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 categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science connection constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry goodwillie-calculus 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 homology homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory internal-categories k-theory lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory 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 stack string-theory 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.
    • CommentAuthorDexter Chua
    • CommentTimeSep 23rd 2016
    • (edited Sep 23rd 2016)

    Added some more explicit constructions in limits and colimits by example, which are hopefully more pedagogically appropriate, as Urs suggested in the discussion. A lot more details are to be filled in.

    • CommentRowNumber2.
    • CommentAuthorDylan Wilson
    • CommentTimeSep 23rd 2016

    One fun thing would be to show by explicit example how each of the colimits in Set can be computed as a limit. For example: the empty set is one of the elements in the set of subsets of the one element set, quotients by an equivalence relation are modeled by the set of cosets, and coproducts over an index set A can be modeled in the following way: take the product of all the power sets and look at those A-tuples which are the empty set everywhere except one place where it’s a singleton. Then add a link to ’exponential object’ and ’topos’ or whatever…

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 24th 2016

    …as a limit involving power objects. Since the barest definition of elementary topos doesn’t mention colimits (power objects and finite limits only), all existing colimits can be built using these (only finite, or all, depending on taste).

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 24th 2016

    Would it be out of place to mention this work in progress, indicating how to prove this fact about an elementary topos using reasoning closer to the naive set theory? An Elementary Approach to Elementary Topos Theory. There are still some parts to fill in, but I’ve put it aside for a few months. May get back to it soon.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 24th 2016

    I was thinking of your old write-up Todd, so I’m glad this has progressed so much more! I seriously think this should eventually be published in a more traditional manner.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 25th 2016

    Thanks for your encouragement, David!

  1. At limits and colimits by example, there was an old query box by Mark C. His observation was right and I changed the entry accordingly.

    In the section on general colimits, under colimits, it stated that

    The equivalence relation \sim is that which is generated by

    ((xF(d))(xF(d)))if((f:dd)withF(f)(x)=x). ((x \in F(d)) \sim (x' \in F(d')))\quad if \quad (\exists (f : d \to d') with F(f)(x) = x') \,.

    If DD is a filtered category then the relation \sim already is an equivalence relation.

    Is this true? Consider a category with two objects d,dd, d' and a single non-identity morphism f:ddf: d \rightarrow d'. Then this is a directed category.

    Consider the functor F:DSetF: D \rightarrow Set taking both dd and dd' to singletons.

    Then, it seems to me the unique xF(d)x \in F(d) would be related to the unique xF(d)x' \in F(d'), but the opposite relation would not hold, since there is no morphism ddd' \rightarrow d in DD. Hence symmetry fails and this is not an equivalence relation.

    Am I missing something?

    EDIT: I think the definition should say:

    The equivalence relation \sim is that which is generated by

    ((xF(d))(xF(d)))if(d,(f:dd),(g:dd)withF(f)(x)=F(g)(x)). ((x \in F(d)) \sim (x' \in F(d')))\quad if \quad (\exists d'', (f : d \to d''), (g: d' \to d'') with F(f)(x) = F(g)(x')) \,.

    This matches the definition of a direct limit on Wikipedia.

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)