ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 661general definition of weighted limit with many good formal properties. In this section werecall this definition and some of these good properties; in the next section we translatesome examples into more concrete terms for indexed V -categories.8.1. Definition. Let J : K −↦−→ A be a V -profunctor and f : A → C a V -functor. A J-weighted colimit of f consists of a V -functor l: K → C together with an isomorphismC (l, 1) ∼ = J ⊲ C (f, 1) (8.2)of profunctors C −↦−→ K. (Recall Remark 5.27.)If instead J : A −↦−→ K, then a J-weighted limit of f consists of a V -functor l: K → Xtogether with an isomorphismof profunctors K −↦−→ C .C (1, l) ∼ = C (1, f) ⊳ J.In general, K, A, and C could be any large V -categories. However, in our examples,most often K and A will be small or set-small, while C will be a V -fibration (hence ourchoice of typefaces).We will consider many examples in §9, but we should at least mention the followingone here, to clarify why this is a reasonable definition of “weighted limit”.8.3. Example. If V = Fam(V) and we take K to be the unit V -category δ1 and Aa small V -category, then J is simply a diagram on A. If C is the indexed V -categoryconstructed from a (possibly large) V-enriched category C as in Example 4.2, then theabove definitions reduce to the usual notion of weighted limit and colimit. Specifically,if we assume l to be an indexed V -functor, then it is just an object of C, and (in thecolimit case) the isomorphism (8.2) means thatC(l, x) ∼ = [A op , C](J, C(f−, x))for all x ∈ C, which is the usual definition of l being a J-weighted colimit of f.By contrast with the above classical situation, in general it turns out to be very usefulto allow K to be an arbitrary V -category. Here is one example of what such more general“limits” include.8.4. Example. Let j : K → A and take J = A(1, j). Then for any f : A → C , we haveA(1, j) ⊲ C (f, 1) ∼ = C (fj, 1)by Lemma 5.29. Hence fj is always a A(1, j)-weighted colimit of f. Dually, fj is alsoalways an A(j, 1)-weighted limit of f.In particular, if j is the identity functor of A, then A(1, 1) = A is the identity profunctorof A, and f is its own A-weighted (co)limit.One real advantage of allowing arbitrary profunctors as weights is that a given profunctorcan be used both as a weight for limits and a weight for colimits. This symmetryis necessary in order to even state the following fact, which will be very useful.