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Sridhar Ramesh (Feb 16 2021 at 23:46):Hello. This is my first use of Zulip, so forgive me if I am posting this in the wrong place or the wrong way. I am interested in knowing of any reference to cite for the following folklore result: If T is a category with finite limits, and t is an object in T, then the free extension of T (qua category-with-finite-limits) with a morphism from 1 to t is given by the slice category T/t.
(Similar results follow for other notions extending the notion of a category with finite limits which are also suitably preserved by pullback to slice categories; e.g., for elementary toposes. And an analogue result holds for categories with finite products, where we do not use the full slice category but only its full subcategory on slices which are projections from products. And again for notions extending this in the appropriate way; e.g., for cartesian closed categories, where the analogue of this result is sometimes called "functional completeness" and can be cited to Lambek. But I am specifically interested in citing a reference for this result for categories with finite limits.)
Jules Hedges (Feb 17 2021 at 00:07):I could swear I've seen this somewhere, but I could be imagining it. Maybe in Handbook of Categorical Algebra... but I'm not very confident
Jules Hedges (Feb 17 2021 at 00:10):I was thinking about that in the context that the version for just finite products is the co-kleisli category of the comonad, whereas is the co-EM category of the same comonad. Not sure if having an equivalent version helps for reference hunting
Jules Hedges (Feb 17 2021 at 00:12):(If I remember correctly that fact is mentioned in passing in Introduction to Higher Order Categorical Logic, but it doesn't take the last step to what you're looking for)
Sridhar Ramesh (Feb 17 2021 at 02:43):Yes, Introduction to Higher-Order Categorical Logic gives under the name "functional completeness" the version of the fact for cartesian closed categories using the Kleisli category/projection slice category, but not the version for categories with finite limits using the Eilenberg-Moore category/full slice category, as far as I can tell.
(While we're bringing up these other facts, it might also be useful to have references to cite for the fact that the full slice category is the Eilenberg-Moore category and the slice category of just projections is the Kleisli category for the comonad. But I'm not so concerned about that.)
Nathanael Arkor (Feb 17 2021 at 02:55):The reference for the characterisation of the (simple) slice categories as categories of (free) coalgebras is Exercise 1.3.4 of Jacobs's Categorical logic and type theory.
Nathanael Arkor (Feb 17 2021 at 02:56):I would expect the finitely complete case to be covered in this book too, but I don't remember off the top of my head.
Nathanael Arkor (Feb 17 2021 at 02:59):It's Proposition 1.10.15(ii) ibid.
Nathanael Arkor (Feb 17 2021 at 03:00):
sarahzrf (Feb 17 2021 at 03:03):ha, i knew it had to be in either the elephant or jacobs
sarahzrf (Feb 17 2021 at 03:04):lol @ the proof offered image.png
Sridhar Ramesh (Feb 17 2021 at 05:04):Thank you, Nathanael!
dusko (Jun 13 2021 at 17:40):the statement in terms of extension is in SGA4: that is how it is proved that is a topos if is. it is equivalent to the statement that is a final resolution of the comonad . it is funny that grothendieck worked out the proof in detail (expose IV i think), but it became too easy later. i am pretty sure that lambek referred to grothendieck for this, and would expect that it is spelled out in lambek scott, if you read far enough.