nLab weak enrichment

Idea

An analog of enriched category in higher category theory. Enrichment in a monoidal category $(V,\otimes,I)$ requires composition morphisms

$$Hom (A,B) \otimes Hom(B,C) \to Hom(A,C)$$

and identity morphisms

$$I \to Hom(A,A)$$

in $V$ satisfying the axioms of associativity and unitality. To weakly enrich, we wish to have the axioms of associativity and unitality hold up to coherent 2-morphisms. To accomplish this we can replace $V$ with a monoidal bicategory which has sufficient structure to accommodate this. This describes one approach to weakly enriching in the 2-dimensional case. For higher values of n we need to replace $V$ with a weak n-category.

Definition

A category weakly enriched in a monoidal bicategory $W$ is called a $W$-bicategory. The data of a $W$-bicategory $C$ includes the data of an ordinary enriched category in the decategorification of $W$. In addition, $C$ must contain associator and unitor 2-morphisms in $W$ which fill the appropriate diagrams of composition and unit 1-morphisms. These 2-cells are required to satsify higher dimensional coherence axioms.

Note that this should not be confused with a category enriched in a bicategory which allows for multiple bases of enrichment. A $Cat$-bicategory is an ordinary bicategory. Many examples come from weakly enriching in the monoidal 2-category $V$-$Cat$ of categories enriched in $V$.

In the context of (∞,1)-category theory see at enriched (∞,1)-category.

In the context of (∞,1)-operad theory see (Lurie, def. 4.2.1.12):

Write $\mathcal{LM}^\otimes$ the operad for modules over an algebra regarded as an (∞,1)-operad, regarded as the (∞,1)-category of operators. Similarly write $\mathcal{Ass}^\otimes$ for the (∞,1)-category of operators of the associative operad.

Definition

For $\mathcal{V}^\otimes \to \mathcal{Ass}^\otimes$ exhibiting a planar (∞,1)-operad, a weak enrichment of an (∞,1)-category $\mathcal{C}$ over $\mathcal{C}^\otimes$ is a fibration of (∞,1)-operads

$$q \colon \mathcal{O}^\otimes \to \mathcal{LM}^{\otimes}$$

which exhibits $\mathcal{C}$ as a module over $\mathcal{V}^\otimes$ in that it is equipped with equivalences

$$\mathcal{V}^\otimes \simeq \mathcal{O}^\otimes_{\mathfrak{a}}$$

and

$$\mathcal{C} \simeq \mathcal{O}^\otimes_{\mathfrak{n}} \,.$$

(Lurie, def. 4.2.1.12)

maybe better: weak tensoring?

Related concepts

• homotopical enrichment;

• enriched (∞,1)-category

References

$W$-bicategories are briefly explained in Section 4 of

• Richard Garner, Nick Gurski, The low-dimensional structures formed by tricategories, arXiv:0711.1761

Section 4.2.1 of

• Jacob Lurie, Higher Algebra

• David Gepner, Rune Haugseng, Enriched ∞-categories via non-symmetric ∞-operads, arXiv:1312.3178

Discussion

This was originally at bicategory:

Sebastian: Is there a formal meaning of weak enrichment?

John Baez: Yes there is; indeed Clark Barwick is writing a huge book on this.

Sebastian: If not, is there at least a method how to get the definition of a weak $n$-category if I know the definition of a (strict) $n$-category?

John Baez: that sounds harder! That’s like pushing a rock uphill. It’s easier to go down from weak to strict.

Sebastian: Of course, I have recognised that there are actually different definitions of what a weak n-category should be… so to give my question a bit more precision: How do I get a definition of a weak $n$-category that is as close as possible to the definition of a strict $n$-category? The weak $n$-category should be what you call “globular”, I think. (Are there different definitions of globular (weak or strict) n-categories?)

John Baez: globular strict n-categories have been understood since time immemorial, or at least around 1963, and there is just one reasonable definition. Globular weak n-categories were defined in the 1990s by Michael Batanin, and his theory relates them quite nicely to the globular strict ones. But there is also a different definition of globular weak n-categories due to Penon. It had a mistake in it which has now been fixed.

From the preface to Towards Higher Categories:

There is a quite different and more extensively developed operadic approach to globular weak infinity-categories due to Batanin (Bat1, Str2), with a variant due to Leinster (Lein3). Penon (Penon) gave a related, very compact definition of infinity-category; this definition was later corrected and improved by Batanin (Bat2) and Cheng and Makkai (ChMakkai).

You can get the references there.

I think this discussion should be moved over to some page on n-categories, since it’s not really about bicategories.