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Recall that a functor $G\colon A\to X$ is called monadic if it has a left adjoint $F$ for which the Eilenberg--Moore comparison functor $K\colon A\to X^{\mathbb{T}}$ is an equivalence of categories, where $\mathbb{T}$ is the monad in $X$ defined by the adjunction $\langle F,G,\ldots\rangle\colon X\rightharpoonup A$, and $X^{\mathbb{T}}$ is the category of $\mathbb{T}$-algebras in $X$.

This means that a monadic functor is the forgetful functor $G^{\mathbb{T}}\colon X^{\mathbb{T}}\to X$ up to composition with an equivalence of categories (the comparison functor $K$). Now, it can be verified that $G^{\mathbb{T}}$ creates
limits (Ex. 6.2.2 of Mac Lane). If the comparison functor is an isomorphism, then it is straightforward to verify that $G$ creates limits. In fact, I think that even if it is only assumed that $K$ is an equivalence for which the object function is surjective, then $G$ creates limits.

However, in Proposition 4.4.1 on p. 178 of Mac Lane--Moerdijk, it is stated that any monadic functor creates limits. The proof starts with the following words (with minor omissions):

Let $G$ be monadic. Then by definition, $G$ is the forgetful functor $G^\mathbb{T}$ up to an equivalence of categories. It thus suffices to show that such a forgetful functor $G^\mathbb{T}$ creates limits.

I simply do not understand this statement: In general, the composition of an equivalence and a functor that creates limits need not create limits. For example, the identity $\mathbf{Set}\to\mathbf{Set}$ creates limits, and for any skeleton $X$ of $\mathbf{Set}$ the inclusion $X\subseteq \mathbf{Set}$ is an equivalence. Let $X$ be some skeleton of $\mathbf{Set}$ (for which I am happy to assume any necessary axiom of choice), and take a one-element set $1$ that is not in $X$. Then $1$ is a limit of the functor obtained by composing the unique functor from the empty category to $X$ with $X\subset\mathbf{Set}\stackrel{\operatorname{Id}}{\to}\mathbf{Set}$, but $1$ has no lifting in $X$.

So it seems that there are 4 possibilities:

  1. The above Proposition 4.4.1, as stated, is wrong. There is a counter example where a monadic functor (for which the comparison functor is not an isomorphism) does not create limits.

  2. The proof in ML-M covers just some of the cases, and for the other cases it is not known if the assertion is true (namely, for a monadic functor $G$ for which the comparison functor is not an isomorphism, it is not known whether in general $G$ creates limits).

  3. The proposition is correct because the comparison functor has some additional special property (e.g., its object function must be surjective whenever it is an equivalence).

  4. (Most likely) I am wrong, and the quoted argument from Mac Lane--Moerdijk is correct.

I would like to note that in Theorem 3.4.2 on p. 105 of Barr-Wells, it is only claimed that monadic functors reflect limits.


Which one of the above 4 possibilities is true? In essence, my question is: If $G$ is monadic and the comparison functor is an equivalence that is not an isomorphism, does $G$ create limits?

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2 Answers

up vote 2 down vote accepted

Mac Lane-Moerdijk slipped up; they really should have said "reflects limits". Now it's true that the forgetful functor from the literal category of algebras to the underlying category creates limits (according to the definition of "creates" in CWM), but this notion doesn't transfer across equivalences, as you have observed.

An example where the distinction is important is the statement that for a topos E, the power object functor P: E^{op} --> E is monadic. It wouldn't make much sense to say that this creates limits in Mac Lane's sense of the term.

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Thank you so much for this answer! I've been banging my head against the wall trying to figure out where I was wrong, because I just couldn't believe that such an inaccuracy appears in ML-M. –  user2734 May 13 '10 at 4:48
I think from the fact that this inaccuracy crept in, one should conclude that the CWM definition of "creates limits" is unnatural, so unnatural that Mac Lane himself forgot that he'd used it instead of the natural version (which is invariant under equivalence)! –  Mike Shulman May 13 '10 at 6:30
That's one possibility, and I hope that's the explanation. I can think of another explanation in the case of Mac Lane: he could be very stubborn, and had a famously strong temper (Ieke has spoken publicly about the heat of some of their writing sessions, adding that with Mac Lane you could always start again fresh the next day, but maybe he didn't care to fight this one out). It's a beautiful book, though, written with a lot of consideration for the reader, and knowing both authors, one can feel the very collaborative nature of it. –  Todd Trimble May 13 '10 at 11:11
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According to the remarks on the nLab's created limit page, Categories for the Working Mathematician has a nonstandard definition of creating limits; the standard definition only requires lifts up to isomorphism.

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Thank you for your answer. Calling this definition "non-standard" is a little strange: first, ML-M themselves use a reference to CWM for the definition of creation (so, is their proposition, as stated, correct?) Second, this is the definition in the Joy of Cats (p. 227), and, if I understand the notation correctly , Barr-Wells (p. 37) also use this definition. If that's not standard, then what is? –  user2734 May 12 '10 at 14:43
I would regard it as not so much a question of "standard vs nonstandard" as of "silly vs useful." In practice, we care only about categories up to equivalence. Why would you want a definition of "creates limits" which is not invariant under equivalence of categories? –  Mike Shulman May 13 '10 at 6:07
Regarding the proposition: the statement is correct if you are consistent with your meaning of "monadic" and of "creates." If you take them both in the sensible up-to-equivalence sense, then it is true that any monadic functor creates limits. If you take them both in the stricter sense (which is what CWM does, i.e. a "monadic" functor is one where the comparison is an isomorphism), then it is still true. You only get into trouble when you try to mix them. –  Mike Shulman May 13 '10 at 6:25
@Mike Shulman: I do not claim that the "old" definition is somehow better. But I believe that anyone that is learning from the standrad (!) textbooks and then, say, chooses not to read the proof in ML-M and just use the assertion "as is" might have the wrong conclusion. Such a person may be assuming that monadic functors create limits in the old way, which is wrong. I beleive (but perhaps I'm wrong) that many mathematicians that are not categorists (but do use category theory) are not necessarily aware that a different, probably more meaningful, definition of "creates" is out there. –  user2734 May 13 '10 at 6:30
..and thank you for the second comment. But isn't this "mixture" exactly what happens in ML-M? Also, I believe that Barr-Wells define creates a-la CWM, and "monadic" with equivalence, but they don't get into trouble because they just do not claim that monadic implies creates. –  user2734 May 13 '10 at 6:32
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