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?


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.
  • $\begingroup$ 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

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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.