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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 8th 2011
   • (edited Oct 8th 2011)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexA <a href="http://mathoverflow.net/questions/77498/the-binary-product-preserves-pushouts/77501#77501">question</a> appeared on Math Overflow which I answered, but I'm not completely satisfied with my answer, and I wondered whether any of you had seen this before. In some generality, the question is this. Let $E$ be a cocomplete pretopos (say), and let $F$ be a finitary polynomial endofunctor on $E$ without constant term. Then, if I'm not mistaken, the canonical arrow $\sum_i F(A_i) \to F(\sum_i A_i)$ is monic. Is something like this true for wide pushouts in place of coproducts, i.e., is the canonical arrow $(\sum_i)_{F P} F(A_i) \to F((\sum_i)_P A_i)$ monic, for any cone from $P$ to the collection of the $A_i$? I'm pretty sure that it's going to be true, but I don't think I'm thinking all that clearly this morning. This seems like a handy little lemma to have archived at the Lab...

   A question appeared on Math Overflow which I answered, but I’m not completely satisfied with my answer, and I wondered whether any of you had seen this before.

   In some generality, the question is this. Let be a cocomplete pretopos (say), and let be a finitary polynomial endofunctor on without constant term. Then, if I’m not mistaken, the canonical arrow is monic. Is something like this true for wide pushouts in place of coproducts, i.e., is the canonical arrow monic, for any cone from to the collection of the ? I’m pretty sure that it’s going to be true, but I don’t think I’m thinking all that clearly this morning.

   This seems like a handy little lemma to have archived at the Lab…

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeOct 8th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIt looks to me as though your proof at MO should work in the generality of a cocomplete pretopos, if you do the zigzag thing appropriately internally. Offhand, I don't recall seeing this anywhere before.

   It looks to me as though your proof at MO should work in the generality of a cocomplete pretopos, if you do the zigzag thing appropriately internally. Offhand, I don’t recall seeing this anywhere before.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 9th 2011
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   Author: Todd_Trimble
   Format: MarkdownItexThanks. I think I was worried over nothing. In any case, I did a little [write-up](http://ncatlab.org/toddtrimble/published/Lemma+on+wide+pushouts) on my web which hopefully does the job.

   Thanks. I think I was worried over nothing. In any case, I did a little write-up on my web which hopefully does the job.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeOct 9th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexNice! I was wondering last night whether the category has to be an infinitary-pretopos, or whether a cocomplete pretopos suffices -- and also whether every cocomplete pretopos is an infinitary-pretopos, i.e. whether arbitrary coproducts in a pretopos are necessarily stable and disjoint. Is there a nice argument for why finite power functors preserve filtered colimits? Perhaps that is where we have to assume something (like extensivity) about the colimits being constructed in a "set-like way"?

   Nice! I was wondering last night whether the category has to be an infinitary-pretopos, or whether a cocomplete pretopos suffices – and also whether every cocomplete pretopos is an infinitary-pretopos, i.e. whether arbitrary coproducts in a pretopos are necessarily stable and disjoint.

   Is there a nice argument for why finite power functors preserve filtered colimits? Perhaps that is where we have to assume something (like extensivity) about the colimits being constructed in a “set-like way”?

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 10th 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexSo I was construing a finite power functor as $$C \stackrel{\Delta}{\to} C^n \stackrel{prod}{\to} C$$ where $\Delta$ preserves filtered colimits because it is left adjoint to $prod$, and $prod$ preserves filtered colimits because of the interchange between finite limits and filtered co... ohhh, dang, you're right, I misapplied the famous theorem. Hm! Back to the old drawing board... I'd love to think about the other questions, too, but today it feels like I have a big mess of confusions of various sorts, all crying for attention. :-)

   So I was construing a finite power functor as

   where preserves filtered colimits because it is left adjoint to , and preserves filtered colimits because of the interchange between finite limits and filtered co… ohhh, dang, you’re right, I misapplied the famous theorem. Hm! Back to the old drawing board…

   I’d love to think about the other questions, too, but today it feels like I have a big mess of confusions of various sorts, all crying for attention. :-)

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 10th 2011
   • (edited Oct 10th 2011)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexOkay, let me try again. If $C$ is an $\infty$-pretopos, then $c \times -: C \to C$ preserves arbitrary colimits. So now let $J$ be a filtered category, and let $F, G$ be functors $J \to C$. Then $$(colim_{j: J} F(j)) \times (colim_{k: J} G(k)) \cong colim_{k: J} (colim_{j: J} F(j)) \times G(k) \cong colim_{j: J} colim_{k: J} (F(j) \times G(k)) \cong colim_{(j, k): J \times J} F(j) \times G(k) \cong colim_{j: J} F(j) \times G(j)$$ where the last isomorphism comes about because $\Delta: J \to J \times J$ is final. So $prod: C \times C \to C$ preserves filtered colimits. Does that work? (Maybe I should say infinitary pretopos instead of $\infty$-pretopos?)

   Okay, let me try again. If is an -pretopos, then preserves arbitrary colimits. So now let be a filtered category, and let be functors . Then

   where the last isomorphism comes about because is final. So preserves filtered colimits. Does that work?

   (Maybe I should say infinitary pretopos instead of -pretopos?)

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeOct 10th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAh, that looks good! So that actually says that any functor of two variables which preserves filtered colimits in each variable separately preserves filtered colimits in both variables together. I knew that for reflexive coequalizers, but I don't recall seeing it for filtered colimits before, although I could easily be misremembering. ("Infinitary pretopos" is an nLabism, intended to express the same thing as the more common "$\infty$-pretopos", but also avoid a terminological collision with "$\infty$-topos", where the $\infty$ denotes the category-level rather than the cardinality of extensivity. You can use it or not, as you like. (-: )

   Ah, that looks good! So that actually says that any functor of two variables which preserves filtered colimits in each variable separately preserves filtered colimits in both variables together. I knew that for reflexive coequalizers, but I don’t recall seeing it for filtered colimits before, although I could easily be misremembering.

   (“Infinitary pretopos” is an nLabism, intended to express the same thing as the more common “-pretopos”, but also avoid a terminological collision with “-topos”, where the denotes the category-level rather than the cardinality of extensivity. You can use it or not, as you like. (-: )