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1. • CommentRowNumber1.
   • CommentAuthorJohn Baez
   • CommentTimeMay 23rd 2018
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   Author: John Baez
   Format: MarkdownItexI think I used to know a name for categories where homming out of the terminal object preserved binary coproducts (or better yet, finite coproducts). Is there a name for this - and more importantly, what are some simple conditions that explain why this often occurs? All that comes to mind is [[extensive categories]], but that's not quite right.

   I think I used to know a name for categories where homming out of the terminal object preserved binary coproducts (or better yet, finite coproducts). Is there a name for this - and more importantly, what are some simple conditions that explain why this often occurs?

   All that comes to mind is extensive categories, but that’s not quite right.

2. • CommentRowNumber2.
   • CommentAuthorThomas Holder
   • CommentTimeMay 23rd 2018
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   Author: Thomas Holder
   Format: MarkdownItexHow about having a look at [[Freyd cover]] !?

   How about having a look at Freyd cover !?

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeMay 23rd 2018
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   Author: Mike Shulman
   Format: MarkdownItexMaybe check out [[connected object]] and [[indecomposable object]] and [[well-pointed topos]]? I don't think I know a name for exactly that condition though.

   Maybe check out connected object and indecomposable object and well-pointed topos? I don’t think I know a name for exactly that condition though.

4. • CommentRowNumber4.
   • CommentAuthorRichard Williamson
   • CommentTimeMay 23rd 2018
   • (edited May 23rd 2018)
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   Author: Richard Williamson
   Format: MarkdownItex[Edited to remove something I thought too quickly about!] A simple condition might be if there is some coproduct preserving functor to Set with a left adjoint which preserves terminal objects. This is the case for Cat for example.

   [Edited to remove something I thought too quickly about!]

   A simple condition might be if there is some coproduct preserving functor to Set with a left adjoint which preserves terminal objects. This is the case for Cat for example.

5. • CommentRowNumber5.
   • CommentAuthorJohn Baez
   • CommentTimeMay 23rd 2018
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   Author: John Baez
   Format: MarkdownItexThanks!! I'm looking at categories like $Set$, $Cat$, $Poset$, $Graph$. I'm wanting to use the fact that these have a full subcategory equivalent to $FinSet$, given by $1$ and its finite coproducts. This follows from the fact that these categories have a terminal object, have finite coproducts, and that homming out of $1$ preserves finite coproducts. Or even: they have a terminal object, finite coproducts of the terminal object, and there are exactly $n$ morphisms from $1$ to the $n$-fold coproduct $1 + \cdots + 1$.

   Thanks!!

   I’m looking at categories like $Set$, $Cat$, $Poset$, $Graph$. I’m wanting to use the fact that these have a full subcategory equivalent to $FinSet$, given by $1$ and its finite coproducts. This follows from the fact that these categories have a terminal object, have finite coproducts, and that homming out of $1$ preserves finite coproducts. Or even: they have a terminal object, finite coproducts of the terminal object, and there are exactly $n$ morphisms from $1$ to the $n$-fold coproduct $1 + \cdots + 1$.

6. • CommentRowNumber6.
   • CommentAuthorRichard Williamson
   • CommentTimeMay 23rd 2018
   • (edited May 23rd 2018)
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   Author: Richard Williamson
   Format: MarkdownItexMike and Todd, for instance, have a good nose for these things and will be able to give a more definitive opinion, but I think the condition I mentioned in #4 expresses fairly well what is happening in these examples. Indeed, all of these cases are rather special, in that we have a kind of enriched or higher well-pointedness. In Cat, for instance, if we take not the set but the category $Hom(1,A)$, then we recover $A$. In such cases, it is immediate that, at the enriched or higher level, we have preservation of coproducts (or indeed anything!). So all we need is to be able to truncate back to a set, in such a way that coproducts are preserved. This functor is the one that is needed for the condition in #4. In Cat, for instance, it is given just by throwing away arrows, the left adjoint being the functor which views a set as a discrete category.

   Mike and Todd, for instance, have a good nose for these things and will be able to give a more definitive opinion, but I think the condition I mentioned in #4 expresses fairly well what is happening in these examples. Indeed, all of these cases are rather special, in that we have a kind of enriched or higher well-pointedness. In Cat, for instance, if we take not the set but the category $Hom(1,A)$, then we recover $A$. In such cases, it is immediate that, at the enriched or higher level, we have preservation of coproducts (or indeed anything!). So all we need is to be able to truncate back to a set, in such a way that coproducts are preserved. This functor is the one that is needed for the condition in #4. In Cat, for instance, it is given just by throwing away arrows, the left adjoint being the functor which views a set as a discrete category.

7. • CommentRowNumber7.
   • CommentAuthorRichard Williamson
   • CommentTimeMay 23rd 2018
   • (edited May 23rd 2018)
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   Author: Richard Williamson
   Format: MarkdownItexAs an aside, it may be that the characterisation of $FinSet$ in terms of $1$ and finite coproducts may in one sense be a bit misleading, even if it is sufficient. This is because it doesn't seem to generalise very well to higher categorical levels. We have discussed somewhere on the nForum before, for instance, the question of the correct analogue of $FinSet$ for $\infty$-groupoids. Here (homotopy) coproducts would certainly not seem the correct thing, as one surely wants something more than discrete homotopy types. But something like the closure of $1$ under all finite (homotopy) limits and colimits might be reasonable. One could still characterise $FinSet$ as the closure of $1$ under all finite limits and colimits, and the argument of #6 still works, upgrading the condition of #4 to preserve all finite colimits (finite limits being automatic).

   As an aside, it may be that the characterisation of $FinSet$ in terms of $1$ and finite coproducts may in one sense be a bit misleading, even if it is sufficient. This is because it doesn’t seem to generalise very well to higher categorical levels. We have discussed somewhere on the nForum before, for instance, the question of the correct analogue of $FinSet$ for $\infty$-groupoids. Here (homotopy) coproducts would certainly not seem the correct thing, as one surely wants something more than discrete homotopy types. But something like the closure of $1$ under all finite (homotopy) limits and colimits might be reasonable. One could still characterise $FinSet$ as the closure of $1$ under all finite limits and colimits, and the argument of #6 still works, upgrading the condition of #4 to preserve all finite colimits (finite limits being automatic).

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeMay 24th 2018
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   Author: Mike Shulman
   Format: MarkdownItexIf you just want a name for the condition, the best I can think of is "the terminal object is connected". The definition at [[connected object]] refers to infinitary coproducts, but there's a proof there that in an infinitary-extensive category (like your examples) this is equivalent to the binary-coproduct version. As for why it happens, well, sometimes you might have a [[connected topos]], such as the category of presheaves on a category with a terminal object. More generally, you might have a category with a forgetful functor to $Set$ that has [[discrete objects]], i.e. a fully faithful left adjoint. This [happens](https://ncatlab.org/nlab/show/discrete+object#Fibrations) more often than you might think: whenever the forgetful functor is an opfibration and preserves the initial object. I think that's the case in all the examples you mentioned.

   If you just want a name for the condition, the best I can think of is “the terminal object is connected”. The definition at connected object refers to infinitary coproducts, but there’s a proof there that in an infinitary-extensive category (like your examples) this is equivalent to the binary-coproduct version. As for why it happens, well, sometimes you might have a connected topos, such as the category of presheaves on a category with a terminal object. More generally, you might have a category with a forgetful functor to $Set$ that has discrete objects, i.e. a fully faithful left adjoint. This happens more often than you might think: whenever the forgetful functor is an opfibration and preserves the initial object. I think that’s the case in all the examples you mentioned.