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Patrick Nicodemus (Oct 07 2023 at 17:24):There are apparently two uses of the term "truncation."
One is used in the context of simplicial homotopy theory and refers to forgetting higher morphisms which join things. It refers to precomposition of a simplicial set with the inclusion .
The other is used in higher category theory and refers to collapsing higher cells. It is more closely analogous to the coskeleton of a simplicial set, as I understand it.
Is there a clear and standard term which would refer to the underlying category of 1-cells of a 2-category when the 2-cells are forgotten? I am afraid the word "truncation" is ambiguous as it can refer to two distinct forms of coercing a 2-category to a 1-category. This is most analogous to the 1-skeleton.
Nathanael Arkor (Oct 07 2023 at 17:49):Is there a clear and standard term which would refer to the underlying category of 1-cells of a 2-category when the 2-cells are forgotten?
Yes, the "underlying category".
Morgan Rogers (he/him) (Oct 08 2023 at 09:36):@Nathanael Arkor is that really unambiguous?
Nathanael Arkor (Oct 08 2023 at 10:15):It's the standard terminology (e.g. see A 2-categories companion or Basic Concepts of Enriched Category Theory). I also can't think of another reasonable interpretation of the term, but perhaps you have something in mind?
Morgan Rogers (he/him) (Oct 08 2023 at 11:23):The fact that it's standard is probably more important.
For a bicategory, the only reasonable way to recover a category is to quotient out 2-isomorphisms, so it's not unreasonable to expect that one might erroneously assume this is what you mean by 'underlying category'.
Nathanael Arkor (Oct 08 2023 at 11:32):If someone used the term "underlying category" in regards to a bicategory, I would want them to clarify what they meant. I think it's only reasonable to use "underlying X" for a structure that may be defined to be an "X" equipped with extra structure/properties.
John Baez (Oct 08 2023 at 12:51):Yeah, people usually like to emphasize that a bicategory or higher weak n-category doesn't have an underlying category in the same sense a 2-category or higher strict n-category has one, since the 1-morphisms don't compose associatively 'on the nose'. So talking about the underlying category of a bicategory or higher weak n-category is likely to cause confusion even if you have some good definition of it.
John Baez (Oct 08 2023 at 12:53):'Truncation' seems to have a standard meaning in homotopy type theory, and 'decategorification' has a fairly standard meaning in n-category theory. I think most people who play these games would know what I meant if I said some category was the decategorification of some bicategory: it means that to get it, we throw out the 2-morphisms and identify isomorphic 1-morphisms.