Not signed in (Sign In)

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 book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation 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-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum 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 sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft 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.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2010

    added to exact functor the characterization of left exact functors as those preserving terminal object and pullbacks. This was previously stated only at finitely complete category.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeDec 3rd 2011
    • (edited Dec 3rd 2011)

    I find the discussion at exact functor a bit strange. Usually a functor is DEFINED to be left exact if it preserves finite limits. That is a clear precise definition… but is given as being just the ’idea’ whilst the ’definition’ is in terms of the characterisation (already essentially in Grothendieck’s Bourbaki seminar(195) in 1960 by the way) that this corresponded to a certain comma category being (co)filtering. This property is fine fro generalisation, but it seems very odd to give it as the definition. Does anyone else agree with this or am I being a traditionalist!!!!

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeDec 3rd 2011

    I agree. I would only use the term “left exact functor” for a finite-limit preserving functor between categories that have finite limits. The more general notion I would call flat.

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeDec 4th 2011
    • (edited Dec 4th 2011)

    I would agree with your extra point on flat, Mike. Mine was a bit more pickie! I think we should avoid having the deep plunge into abstraction too soon in entries. (This could be seen as a criticism on nLab by some in the recent discussion.) The question is in any given context / entry should a pedagogic or perhaps historical perspective come fairly early on (even if there is a historical paragraph later on perhaps). I have to write up something on Grothendieck’s Galois theory for another piece of work and have been looking back at his Techniques de descent. (Very clearly written) I will try to restructure exact functor later when I have finished what I have to do. (I also think that Grothendieck’s Galois theory needs a bit a work, but cannot see yet what needs adding.)

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeDec 5th 2011

    I have made some readjustments to exact functor.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeDec 6th 2011

    Thanks! I tried to clarify further.

    • CommentRowNumber7.
    • CommentAuthorvarkor
    • CommentTimeApr 20th 2023

    Add redirects for “finitely continuous (functors)”.

    diff, v33, current