Enriched Category

Let \(\mathcal{C}\) be a \(2\)-category. When \(X\) is a set, we can define \(\mathcal{C}\)-enriched category on \(X\) as a lax \(2\)-functor \(K_X \to \mathcal{C}\), where \(K_X\) is an indiscrete category on \(X\). We can generalize this definition and view every lax \(2\)-functor as an enrichment. This definition includes monads and coincides with Grothendieck’s relative point of view.

This suggests how to define enriched bicategory: that is, as a lax functor between tricategories. But as we don’t have a good notion for tricategories1, so let’s just confine ourselves to bicategories2.

Enriched Functors

Then what are enriched functors? Well, it gets a little bit stranger. An enriched functor between \(F:C\to B\) and \(G:D\to B\) is a tuple \((H, \eta)\) where \(H\) is a lax functor from \(C\) to \(D\) and \(\eta\) is an oplax natural transformation3 between lax functors \(F\to HG\), lax inverse natural transformation. This isn’t that unnatural, considering the definition of slice 2-categories, is it?

Enriched Natural transformation

The above definition does not give the ordinary notion of enriched functor.4 But with appropriate enriched natural transformation, the above definition gives functor category equivalent to the ordinary enriched functor category.

  1. Lax functor plays an important role. Without it, we cannot even define (elementary) 3-topos. But \(2Cat\) does not have information about it. ↩

  2. Most, if not all, of the discussions here can be easily generalized to \((\infty,2)\)-categories. Though we won’t do it. ↩

  3. Here, the direction of lax and oplax follows that of nLab. I think that it might be more natural to define them inversely. ↩

  4. When the base bicategory \(B\) is delooping of \(Set\), it gives the usual notion of functors. But in general, it differs from the usual notion of enriched functor. ↩