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ADITTYA CHAUDHURI (Dec 26 2020 at 22:11):A Simplicial group is defined as a simplicial object in the category of groups. From the definition of group object in a category, I think it readily follows(though I have not explicitly verified all the conditions) that every group object in sSets, the category of simplicial sets is actually a simplicial group. So , every -group(group object in the category of Kan complexes) is indeed a simplicial group.
My question is about the existence of a (full or partial) converse of the above fact that is
Is every simplicial group a group object in sSets?
My intuition is saying that the answer is to my question is "No".
What can be a good counter example to my question?
Reid Barton (Dec 27 2020 at 00:06):Actually group objects in simplicial sets are the same as simplicial groups. This happens quite generally.
Todd Trimble (Dec 27 2020 at 00:06):Why do you have that intuition? In the language I am accustomed to, the answer is an obvious "yes". It really comes down to the observation that limits in presheaf categories, finite products in particular, are computed objectwise.
John Baez (Dec 27 2020 at 01:31):General abstract nonsense shows that for any category X with finite products, "group objects in the category of simplicial objects in X" is are equivalent to "simplicial objects in the category of group objects in X".
John Baez (Dec 27 2020 at 01:33):The proof relies on absolutely nothing about the concepts "simplicial object" and "group object" except that these concepts can be described using Lawvere theories. (For simplicial objects it's overkill to use a Lawvere theory, but for group objects it's just right.)
John Baez (Dec 27 2020 at 01:33):I'd be happy to explain if someone wants.
Matteo Capucci (he/him) (Dec 27 2020 at 08:47):Pedantic mod hat on @ADITTYA CHAUDHURI: please open new topics for new discussions, don't recycle (very) old ones
ADITTYA CHAUDHURI (Dec 27 2020 at 09:57):@Todd Trimble I apologise for my wrong intuition. At that time , I was actually not getting why all the structure maps in a simplicial group for each should always come from morphisms of simplicial sets. Now I think I can understand my mistake. Thank you Sir.
ADITTYA CHAUDHURI (Dec 27 2020 at 09:58):@John Baez Thank you Sir.
ADITTYA CHAUDHURI (Dec 27 2020 at 09:59):John Baez said:
I'd be happy to explain if someone wants.
Sir, it would be really great if you explain! Thanks in advance.
ADITTYA CHAUDHURI (Dec 27 2020 at 09:59):@Reid Barton Thank you Sir.
ADITTYA CHAUDHURI (Dec 27 2020 at 10:00):Matteo Capucci said:
Pedantic mod hat on ADITTYA CHAUDHURI: please open new topics for new discussions, don't recycle (very) old ones
I got it. I apologise for the inconvenience caused.
Matteo Capucci (he/him) (Dec 27 2020 at 11:11):ADITTYA CHAUDHURI said:
Matteo Capucci said:
Pedantic mod hat on ADITTYA CHAUDHURI: please open new topics for new discussions, don't recycle (very) old ones
I got it. I apologise for the inconvenience caused.
No worries :) just a reminder for the future
Todd Trimble (Dec 27 2020 at 11:41):Of course it's no problem, Adittya.