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The following result seems to be well known:

If T is a (V-)monad on a (V-)category C, then the forgetful functor $U^T \colon C^T \to C$ creates

  1. any limits that exist in C, and
  2. any colimits that exist in C and are preserved by T.

But I don't know of a published proof for general V. Toposes, Triples and Theories proves (1) for V = Set and leaves (2) for the special case of coequalizers as an exercise. Kelly's book doesn't mention monads at all. Lack, in Codescent objects and coherence, says that 'of course' (2) is true for V = Cat, but doesn't give a citation, while Blackwell, Kelly & Power, in 2-dimensional monad theory, say that (1) is 'well known' but don't give a reference either. And so on.

My questions are:

  1. Has a complete proof of this result (assuming it's true as stated) been published?
  2. Has its bicategorical analogue been treated?
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    $\begingroup$ For $V = Set$ there is a complete proof in the Handbook of Categorical Algebra Volume 2, 4.3 (Borceux). I would be surprised if the proof didn't work for general $V$... $\endgroup$
    – Dylan Wilson
    Commented Jul 22, 2013 at 21:43
  • $\begingroup$ Ah, thank you, I had forgotten to try Borceux. It's hard to believe that the enriched case has never been published, though. $\endgroup$
    – Finn Lawler
    Commented Jul 22, 2013 at 22:15
  • $\begingroup$ Maybe not helpful, but here are two related results. Linton's paper ``Coeequalizers in categories of algebras'', Springer $\endgroup$
    – Peter May
    Commented Jul 22, 2013 at 22:50
  • $\begingroup$ Tried to edit my incomplete message, but the editing function is too unfriendly (times you out), so continuing from above: Lecture Notes 80(1969), 75-90) shows that if C^T has coequalizers, then it is cocomplete. Proposition II.7.4 in EKMM (math.uchicago.edu/~may/BOOKS/EKMM.pdf) shows that if T preserves reflexive coequalizers then C^T is cocomplete, and the proof displays the colimits in C^T as coequalizers in C. Neither is written in an enriched setting, but at least the latter should adapt, I think. $\endgroup$
    – Peter May
    Commented Jul 22, 2013 at 23:07
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    $\begingroup$ Thanks, Peter. I was aware of Linton's paper all right, but what I'm really looking for is a citeable statement and proof of the (enriched case of the) more general result above. $\endgroup$
    – Finn Lawler
    Commented Jul 23, 2013 at 0:38

2 Answers 2

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A result slightly weaker than your point 1. (creation of limits) for general enriched categories can be found in

Eduardo J. Dubuc, Kan extensions in Enriched Category Theory, Lecture Notes in Mathematics, Volume 145, 1970, Chapter II

where a few special cases are proved. Proposition II.4.7, Dubuc proves that powers (also known as cotensors) are created, and Proposition II.4.8 covers conical limits and ends. (The proof of the first proposition is very detailed, while the second is mostly left as an exercise.)

Taken together, these results show that the category of algebras is complete if the base is. I do not know a reference for creation of a general weighted limit (which might exist even if the category doesn't have copowers, products, and ends), nor for the creation of colimits that are preserved by $T$.

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  1. The creation of weighted limits by the forgetful functor from the $\mathscr V$-category of algebras for an enriched (relative) monad is proven in Proposition 2.5 of Arkor–McDermott's Relative monadicity. I'm not aware of a reference for the creation of colimits.

  2. The creation of bilimits by the forgetful pseudofunctor for a pseudomonad is proven in Theorem 6.3.1.6 of Osmond's A categorical study of spectral dualities. (The creation of conical bicolimits that are preserved by the pseudomonad is also claimed in Lemma 2.7 of Osmond's Codescent and bicolimits of pseudo-algebras without proof.)

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