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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.
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  • $\begingroup$ 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. $\endgroup$ – Jonathan Beardsley Jul 1 '18 at 17:02
  • $\begingroup$ @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? $\endgroup$ – B. W. Jul 2 '18 at 10:08
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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.

The above is if you have fixed a color set. If you haven't, then the relevant papers to check out are "A Model Structure for Enriched Coloured Operads" by Caviglia, and " Dwyer-Kan Homotopy Theory of Algebras over Operadic Collections" by Yau.

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  • $\begingroup$ Thanks. Is there any result on presentability when color sets are not fixed? $\endgroup$ – B. W. Jul 6 '18 at 12:24
  • $\begingroup$ 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 $\endgroup$ – David White Jul 7 '18 at 4:05
  • $\begingroup$ 4.2.5 in [Yau] seems to be a theorem for enriched categories. $\endgroup$ – B. W. Jul 7 '18 at 4:39
  • $\begingroup$ The main result of [Yau] (6.5.5) includes the operad case, but I cannot find whether his ``Dwyer-Kan model structure'' is combinatorial. $\endgroup$ – B. W. Jul 7 '18 at 5:10

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