A Characterization of Representable Intervals

In this note we provide a characterization, in terms of additional algebraic structure, of those strict intervals (certain cocategory objects) in a symmetric monoidal closed category E that are representable in the sense of inducing on E the structure of a finitely bicomplete 2-category. Several examples and connections with the homotopy theory of 2-categories are also discussed. Introduction Approached from an abstract perspective, a fundamental feature of the category of spaces which enables the development of homotopy theory is the presence of an object I with which the notions of path and deformation thereof are defined. When dealing with topological spaces, I is most naturally taken to be the closed unit interval [0, 1], but there are other instances where the homotopy theory of a category is determined in an appropriate way by an interval object I. For example, the simplicial interval I = ∆[1] determines — in a sense clarified by the recent work of Cisinski [2] — the classical model structure on the category of simplicial sets and the infinite dimensional sphere J is correspondingly related to the quasi-category model structure studied by Joyal [5]. Similarly, the category 2 gives rise to the natural model structure — in which the weak equivalences are categorical equivalences, the fibrations are isofibrations and the cofibrations are functors injective on objects — on the category Cat of small categories [7]. This model structure is, moreover, well-behaved with respect to the usual 2-category structure on Cat (it is a model Cat-category in the sense of [10]). One special property of the category 2, which is in part responsible for these facts, is that it is a cocategory in Cat. In this paper we study, with a view towards homotopy theory, one (abstract) notion of strict interval object — namely, a cocategory with object of coobjects the tensor unit in a symmetric monoidal closed category — of which 2 is a leading example. Every such interval I gives rise to a 2-category structure on its ambient category and it is our principal goal to investigate certain properties of the induced 2-category structure in terms of the interval itself. In particular, our main theorem (Theorem 2.10) gives a characterization of those strict intervals I for which the induced 2-category structure is finitely bicomplete in the 2-categorical sense. A strict interval I with this property is said to be representable and the content of Theorem 2.10 is that a strict interval I is representable whenever it is a distributive lattice with top and bottom elements which are, in a suitable sense, its generators. We note here that neither the closed unit interval in the category of spaces nor the simplicial interval in the category of simplicial sets are examples of strict interval Date: March 22, 2009. 2000 Mathematics Subject Classification. Primary: 18D05, Secondary: 18D35. 1