homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
A -truncated object in an n-category is an object which “behaves internally like a -category”. More precisely, since an object of an -category can behave at most like an -category, a -truncated object behaves like a -category. More generally, a -truncated object in an (n,r)-category is an object which behaves internally like a -category.
Let be an -category, where and can range from to inclusive. An object is -truncated if for all objects , the -category is in fact a -category.
In a 1-category:
In a 2-category:
In an -category, the -truncated objects (which are automatically -truncated) are also called -types. See n-truncated object of an (∞,1)-category.
If the -category has sufficient exactness properties, then the -truncated objects form a reflective subcategory. More generally, in such a case there is a factorization system such that is the category of -truncated objects. (Note that this is not a reflective factorization system, but it is often a stable factorization system.) For example:
In any category with a terminal object, the subcategory of terminal objects is reflective, and corresponds to the factorization system all morphisms, isomorphisms.
In a regular category, the category of subterminal objects is reflective, and corresponds to the factorization system regular epimorphisms, monomorphisms.
In a regular 2-category, the same holds true, where is the class of regular -epimorphisms (eso morphisms) and the class of -monomorphisms (ff morphisms). With additional exactness conditions, the categories of -truncated, -truncated, and -truncated objects are also reflective; see here.
In an (∞,1)-topos, the -truncated objects are reflective and we have the (n-connected, n-truncated) factorization system.
Last revised on November 24, 2023 at 06:34:33. See the history of this page for a list of all contributions to it.