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).

nLab > Latest Changes: Examples of why isomorphism of categories is too strong.

Bottom of Page

1 to 8 of 8

  1.  
    • CommentRowNumber1.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 3rd 2017

    The next-to-latest revision of equivalence of categories had a “query” to add an “intuitively clear” example why the notion of strict isomorphism of categories is too strong to be useful. I cannot think of a better example than the category of pointed sets versus the category of partial functions. In particular since even readers who have never learned category theory are likely to have been weaned on partial functions. I have therefore started to anser to this “query” with a condensed exposition of this example. The exposition had to broken off for the time being though. I intend to finish it tomorrow, complete with a proof that the categories are not isomorphic and a brief intuitive argument why they are (to be considered) the same nevertheless.

    Comments on whether you agree to use this example appreciated.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 3rd 2017
    • (edited Jun 3rd 2017)

    We have that equivalence described at partial function.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 3rd 2017

    I’m not sure that giving an example of an equivalence (that is not necessarily an isomorphism) really constitutes an adequate explanation of why strict isomorphism is too strong a notion.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 3rd 2017

    It might be worth looking at the source for equivalences, which I think is the Tohoku paper of Grothendieck.

    • CommentRowNumber5.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 4th 2017

    @David_Corfield. Thanks for pointing out. Then I will perhaps just briefly summarize this in words and link to the existing treatment. Also, an even more “bell-ringing” example, intuitive for many readers, would be the usual example that the category “of integer dimensions”, with real matrices as the morphisms is equivalent but not isomorphic to the category of finite-dimensional real vector spaces, and I now intend to mention this, too, as an exercise in exposition.

    Something intervened today, though, so continuing this little section in equivalence of categories will have to wait till tomorrow.

    • CommentRowNumber6.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 7th 2017

    Not to leave this unfinished for too long, it was now decided to not use the partial-function-example as the requested intuitive example, but rather the usual trivial example of the category of finitely dimensional vector spaces versus the category of finite dimensions. This will be known to most readers. The partial-functions-example is briefly mentioned after that, with a link.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJul 21st 2017

    Thanks for adding these examples! However, I think the discussion that went along with them is unnecessarily wordy and conveys hardly any information aside from the distinction of size, so I condensed it greatly. Also, I generally dislike subsections that are the only subsection in their section, so I removed the section heading and incorporated them as bullet points in the containing section “Isomorphism”.

    • CommentRowNumber8.
    • CommentAuthorPeter Heinig
    • CommentTimeJul 21st 2017

    Re #7. Thanks for the edits and advice. I was aware that this is almost entirely about size. I also like it better this way.

1 to 8 of 8

  1.