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Let $U: Alg_T \to C$ be the forgetful functor from the category of algebras of $T: C \to C$ ($T$ could be a monad; I'm happy to think about the simpler case where $T$ is just an endofunctor or pointed endofunctor). Then a very simple diagram chase shows that

$U$ creates limits.

In other words, if I have a diagram in $Alg_T$ whose image in $C$ has a limit, there is an obvious way to lift the limit cone to $Alg_T$, and a straightforward diagram chase verifies that indeed, this is a limit in $Alg_T$.

This is all very easy, but I'm unhappy with the state of affairs for a few reasons:

  1. I'm very lazy and I hate doing diagram chases.

  2. This proof does not obviously generalize to other contexts. For one thing, I have to rehash variations on the same diagram chase for each of the cases where $T$ is a monad, an endofunctor, a pointed endofunctor, etc. For another, even if I stick with just monads, say, I have to rehash the same diagram chase if I want to generalize to other contexts such as enriched or internal category theory. Of course, doing the same work over and over is supposed to mean there's a bigger picture I'm missing.

  3. I find it remarkable that in order for $U$ to create limits, one need not assume any kind of limit-preservation hypotheses about $T$. There's something to be explained, and the proof via diagram chase doesn't accomplish that.

The second point may have real weight -- I haven't checked very diligently, but it seems that it might not actually be known whether this this theorem remains true in full generality in the enriched context, for example! (Although a moment's reflection makes me think I could run the same diagram chase with no fuss if it weren't for point (1) above.) So here's my

Question: What is the conceptual reason for which monadic functors (and other forgetful functors from "categories of algebras") create limits?

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    $\begingroup$ One point is that the Eilenberg-Moore object of a monad $T$ on an object of a 2-category $\mathcal K$ is the lax limit of $T$, when viewed as a lax functor from the terminal 2-category. Since representables preserve lax limits, this implies that for any $X$, if $T$'s base is $Y$, then $\mathcal K(X,\mathrm{EM}(T))$ is the EM category of the monad $\mathcal K(X,T)$ on the ordinary category $\mathcal K(X,Y)$. For reasonable examples of $\mathcal K$, limits in $Y$ are hopefully some kind of thing that turns into a limit in a category upon applying a representable functor, no? $\endgroup$ – Kevin Carlson Feb 17 at 2:47
  • $\begingroup$ The above is more relevant for generalizing the 2-category in which you take a monad, rather than generalizing away from monads. I'm not sure exactly how, but surely there must be a lax limit definition for these other categories of algebras too. $\endgroup$ – Kevin Carlson Feb 17 at 2:50
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    $\begingroup$ @KevinCarlson Hmm... there's something delicate here, maybe a level-shifting thing. The co-EM object for a comonad is an oplax limit, and its creates colimits rather than limits. I don't think I understood your argument well enough to see where you used that the limit is lax rather than oplax. $\endgroup$ – Tim Campion Feb 17 at 4:12
  • $\begingroup$ I don’t think I did. Rather the schematic argument is that forgetful functors out of EM/coEM categories for monads/comonads/endomorphims should generally create whatever (co)limits they do in $\underline{\mathbf{Cat}}$, since both lax and oplax limits in a 2-category are created by representables, as are at least certain notions of limit in an object in a 2-category. Forone specific example, $x$ has (co)limits indexed by a category $J$ if the canonical morphism $x\to x^J$ into the cotensor by $J$ has a right (left) adjoint. The latter argument certainly needs a more precise formulation. $\endgroup$ – Kevin Carlson Feb 17 at 8:23
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The heart of the diagram chase is: All the operations and equations involved in an algebra structure on X have target X. (More generally: the target of each operation/equation could be any limit-preserving functor of the algebra-specified-so-far.)

For each of the kinds of “algebra” you describe, the structure is built up by sequentially adding operations and equations.

Each “add an operation” step fits into the template of taking an inserter. You have two categories and two parallel functors $\newcommand{\C}{\mathbf{C}}\newcommand{\D}{\mathbf{D}}F, G : \C \to \D$; then the inserter category $\newcommand{\Ins}{\mathrm{Ins}}\Ins(F,G)$ consists of objects $X$ of $\C$ equipped with a map $x : FX \to GX$, and a map $f:(X,x) \to (Y,y)$ is a map $f:X \to Y$ in $\C$ satisfying $(Gf)x = y(Ff)$.

(If $T$ is a monad on $\newcommand{\E}{\mathcal{E}}\E$, then the first stage of defining monad algebras is equipping objects with a map $TX \to X$, i.e. taking the inserter of $\Ins(T,1_\E)$.)

Similarly, each “add an equation” stage fits into the template of taking an equifier: you have categories and functors $F,G : \C \to \D$ as before, plus natural transformations $\alpha,\beta : F \to G$, and the equifier $\newcommand{\Eqf}{\mathrm{Eqf}}\Eqf(\alpha,\beta)$ is the full subcategory of $\C$ of objects $X$ for which $\alpha_X = \beta_X$.

(Back with our monad $T$ on $\E$: after adding the operation, getting the category $\Ins(T,1_\E)$ with a forgetful functor $U$ to $\E$, we can now impose the associativity axiom by taking the equifier for the functors $T^2U, U : \Ins(T,1_\E) \to \E$, with the natural transformations sending $(X,x)$ to the maps $x(Tx), x\mu_X : T^2 X \to X$.)

So monad algebras, and their forgetful functor, can be built up as $\newcommand{\Alg}{\mathrm{Alg}}\Alg_\E(T) = \E_3 \to \E_2 \to \E_1 \to \E_0 = \E$, where each step $\E_{n+1} \to \E_n$ “adds an operation or equation”, i.e. is the forgetful functor from an inserter or equifier category. And moreover, “the target of each operation/equation on $X$ is just $X$”, i.e. the target functor $G$ of each inserter/equifier is the composite forgetful functor $\E_n \to \E$. (And algebras for endofunctors or pointed endofunctors can be built up in the same way.)

Now for creation of limits:

Proposition.

  1. Given $F,G : \C \to \D$, the forgetful functor $\Ins(F,G) \to \C$ creates all limits that exist in $\C$ and are preserved by $G$. Dually, it creates all colimits that exist in $\C$ and are preserved by $F$.

  2. Given $\alpha, \beta : F \to G : \C \to \D$, the forgetful functor $\Eqf(\alpha,\beta) \to \C$ creates all limits that exist in $\C$ and are preserved by $G$. Dually, it creates all colimits that exist in $\C$ and are preserved by $F$.

The proof of this proposition is that diagram chase you don’t want to do — it has to be done sooner or later. But once it’s done here, it applies to give creation of limits for monad algebras, endofunctors, etc, by their presentation in terms of inserters/equifiers as above. It also (dually) gives creation of colimits for coalgebras over comonads, endofunctors, etc.; and also directly gives creation of limits/colimits for various other structures, e.g. monoids/comonoids in a monoidal category, without needing to show they’re (co)monadic.

So I think it quite satisfactorily answers your questions (1) and (3). It doesn’t answer your question (2) — I’m surprised to learn that this doesn’t generalise straightforwardly to the enriched setting, and haven’t worked it through enough to understand why — but it should help clarify what happens there, since the decomposition of algebras in terms of inserters and equifiers still holds, even if the proposition about creation of limits fails.

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    $\begingroup$ +1, this is very instructive. $\endgroup$ – Fosco Feb 18 at 7:40
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    $\begingroup$ Thanks! Regarding the generalization to enriched category theory, it occurs to me that what probably happened in the discussion I linked to is that somebody read about this topic in an old reference from way back before the general definition of a weighted limit was given by Borceux and Kelly (after all, a substantial amount of enriched category theory was originally developed using just conical limits and cotensors) and so there simply wasn't the language available to show this. Surely it must have been noted in the literature at some point since then that it (presumably) works out... $\endgroup$ – Tim Campion Feb 21 at 5:43
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I am not able to give the high-tech answer you are clearly hoping for. For me this statement is just a straight forward generalization of the known fact that, say, $\mathsf{Grp} \to \mathsf{Set}$ creates limits, and the same proof can be used. The statement is so basic that you should better watch out if any high-tech answer actually already uses this statement in its proof or in the proofs of the results that are used.

I think the following related example should answer in particular question 3, because it makes visible what is responsible for the limit creation, namely a second functor.

Let $T : \mathcal{A} \to \mathcal{C}$, $S : \mathcal{A} \to \mathcal{C}$ be two functors. Consider the inserter category $\mathrm{Ins}(T,S)$ with object class $\{(A,h) : A \in \mathcal{A}, h : T(A) \to S(A)\}$ and the evident morphisms. We have the forgetful functor $E : \mathrm{Ins}(T,S) \to \mathcal{A}$.

Lemma. If $S$ preserves limits, then $E$ creates limits.

In particular, when $T : \mathcal{C} \to \mathcal{C}$ is an endofunctor, then the forgetful functor $\mathrm{Alg}(T) = \mathrm{Ins}(T,\mathrm{id}_{\mathcal{C}}) \to \mathcal{C}$ creates limits.

When $\mathbf{T}=(T,\eta,\mu)$ is a monad, then the inclusion $\mathrm{Alg}(\mathbf{T}) \hookrightarrow \mathrm{Alg}(T)$ creates limits as well. Hence, $\mathrm{Alg}(\mathbf{T}) \to \mathcal{C}$ creates limits.

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From an abstract point of view, the reason is that the monad $T$ always preserves any limits that exist colaxly and colax preservation is what is required.

(This answer is closely related to Peter's answer, but describes some published results on the topic.)

The following is Proposition 4.11 of Limits for Lax morphisms by Steve Lack.

If $T$ is a $2$-monad on a $2$-category $C$ then the forgetful $2$-functor $U:T-Alg_c \to C$ from strict algebras and colax morphisms to the base creates lax limits.

Note the switch between lax limits and colax morphisms. The sense of creation is that the projections from the limit should be strict maps.

The Eilenberg-Moore object of a monad or the category of algebras for a pointed endofunctor are both examples of lax limits.

One can take $T$ the $2$-monad for categories with a class $D$ of limits and the result then applies to your setting.

Another instance would be to take as $T$ the $2$-monad for monoidal categories. Then the result becomes that if you have an opmonoidal monad on a category, it lifts to a monoidal structure on the category of algebras.

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    $\begingroup$ Let's see if I've got this -- Let $\mathbb T: Cat \to Cat$ be the 2-monad for categories with $D$-limits. Then $\mathbb T-Alg_c$ is the 2-category of categories with $D$-limits and all functors between them. By Lack's result the forgetful functor $\mathbb T-Alg_c \to Cat$ creates lax limits. In particular, a monad $T: C \to C$ in $\mathbb T-Alg_c$ is the same as a monad in $Cat$ whose underlying category $C$ has $D$-limits. So Lack's result says that the EM object for $T$ in $\mathbb T-Alg_c$ is the usual category of algebras $C^T$ as in $Cat$. In particular, $C^T$ has $D$-limits. Neat! $\endgroup$ – Tim Campion Feb 18 at 6:56
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    $\begingroup$ Of course, this is a bit weaker than than the result I asked about, in a few respects: (1) we get that the existence of certain limits in $C^T$, but not the full strength of "creating a limit" (2) we need to assume that $C$ has all $D$-limits, not just the limit of a particular diagram (3) we don't get the case of algebras for an endofunctor. I think (1) is not a big deal since one can show separately that the forgethful functor preserves limits and is conservative. (2) and (3) are more significant weakenings. Nevertheless, it's really great to get this far at such an abstract level! $\endgroup$ – Tim Campion Feb 18 at 6:59
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    $\begingroup$ Yes, that's right. With regards (3), it does cover the case of an endofunctor too -- that's the lax limit of an endomorphism. I forgot to mention this. I agree it doesn't answer your (2). With regards (1) one can certainly capture creation in the 2-monad sense too though, much as for (2), it won't concern an "individual limit". $\endgroup$ – john Feb 18 at 12:11
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    $\begingroup$ An especially cute thing about this is that it's an example of the microcosm principle: we show that categories of algebras for 1-monads (i.e. monads in 2-categories) create limits by showing that 2-categories of algebras for 2-monads (i.e. monads on 2-categories, i.e. monads in 3-categories) create limits. $\endgroup$ – Mike Shulman Feb 22 at 0:46
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    $\begingroup$ You can deduce the corresponding result for limits of a particular diagram using the Yoneda embedding, which preserves and reflects all existing limits. A monad $T$ on $C$ induces by left Kan extension a monad $\hat{T}$ on the presheaf category $\hat{C}$, and the latter is complete. Thus the category of $\hat{T}$-algebras has all limits created in $\hat{C}$, and a $\hat{T}$-algebra is a $T$-algebra precisely when its underlying object in $\hat{C}$ is representable. See Prop. 5.6 of arxiv.org/abs/1104.2111 for a similar argument. $\endgroup$ – Mike Shulman Feb 22 at 0:47
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This is just to flesh out the approach using inserters and equifiers discussed in the answers, in a way that doesn't quite go down to the level of diagram chasing. I fear, however, that some things implicit in this argument woule require a diagram chase to carefully check.

Also, there's something I still find mysterious: what is it about inserters and equifiers which makes it so there is a particular leg of the limit cone and particular elements of the diagram such that certain hypotheses on these elements in the diagram ensure the forgetful functor down the special leg creates limits? To put a finer point on it: can we give a better description of the class of restriction functors among limits which (under certain partial limit preservation conditions) create limits? A better description than "whatever can be built up from inserters and equifiers"?

Lemma: Let $F,G: C\rightrightarrows D$ be functors, and let $Ins(F,G)$ be the inserter. Then the forgetful functor $Ins(F,G) \to C$ creates any limits that $G$ preserves.

For the proof, note that in in general, if $(c',\phi'), (c,\phi) \in Ins(F,G)$, then

$$ Hom((c',\phi'), (c,\phi)) = Hom(c',c) \times_{Hom(Fc',Gc)^2} Hom(Fc',Gc)$$

where the pullback is over the diagonal map.

Proof: Consider a diagram $(c_i, F(c_i) \xrightarrow {\phi_i} G(c_i))_{i \in I}$ in $Ins(F,G)$ such that $G(\varprojlim_i c_i) = \varprojlim_i G(c_i)$. Then $(F(\varprojlim_i c_i) \to F(c_i) \xrightarrow{\phi_i} G(c_i))_{i \in I}$ is a cone, and so induces a map $\phi: F(\varprojlim_i c_i) \to \varprojlim_i G(c_i) = G(\varprojlim_i c_i)$. We claim that $(\varprojlim_i c_i , \phi)$ is a limit of our diagram. Indeed,

$$Hom((c',\phi'), (c,\phi)) = Hom(c',c) \times_{Hom(Fc',Gc)^2} Hom(Fc',Gc) \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \varprojlim_i Hom(c',c_i) \times_{\varprojlim_i Hom(Fc',Gc_i)^2} \varprojlim_i Hom(Fc',Gc_i) \\ \qquad \qquad \qquad \qquad \qquad \qquad = \varprojlim_i (Hom(c',c_i) \times_{Hom(Fc',Gc_i)^2} Hom(Fc',Gc_i)) \\ \qquad \quad = \varprojlim_i Hom((c',\phi'),(c_i,\phi_i))$$

where we have used that limits commute with limits.

Lemma: Let $\phi,\psi: F \rightrightarrows G : C \rightrightarrows D$ be a diagram of categories, and let $Eq(\phi,\psi)$ be its equifier. Then the full subcategory inclusion $Eq(\phi,\psi) \to C$ is closed under any limits preserved by $G$.

Proof: This boils down to the limit of equal morphisms being equal.

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    $\begingroup$ Well, PIE limits are well studied and have nice characterizations as in Bourke and Garner’s paper on flexible, semi flexible, and pie. However I’m not sure which legs of PIE limits are supposed to create limits...certainly one needs all the legs for products. More familiar is a result that flips roles around-the forgetful functor out of algebras for a 2-monad, with the pseudo morphisms, itself creates PIE limits! $\endgroup$ – Kevin Carlson Feb 18 at 6:35
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    $\begingroup$ @KevinCarlson Yeah, this business about assuming certain parts of the diagram preserve the limit is hard to put in a larger context. In the paper of Lack mentioned by john in his answer above, this pattern is repeated: he shows that $U: T-Alg_c \to C$ creates any comma objects where the appropriate leg is strong, etc. Maybe there's something to say about weighted $\mathcal F$-enriched limits... BTW does the forgetful functor from algebras for a 2-monad create even all flexible limits? $\endgroup$ – Tim Campion Feb 21 at 5:53
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    $\begingroup$ Steve and I characterized the limits that lift to $\mathcal{F}$-categories of algebras and pseudo/lax/colax morphisms for a 2-monad in arxiv.org/abs/1104.2111 -- which was in fact the original purpose of introducing $\mathcal{F}$-categories. $\endgroup$ – Mike Shulman Feb 22 at 0:49

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