The n-Category Café

When we first learn about category theory, we usually work with categories that have a set (or class) of objects and a set (or class) of morphisms. Later, we may learn that this can be generalized in several ways:

1. If $V$ is a monoidal category, then a $V$-enriched category has a set of objects, but hom-objects $hom(x,y)$ that lie in $V$. When $V=Set$, we recover the original notion of (locally small) category.

2. If $C$ is a category with pullbacks, then a $C$-internal category has a $C$-object of objects and a $C$-object of morphisms. When $C=Set$, we recover the original notion of (small) category.

3. If $S$ is any category, then an $S$-indexed category has, for each object $X\in S$, an ordinary category of “$X$-indexed objects”.

Each of these kinds of category has its own “category theory”, with all the usual stuff: limits, colimits, adjunctions, monads, etc. But the three cases are fairly different from each other. Internal categories can be identified with certain indexed ones, but it’s harder to compare the enriched and indexed cases. When I first learned about all of these, I found it unsatisfying that we have two or three useful generalizations of the notion of category, but that they seem to be incomparable.

Often, however, when one important concept can be generalized in seemingly-different ways, there is actually a further generalization which includes them all, and that is the case here. The very general generalization is the notion of category enriched in a bicategory, or even more generally category enriched in a virtual equipment (i.e. fc-multicategory). The former has been studied by various category theorists off and on for years; the latter was first proposed by Tom.

The point of the new paper is that there’s a less general generalization than this, which still includes all three basic examples, and which is worth studying in its own right. In this case, what we enrich in is an indexed monoidal category, a.k.a. a monoidal fibration. If you’ve been following my blog posts for a couple of years (e.g. this one and this one), then you may know that these things are close to my heart.

An indexed monoidal category consists of a category $S$ (which here we assume to have finite products) and a pseudofunctor $V:S^{op}\to MonCat$. Given such a beast, there is a fairly natural definition of a $V$-enriched category: its objects are in, or indexed over, $S$, while its homs are objects of the “fiber categories” $V(X)$. Thus, we can think of it as simultaneously indexed over $S$ and enriched over $V$, hence the name “enriched indexed categories”. When $S$ has pullbacks and $V(X) = S/X$, then this gives internal and indexed categories (depending on the “size” we choose to use), while when $S=Set$ and $V(X) = V^X$ for an ordinary monoidal category $V$, this gives classical enriched categories.

An indexed monoidal category also gives rise to a bicategory, or more generally to a (virtual) equipment, as I showed way back in this paper. Thus, the very general theory can be applied. This is nice, because it automatically produces formally well-behaved notions of limit, colimit, presheaf, etc. However, in the special case of an indexed monoidal category, we can reinterpret these formal notions in more explicit and familiar terms, which turn out to combine standard ideas from enriched category theory and indexed category theory in very natural ways.

For instance, one way to think of an enriched indexed category (which is not the one that you get automatically out of the bicategory/equipment setup) is that it consists of a $V(X)$-enriched category $C(X)$ for each object $X\in S$, which we may call the “fiber over $X$”, together with appropriately defined “restriction functors”. When we unravel the abstract notion of limit with respect to this description, we get examples like the following:

• Weighted limits in some fiber category $C(X)$, in the usual $V(X)$-enriched sense, which are moreover preserved by the restriction functors (the common requirement on ordinary “fiberwise” limits in indexed category theory).

• Left and right adjoints to the restriction functors satisfying the Beck-Chevalley condition (this is the usual notion of “indexed (co)product” from indexed category theory), which moreover lift to “$V(X)$-enriched adjunctions” in an appropriate sense.

This is why it’s worth studying the less general notion of category: it includes lots of important examples, and it has interesting and useful structure not present in the more general generalization. I find it very pleasing how the abstract equipment-theoretic context automatically tells us the right way to combine “enriched-ness” with “indexed-ness”.

There’s a list of a bunch of examples in the introduction to the paper, and more specific examples appear throughout. Let me close by mentioning two of the examples that motivated me to write the paper.

• In May and Sigurdsson’s parametrized homotopy theory, there are categories of parametrized spectra $Sp_B$ over base topological spaces $B$. These fit together to form a category which is both indexed over $Top$ and enriched over the classical category of spectra. In fact, it is itself a monoidal fibration, and hence we can enrich further over it.

• In Anna Marie Bohmann’s global equivariant homotopy theory, the category of “global spectra” is both indexed over $Grp$ and enriched over $Top$. Moreover, it is most naturally defined by mimicking the classical construction of orthogonal spectra, but in the world of $Grp$-indexed $Top$-enriched categories.