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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.