Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

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 internal-categories k-theory 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 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 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

1. • CommentRowNumber1.
   • CommentAuthorNoam_Zeilberger
   • CommentTimeJun 25th 2015
   • PermaLink
   Author: Noam_Zeilberger
   Format: MarkdownItexI was reading [[connected object]] (linked to from the new article [[class equation]]) and noticed that the definition of a connected object ($Hom(X,-)$ preserves all coproducts) is very similar to the definition of a [[tiny object]] ($Hom(X,-)$ preserves all small colimits). Neither article references the other, though. Would it be worth adding some explicit comparison? Since the definition of tiny object is stronger, are any of the examples in [[connected object#examples]] also examples of tiny objects, or if not, how do they fail to preserve colimits? (I see that there was some related discussion earlier on the nForum at [tiny objects and projective objects](http://nforum.ncatlab.org/discussion/6537/tiny-objects-and-projective-objects/).)

   I was reading connected object (linked to from the new article class equation) and noticed that the definition of a connected object ($Hom(X,-)$ preserves all coproducts) is very similar to the definition of a tiny object ($Hom(X,-)$ preserves all small colimits). Neither article references the other, though. Would it be worth adding some explicit comparison? Since the definition of tiny object is stronger, are any of the examples in connected object#examples also examples of tiny objects, or if not, how do they fail to preserve colimits? (I see that there was some related discussion earlier on the nForum at tiny objects and projective objects.)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 25th 2015
   • (edited Jun 25th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexBoth entries are cross-linked, sort of, via their joint "floating context menu" in the top right, which provides as a pull-down menu the content of _[[compact object - contents]]_. But of course you are invited to add more in-line cross-links where you see the need. Wherever you would have liked to find them, other readers may appreciate the link, too.

   Both entries are cross-linked, sort of, via their joint “floating context menu” in the top right, which provides as a pull-down menu the content of compact object - contents.

   But of course you are invited to add more in-line cross-links where you see the need. Wherever you would have liked to find them, other readers may appreciate the link, too.

3. • CommentRowNumber3.
   • CommentAuthorZhen Lin
   • CommentTimeJun 25th 2015
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexNone of the specific examples listed are tiny in general. I think the only tiny object in $\mathbf{Top}$ is the point, and I think the only tiny object in $G \text{-}\mathbf{Set}$ is $G$ (with the regular action).

   None of the specific examples listed are tiny in general. I think the only tiny object in $\mathbf{Top}$ is the point, and I think the only tiny object in $G \text{-}\mathbf{Set}$ is $G$ (with the regular action).

4. • CommentRowNumber4.
   • CommentAuthorNoam_Zeilberger
   • CommentTimeJun 25th 2015
   • (edited Jun 25th 2015)
   • PermaLink
   Author: Noam_Zeilberger
   Format: MarkdownItex@Urs: aha! I am not used to looking at the Context bar, but that list is very thorough indeed. I will start paying more attention to that feature of the nlab. @Zhen Lin: thanks. For the example of $G\text{-}\mathbf{Set}$, I see that $G$ is tiny because it corresponds to the representable presheaf $\mathbf{B}G(*,-)$. I don't have much intuition for why in general an inhabited transitive $G\text{-}\mathbf{Set}$ need not preserve colimits -- would you mind spelling that out?

   @Urs: aha! I am not used to looking at the Context bar, but that list is very thorough indeed. I will start paying more attention to that feature of the nlab.

   @Zhen Lin: thanks. For the example of $G\text{-}\mathbf{Set}$, I see that $G$ is tiny because it corresponds to the representable presheaf $\mathbf{B}G(*,-)$. I don’t have much intuition for why in general an inhabited transitive $G\text{-}\mathbf{Set}$ need not preserve colimits – would you mind spelling that out?

5. • CommentRowNumber5.
   • CommentAuthorZhen Lin
   • CommentTimeJun 25th 2015
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexIf you have a topos, then any tiny object is connected and projective. But a connected projective object in a presheaf topos must be a retract of a representable presheaf, and in the case of $G\text{-}\mathbf{Set}$ the only such is $G$ with its regular action.

   If you have a topos, then any tiny object is connected and projective. But a connected projective object in a presheaf topos must be a retract of a representable presheaf, and in the case of $G\text{-}\mathbf{Set}$ the only such is $G$ with its regular action.