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Stream: learning: questions

Topic: products and coproducts in slice categories

Paolo Perrone (Sep 06 2023 at 16:03):

Hi all. I'm looking for some theory about limits and colimits in slice categories.
Statements such as "in the slice category under the object X, products are the same as in the base category, and coproducts are given by wide pushouts in the base category".
Does anyone know a good reference for these statements?

Bruno Gavranović (Sep 06 2023 at 16:14):

I'm not sure what's a good reference, but aren't products in the slice equivalent to pullbacks in the base?

Mike Shulman (Sep 06 2023 at 16:15):

"slice under" is another way of saying "coslice".

Mike Shulman (Sep 06 2023 at 16:16):

If you look hard enough you could probably find a reference, but this is the sort of statement I would expect to see as an exercise in category theory 101 and not need a citation.

Paolo Perrone (Sep 06 2023 at 16:32):

One of the things I wonder is, is there a concise statement for what happens for general colimits instead of just coproducts?

Mike Shulman (Sep 06 2023 at 16:40):

The colimit of a $D$-shaped diagram in the coslice is the colimit of the induced $D^{\lhd}$-shaped diagram in the original category, where $D^\lhd$ denotes $D$ with a new initial object freely adjoined.

Kevin Arlin (Sep 06 2023 at 17:00):

The other key general point to have in mind is that all connected colimits in $x/C$ are computed as in $C$, as with all limits. So in a sense coproducts turning into wide pushouts is the only interesting thing that happens.

Mike Shulman (Sep 06 2023 at 17:02):

Right! (Although that gets less true for higher categories; I believe in an $n$-category the only colimits in $x/C$ that are computed as in $C$ are those with $(n-1)$-connected nerve.)

Evan Patterson (Sep 06 2023 at 19:01):

Mike Shulman said:

If you look hard enough you could probably find a reference, but this is the sort of statement I would expect to see as an exercise in category theory 101 and not need a citation.

That's true if the audience is category theorists or other CT-adjacent people but I fairly often find myself writing for audiences where tossing out a fact like this without further explanation wouldn't be so helpful. With that in mind, I have had occasion to find references for these facts.

McLarty's book Elementary categories, elementary toposes has statements and proofs for most of these facts, as part of proving what he calls the "fundamental theorem of topos theory" (that slices of a topos are again toposes).

Evan Patterson (Sep 06 2023 at 19:02):

Mike Shulman said:

The colimit of a $D$-shaped diagram in the coslice is the colimit of the induced $D^{\lhd}$-shaped diagram in the original category, where $D^\lhd$ denotes $D$ with a new initial object freely adjoined.

A reference for exactly this statement, if you want one, is Wyler's book Lecture notes on topoi and quasitopoi, Section 17.

Paolo Perrone (Sep 07 2023 at 17:01):

Thank you Evan, and thanks everyone!