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 bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality education elliptic-cohomology enriched 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 mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories 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 science 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).
  1. Added the property that final functors and discrete fibrations form an orthogonal factorisation system.

    diff, v22, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 28th 2019

    Borceux (Handbook of Categorical Algebra, Volume 1, Definition 2.11.1) defines final functors as those functors for which restriction preserves limits, which is the exact opposite of what the nLab says (i.e., restriction along final functors preserves colimits).

    Is this just Borceux’s idiosyncrasy? His books generally appear to use conventional terminology. Shouldn’t this be mentioned in the article?

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 28th 2019

    Added alternative meaning of “final” to the Idea section. (Making explicit something left somewhat implicit before in the mention of the alternative meaning of “cofinal”.)

    diff, v23, current

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 2nd 2019

    The new version is confusing: “final” appears as a name for two opposite notions!

    Maybe we should offer some guide to the terminology. I already mentioned Borceux’s book terminology: cofinal for colimits, final for limits. Mac Lane’s terminology: final for colimits, initial for limits. What other canonical sources should be considered?

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeAug 3rd 2019

    I think Borceaux is the odd one out. We can mention his terminology, but we shouldn’t suggest that its use be continued.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeAug 3rd 2019

    More extensive and less ambiguous advice about terminology.

    diff, v24, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 3rd 2019

    Added a quote from Johnstone’s Elephant

    diff, v25, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 3rd 2019

    Added another reference point: Lurie’s Higher Topos Theory uses “cofinal”.

    diff, v25, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTime5 days ago

    added (here) statement of the example of the final functor

    ([1]d 0d 1[0])Δ op \big( [1] \underoverset {d_0} {d_1} {\rightrightarrows} [0] \big) \;\xrightarrow{\;\;\;\;}\; \Delta^{op}

    diff, v30, 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)