Conceptual reason that monadic functors create limits?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.