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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 5th 2015
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   Author: Todd_Trimble
   Format: MarkdownItexI just feel like bringing up a trap I fell into recently. A model of ETCS is a well-pointed topos $E$ with natural numbers object $N$ and satisfying AC, right? Being well-pointed, the terminal object $1$ is (externally) projective and connected; I'm assuming here that the meta-logic is classical, as needed. The trap I'd been falling into is thinking that projectivity and connectedness is the same as saying $\hom(1, -): E \to Set$ is right exact, i.e., preserves finite colimits. I don't think that can be right, because it would imply that $\hom(1, -): E \to Set$ preserves the natural numbers object, since natural numbers objects in toposes can be characterized in terms of finite colimits (famous result due to Freyd). If that were the case, then there can be no "nonstandard" elements $1 \to N$ in $E$ (taking $\mathbb{N}$ in $Set$ as our "standard"). So far I haven't found a spot in the nLab where there is an outright error, but I do think there are spots that are at least misleading. Over at [[Freyd cover]], Theorem 1, it says "The terminal object in the initial topos $\mathcal{T}$ is connected and projective, i.e., $\Gamma = \hom(1, -) \colon \mathcal{T} \to Set$ preserves finite colimits." Actually we do prove the stronger result that $\Gamma$ is right exact, but it's that "i.e." which seems a little misleading. Lawvere does make a far-sighted remark on this in his ETCS paper (the [TAC version](http://tac.mta.ca/tac/reprints/articles/11/tr11.pdf)), where he observes (Remark 8, page 31) that $\Gamma$ preserves coequalizers of kernel pairs = equivalence relations, but "it seems unlikely that it is necessary that $[\Gamma]$ be right exact, since as remarked above, the construction of the RST [reflective symmetric transitive] hull involves $N$ and by Gödel’s theorems the nature of the object $N$ may vary from one model $[E]$ of any theory to another model." I call this far-sighted because I don't think he was in possession of Freyd's theorem (1972) or even of topos theory when that was first written.

   I just feel like bringing up a trap I fell into recently. A model of ETCS is a well-pointed topos $E$ with natural numbers object $N$ and satisfying AC, right? Being well-pointed, the terminal object $1$ is (externally) projective and connected; I’m assuming here that the meta-logic is classical, as needed.

   The trap I’d been falling into is thinking that projectivity and connectedness is the same as saying $\hom(1, -): E \to Set$ is right exact, i.e., preserves finite colimits. I don’t think that can be right, because it would imply that $\hom(1, -): E \to Set$ preserves the natural numbers object, since natural numbers objects in toposes can be characterized in terms of finite colimits (famous result due to Freyd). If that were the case, then there can be no “nonstandard” elements $1 \to N$ in $E$ (taking $\mathbb{N}$ in $Set$ as our “standard”).

   So far I haven’t found a spot in the nLab where there is an outright error, but I do think there are spots that are at least misleading. Over at Freyd cover, Theorem 1, it says “The terminal object in the initial topos $\mathcal{T}$ is connected and projective, i.e., $\Gamma = \hom(1, -) \colon \mathcal{T} \to Set$ preserves finite colimits.” Actually we do prove the stronger result that $\Gamma$ is right exact, but it’s that “i.e.” which seems a little misleading.

   Lawvere does make a far-sighted remark on this in his ETCS paper (the TAC version), where he observes (Remark 8, page 31) that $\Gamma$ preserves coequalizers of kernel pairs = equivalence relations, but “it seems unlikely that it is necessary that $[\Gamma]$ be right exact, since as remarked above, the construction of the RST [reflective symmetric transitive] hull involves $N$ and by Gödel’s theorems the nature of the object $N$ may vary from one model $[E]$ of any theory to another model.” I call this far-sighted because I don’t think he was in possession of Freyd’s theorem (1972) or even of topos theory when that was first written.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeApr 6th 2015
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   Author: Mike Shulman
   Format: MarkdownItexThat's a good point and should be emphasized somewhere (everywhere)? I might have fallen into that trap myself on occasion. All finite colimits can be constructed from finite coproducts and coequalizers, and all coequalizers can be constructed from quotients of equivalence relations, but the latter requires some "infinitary" construction (either internal, by using an NNO, or external, by using countable unions/coproducts). Thus, a functor which preserves finite coproducts and epimorphisms, hence also quotients of equivalence relations, still may not preserve all coequalizers.

   That’s a good point and should be emphasized somewhere (everywhere)? I might have fallen into that trap myself on occasion. All finite colimits can be constructed from finite coproducts and coequalizers, and all coequalizers can be constructed from quotients of equivalence relations, but the latter requires some “infinitary” construction (either internal, by using an NNO, or external, by using countable unions/coproducts). Thus, a functor which preserves finite coproducts and epimorphisms, hence also quotients of equivalence relations, still may not preserve all coequalizers.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 6th 2015
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   Author: Todd_Trimble
   Format: MarkdownItexThanks for the comment, Mike -- I don't think I was aware of the remark that preservation of finite coproducts and epis implies preservation of quotients of equivalence relations. I'm trying to think of a good place to put some of the information under discussion.

   Thanks for the comment, Mike – I don’t think I was aware of the remark that preservation of finite coproducts and epis implies preservation of quotients of equivalence relations. I’m trying to think of a good place to put some of the information under discussion.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeApr 7th 2015
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   Author: Mike Shulman
   Format: MarkdownItexHmm, possibly I meant to say that preservation of finite *limits* (as any representable does) and also epis is what implies preservation of quotients. Possibly that requires the domain and codomain to be regular categories. I don't remember the exact statement, but I think something like that is true.

   Hmm, possibly I meant to say that preservation of finite limits (as any representable does) and also epis is what implies preservation of quotients. Possibly that requires the domain and codomain to be regular categories. I don’t remember the exact statement, but I think something like that is true.

5. • CommentRowNumber5.
   • CommentAuthorZhen Lin
   • CommentTimeApr 7th 2015
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   Author: Zhen Lin
   Format: MarkdownItexA functor that preserves pullbacks and regular epimorphisms will also preserve _effective_ quotients of equivalence relations. So if the domain is an effective regular category, then such a functor preserves all quotients of equivalence relations.

   A functor that preserves pullbacks and regular epimorphisms will also preserve effective quotients of equivalence relations. So if the domain is an effective regular category, then such a functor preserves all quotients of equivalence relations.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeApr 7th 2015
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   Author: Mike Shulman
   Format: MarkdownItexThat's it, thanks. Here we're in a topos, so no problem.

   That’s it, thanks. Here we’re in a topos, so no problem.