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-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology combinatorics complex-geometry computable-mathematics computer-science connection constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck 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 infinity integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar 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 stack 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.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2010

    started an Examples-section at natural numbers object

    btw, the natural numbers objects of a topos is unique, up to isomorphism, right? if so, we should say that

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJan 12th 2010

    Yes, of course, since it has a universal property.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2012

    I have expanded a little at Transfer of NNOs along inverse images.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 14th 2013

    I added an explicit definition of parametrized NNO to natural numbers object.

    I have a question: there is mention in that article of a categories with finite products and a parametrized NNO, and the initial such structure FF, and it is written that the canonical maps

    hom F(N k,N)hom Set( k,)\hom_F(N^k, N) \to \hom_{Set}(\mathbb{N}^k, \mathbb{N})

    surject onto the primitive recursive maps. Why couldn’t “surject” be replaced by “biject”?

    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeAug 14th 2013

    Perhaps it’s a question of extensionality? The morphisms N kNN^k \to N are surely constructed by syntactic means, and it’s conceivable that two such morphisms describe the same primitive recursive function in the standard model of arithmetic while not being provably equal (in whatever system of arithmetic corresponds to FF).

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 14th 2013

    Maybe, but I’d really like to see a specific example of that.

    • CommentRowNumber7.
    • CommentAuthorZhen Lin
    • CommentTimeAug 14th 2013

    Hmmm. Well, consider the primitive recursive “function” GG defined by G(n)=1G(n) = 1 if nn codes the proof of a contradiction in Peano arithmetic, and G(n)=0G(n) = 0 otherwise. (This is primitive recursive because no unbounded searches are required to verify the validity of a proof.) In the standard model of arithmetic, GG is the constant 00 function, but we could equally consider the elementary topos corresponding to a model of ZFC+¬Con(PA)ZFC + \neg Con(PA), in which GG is not the constant 00 function. Thus GG and the constant 00 function must be distinct in the category FF.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 14th 2013

    Ah yes, that certainly seems to work. Nice!

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 14th 2013
    • (edited Aug 14th 2013)

    But wait: ZFC proves that PA is consistent (since one can establish ordinal induction up to ε 0\epsilon_0 in ZFC).

    Edit: Well, I think the conclusion must still hold anyway: in the initial structure, it cannot be established that G=0G = 0, since that would amount to provability of that statement within primitive recursive arithmetic. So GG and 00 are distinct there.

    • CommentRowNumber10.
    • CommentAuthorZhen Lin
    • CommentTimeAug 14th 2013

    Oops, quite right. I suppose ZFC+¬Con(ZFC)ZFC + \neg Con(ZFC), with the “function” that verifies proofs of inconsistency of ZFC, is what I meant.

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 28th 2015

    Why is it that a parameterized NNO is the same as being preserved by all the functors to the coKelisli categories for the comonad A×A\times -? I’m trying to see how this works as I’d like to think of an NNO as being an initial algebra for a polynomial endofunctor (arising from 1←1→1+1→1) and these two pictures aren’t meshing.

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)