weak enrichment



Enriched category theory



An analog of enriched category in higher category theory.


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

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

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.


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}}


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

(Lurie, def.

maybe better: weak tensoring?


Section 4.2.1 of


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.

Revised on February 12, 2013 07:58:05 by Urs Schreiber (