# Is the category of enriched operads (co)complete?

Let $V$ be a symmetric monoidal category which is complete and cocomplete. Is the category of small symmetric colored $V$-enriched operads complete and cocomplete? If $V$ is presentable, is it presentable?

### Background

In the case of $V$-enriched categories, corresponding results are proven in the following papers:

• Wolff, $V$-cat and $V$-graph.
• Kelly, Lack, $V$-Cat is locally presentable or locally bounded if $V$ is so.
• In the case that $V$ is co-complete then the category of $V$-enriched symmetric multicategories (aka symmetric colored operads) is co-complete. This is discussed here: arxiv.org/abs/1111.4146, but it really is a straightforward generalization of the statements in the paper by Elmendorff and Mandell cited in that paper. – Jonathan Beardsley Jul 1 '18 at 17:02
• @JonathanBeardsley Thank you. I found [Robertson] just said that the result was shown in [EM]. Could you tell me which part of [EM] she cited? – B. W. Jul 2 '18 at 10:08

The answer to both questions is yes. It's a special case of the fact that, for any colored operad $P$, the category of $P$-algebras is bicomplete (when $V$ is) and is presentable (when $V$ is). For the former statement, you can check out Schwede-Shipley's "Algebras and modules in monoidal model categories", where this is handled in the case of a monad with a condition on the forgetful functor (one that's always satisfied for algebras over a colored operad). For the second statement, I learned this from Lurie's book "Higher Algebra". He discusses it for commutative monoids, and I believe he generalizes that later for other operads. If not, the generalization would be easy.
• Yes, it's Theorem 4.2.5 in the paper by Yau that I cited above. The version for $\infty$-operads is 2.1.4 of Lurie's higher algebra – David White Jul 7 '18 at 4:05
• The main result of [Yau] (6.5.5) includes the operad case, but I cannot find whether his Dwyer-Kan model structure'' is combinatorial. – B. W. Jul 7 '18 at 5:10