited May 28th 2020)

    As a random aside, the difference between #4 and #5 is a quite typical example of a translation of a weaker (foundationally, or alternatively with regard to the structure required of the ’doctrine’ one is working in ), in particular predicative, formulation into a concise but impredicative one!

    • CommentRowNumber7.
    • CommentAuthorHurkyl
    • CommentTimeMay 29th 2020
    • (edited May 29th 2020)

    Yes, #5 was the characterization I was thinking of; that A IA^I is the pullback (in the (,1)(\infty,1) category of \infty-categories) of A×AB×BB IA \times A \to B \times B \leftarrow B^I in the manner you describe. Or maybe that you need to restrict to the core first. It’s clear that this implies full faithfulness, but I’ve confused myself over the details for the reverse implication and haven’t gotten around to looking at it again.

  1. Ah, I hadn’t actually noticed that you were asking about (,1)(\infty,1)-functors! We should add something about this in the ordinary category theory setting to full and faithful functor too.

    • CommentRowNumber9.
    • CommentAuthorHurkyl
    • CommentTimeMay 29th 2020

    I found a reference for the statement, so I’ll cite Cisinski for it.

    diff, v16, current

    • CommentRowNumber10.
    • CommentAuthorHurkyl
    • CommentTimeJun 1st 2020

    Improved the list of characterizations of full faithfulness.

    diff, v17, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)