nLab
weak enrichment

Contents

Context

(,1)(\infty,1)-Category theory

Enriched category theory

Contents

Idea

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

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

and identity morphisms

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

in VV 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 VV 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 VV with a weak n-category.

Definition

A category weakly enriched in a monoidal bicategory WW is called a WW-bicategory. The data of a WW-bicategory CC includes the data of an ordinary enriched category in the decategorification of WW. In addition, CC must contain associator and unitor 2-morphisms in WW 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 CatCat-bicategory is an ordinary bicategory. Many examples come from weakly enriching in the monoidal 2-category VV-CatCat of categories enriched in VV.

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

Definition

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

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

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

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

and

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

(Lurie, def. 4.2.1.12)

maybe better: weak tensoring?

References

WW-bicategories are briefly explained in Section 4 of

Section 4.2.1 of

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 nn-category if I know the definition of a (strict) nn-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 nn-category that is as close as possible to the definition of a strict nn-category? The weak nn-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.

Last revised on August 16, 2019 at 15:35:33. See the history of this page for a list of all contributions to it.