![]() Just as the collection of all ( small) sets is the prototypical example of a category, so the collection of all small n n-categories is the prototypical example of an ( n + 1 ) (n+1)-category.Īctually, if you define things cleverly, then you can get an ( n + 1 ) (n+1)-category of all n n-categories. On the other hand, perfectly strict n n-categories are typically too rigid to play more than a niche role in higher category theory.įinally, it is sometimes useful to subsume degenerate cases in the general pattern of n n-categories, such as 0-categories ( sets), (0,1)-categories ( posets) and even (-1)-categories ( truth values) and (-2)-categories (the terminal category). ![]() In as far as these are still suitably equivalent to the general notion one speaks of semi-strict n-categories and much of higher category theory revolves around identifying and comparing such rigidifications. In this homotopy theoretic-perspective on n n-categories they are naturally understood as (models for) directed spaces (with an n n-fold notion of âdirectionâ) and in principle the topics of directed homotopy theory (and directed homotopy type theory) ought to be thought of as essentially synonymous to ( â, n ) (\infty,n)-category theory, though a genuine perspective of directed homotopy/ directed homotopy type theory is still in its infancy.Ĭonversely, for concrete computations it is at times convenient to have less flexible and more rigid definitions or at least presentations/models of n n-categories. Therefore many authors these days use â â \infty-categoryâ as a pseudonym for ( â, 1 ) (\infty,1)-category and some use â n n-categoryâ as a tacit pseudonym for ( â, n ) (\infty,n)-categories. Particularly important here is the limiting case ( k = â k = \infty) of ( â, n ) (\infty,n)-categories: Since ( â, 0 ) (\infty,0)-categories aka â \infty-groupoids are the bare homotopy types of âspacesâ as considered in homotopy theory, ( â, n ) (\infty,n)-categories are the natural notion of n n-categories internal to homotopy theory (also: internal to homotopy type theory, up some technical issues) where the notoriously intricate coherence laws of higher category theory typically have a particularly natural (if maybe non-explicit) formulation. In broad generality, under n n-categories one understands higher categories with non-trivial higher morphisms up to dimension n â â â. strict â-category, strict â-groupoid.Grothendieck-Maltsiniotis â-categories. ![]() algebraic definition of higher category.simplicial model for weak â-categories.geometric definition of higher category.applications of (higher) category theory.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |