### A Weighted Limits Proof of Monadicity

#### Posted by Emily Riehl

Since the 1970s, it has been known that the Eilenberg-Moore object (EM-object) associated with a monad in a 2-category $\mathcal{K}$ can be characterized as a weighted limit. An element of the limit of a Set-valued diagram can be thought of as a cone over that diagram. Similarly, in the case of a monad in **Cat**, an object of this weighted limit, more commonly called an *algebra* for the monad, is exactly a cone over the monad diagram whose shape is described by the weight.

Using the weighted limits formalism, EM-objects in a 2-category are defined representably. It follows that the monadicity theorem characterizing EM-objects and their monadic adjunction can be deduced representably from the monadicity theorem in **Cat**. Despite this fact, I want to give a direct proof of the 2-categorical monadicity theorem to illustrate that a suitable reinterpretation of Beck’s proof is “all in the weights.” To begin, we’ll use weighted limits to characterize the monadic adjunction. It follows immediately that any adjunction inducing the same monad has a comparison map to the monadic adjunction. Inspecting the weights, we observe that the monadic forgetful functor is conservative. We also show that any object in the EM-object is the colimit of a canonical diagram, itself described by a weighted limit, and the monadic forgetful functor preserves these colimits. If the domain of the comparison map admits certain colimits, this map has a left adjoint. If the forgetful functor is conservative, then the comparison adjunction is an adjoint equivalence.

The point of all this is that the 2-functors defining these weights can be interpreted as *simplicial* functors. The same discussion, with only minimal modifications, then gives a (formal!) proof of the monadicity theorem for homotopy coherent adjunctions of quasi-categories. I’ll be speaking about this in Sydney this week and Warsaw later this month. This is joint work with Dominic Verity.

### The free adjunction

The free monad is a 2-category **Mnd** built from the algebraist’s $\Delta_+$, the free monoidal category containing a monoid. Here it will be convenient to observe that **Mnd** is a full subcategory of the Schanuel-Street free adjunction **Adj**. The 2-category **Adj** has two objects, + and -, with hom-categories

Adj(+,+)=$\Delta_+$,Adj(-,-)=$\Delta_+^{op}$,Adj(-,+)=$\Delta_\infty$Adj(+,-)=$\Delta_{-\infty}$

The categories $\Delta_\infty$ and $\Delta_{-\infty}$ are opposites, defined to be the subcategories of $\Delta$ containing maps that preserve the top or bottom elements, respectively, in each ordinal. A 2-functor **Adj** $\to \mathcal{K}$ is an adjunction in the 2-category $\mathcal{K}$. We’ll use the familiar notation $f : B \rightleftarrows A : u$, $\eta \colon 1_B \Rightarrow uf$, $\epsilon \colon fu \Rightarrow 1_A$. The functor

Adj(+,+)=$\Delta_+\to\mathcal{K}(B,B)$

defines the *monad resolution* associated to the adjunction. The diagram

Adj(-,+)=$\Delta_\infty\to\mathcal{K}(A,B)$

is one of the bar resolutions associated to the adjunction. (If this is unfamiliar, see the displayed equations in section 4 here.)

We define **Mnd** to be the full sub 2-category on the object +. A 2-functor **Mnd**$\to \mathcal{K}$ is a monad in the 2-category $\mathcal{K}$.

### Weights for the monadic adjunction

Write Adj$_+$ and Adj$_-$ for the two representable 2-functors for **Adj** and $W_+$ and $W_-$ for their restrictions to **Mnd**. Suppose $T\colon$**Mnd**$\to\mathcal{K}$ is a monad in $\mathcal{K}$. By the Yoneda lemma, the weighted limit $\{W_+,T\} = B$, the object on which the monad acts. By contrast, the weighted limit $\{W_-,T\}$ is — you guessed it — the EM-object!

You can check this directly in $\mathcal{K}=$**Cat**. The weighted limit is defined to be the equalizer

$\{W_-,T\} \rightarrowtail B^{\Delta_\infty} \rightrightarrows B^{\Delta_+\times\Delta_\infty}.$

One of the maps in the equalizer diagram is the monad resolution. The other is precomposition with ordinal sum $\Delta_+\times\Delta_\infty \to \Delta_\infty$, which defines horizontal composition in **Adj**. It follows representably that this weighted limit defines the EM-object whenever it exists. (For more on this see Ross Street’s *Limits indexed by category-valued 2-functors*.)

The advantage of this description of the weight for the EM-object is that we immediately obtain the monadic adjunction: it is an adjunction between the weights $W_+$ and $W_-$! This adjunction is simply the restriction of the Yoneda embedding

Adj$^{op}\to$[Adj,Cat]$\to$[Mnd,Cat]

Composing with the weighted limit 2-functor

{-,-}:[

Mnd,Cat]$^op\times$[Mnd,$\mathcal{K}]\to\mathcal{K}$

we get an adjunction $\mathcal{K}$ between $B[T]\colon=\{W_-,T\}$ and $B=\{W_+,T\}$, the *monadic adjunction*.

### Conservativity of the monadic forgetful functor

The monadic forgetful functor $u^T\colon B[T] \to B$ is induced from the map $W_+\to W_-$ of weights in [**Mnd**,**Cat**], defined by pre-composing with $[0]\colon - \to +$. Applied to the unique object of **Mnd**, the image is the identity-on-objects inclusion $\Delta_+\to\Delta_\infty$. We can factor $W_+\to W_-$ as a relative cell complex built from two types of cells, which both have the form:

- the product Mnd$_+ \times (C \to D)$ of the representable with a functor $C \to D$.

Here $C \to D$ is either the inclusion of the two endpoints into the walking arrow (used to freely attach the missing map in each hom-set from $[n+1]$ to $[n]$), or it is the surjection from the parallel pair into the walking arrow (used to impose the appropriate composition relations for the attached cells). Importantly both functors are identity-on-objects.

The point is the map of weighted limits $\{W_-,T\}\to\{W_+,T\}$ factors as a composite of pullbacks of products of maps

- {Mnd$_+ \times D$, $T$}$\to${Mnd$_+ \times C$, $T$}

and this, by the Yoneda lemma, is just the induced map between the cotensors $D \pitchfork B \to C \pitchfork B$. Now this map is (representably) conservative because a natural transformation is invertible if and only if its components are isomorphisms, and the functor $C \to D$ is identity-on-objects. Hence, $u^T \colon B[T] \to B$ is conservative.

### Colimit representation of algebras

Any algebra $(b,\beta)$ for a monad $T$ on a category $B$ is the colimit of a canonical $u^T$-split coequalizer diagram. More exactly, there is a reflexive coequalizer diagram in $B[T]$ built from free algebras (though not free algebra maps) whose colimit is $(b,\beta)$, and this diagram admits a splitting upon application of the forgetful functor $u^T\colon B[T] \to B$.

These canonical colimits diagrams are also “all in the weights”! Before making this precise, allow me to change the diagram shape. The reflexive coequalizer diagram defines a final subcategory of $\Delta^{op}$; instead of reflexive coequalizers, we’ll speak of colimits of simplicial objects; colimit cones have shape $\Delta_+^{op}$. Such a colimit cone is *split* if it extends along the natural identity-on-objects inclusion $\Delta_+^op\to\Delta_\infty$.

We’ll describe a weight $W$ for a higher dimensional analog of $u^T$-split augmented simplicial objects. For formal reasons, augmented simplicial objects of this form are colimit diagrams, indeed are absolute colimits. There is a map of weights $W \to W_-$, hence a map of weighted limits, carrying the EM-object to the object of these canonical colimit cones.

It is easy to define the weight $W$ using the principle that the weighted limit functor is cocontinuous in the weights (and contravariant). Namely, $W$ is the pushout

$\begin{matrix}
W_+ \times \Delta_+^{op} & \longrightarrow & W_+ \times \Delta_\infty \\
\downarrow & & \downarrow \\
W_-\times\Delta_+^{op} & \longrightarrow & W
\end{matrix}$
The categories $\Delta_\infty$ and $\Delta_+^{op}$ act on the restricted representables using composition in **Adj** (ordinal sum). This action defines a cone under the pushout diagram with summit $W_-$ and hence the desired map $W \to W_-$.

The functor $W_- \times \Delta_+^{op}$ defines the weight for augmented simplicial objects in $B[T]$. By cocontinuity, $\{W,T\}$ is the pullback $\begin{matrix} \{W,T\} & \longrightarrow & \Delta_+^{op} \pitchfork B[T] \\ \downarrow & & \downarrow u^T\\ \Delta_\infty \pitchfork B & \longrightarrow & \Delta_+^{op} \pitchfork B \end{matrix}$ i.e., it is the object of augmented simplicial objects in $B[T]$ whose images under $u^T$ admit a splitting. Thus $\{W,T\}$ is the object of $u^T$-split augmented simplicial objects in $B[T]$. Using the (locally posetal) 2-category structure on $\Delta_+$, there is a 2-cell

$\begin{matrix} W_- \times \Delta^{op} & \longrightarrow & W \\ \downarrow & \overset{\Downarrow}{\nearr} & \\ W_- & \end{matrix}$ that defines an absolute left extension diagram. Furthermore, this universal property is equationally witnessed and so preserved by any 2-functor. In particular, taking weighted limits and restricting along $\{W_-,T\} \to \{W,T\}$ we get an absolute left lifting diagram

$\begin{matrix} & & B[T]=\{W_-,T\} \\ & {\mathrm{id}}\underset{\Uparrow}{\nearr} &\downarrow\mathrm{const} \\ B[T]=\{W_-,T\} & \longrightarrow & \Delta^{op}\pitchfork B[T]=\{W_-\times\Delta^{op},T\} \end{matrix}$ which we interpret as presenting the elements of the EM-object as colimits of diagrams of shape $\Delta^{op}$. Furthermore, the monadic forgetful functor preserves these colimits, essentially by functoriality of our definition of the maps between the weights.

I realize this is all a bit sketchy. The best I can do here is give a reference: this argument is very similar to the discussion in section 5.3 of this paper.

### Comparison with the monadic adjunction

Of course, monadicity is about comparing an adjunction $f : B \rightleftarrows A : u$ with monad $uf=T$ with the monadic adjunction $f^T : B \rightleftarrows B[T] : u^T$. Recall adjunctions are diagrams of shape **Adj** in $\mathcal{K}$. A general result, immediate from the defining universal properties, says that the weighted limit of a restricted diagram is isomorphic to the limit of the original diagram weighted by the left Kan extension of the weight. In particular, writing $H\colon$**Adj**$\to\mathcal{K}$ for our adjunction, its monad is $\mathrm{res}H\colon$**Mnd**$\to\mathcal{K}$. Hence, its EM-object is the weighted limit

{$W_-$, res $H$} = {lan$W_-$, $H$} = {lan res Adj$_-$, $H$}

recalling that $W_-$ was defined to be the restriction along **Mnd**$\to$**Adj** of the representable Adj$_-$. The weight for the monadic adjunction is now the left Kan extension of the restriction of the Yoneda embedding $y\colon$**Adj**$^{op}\to$[**Adj**,**Cat**]. The limit $\{y,H\}$ returns the adjunction $H$. Hence, the canonical map of weights lan res $y \to y$ induces the comparison from the given adjunction $f : B \rightleftarrows A : u$ to the monadic adjunction $f^T : B \rightleftarrows B[T] : u^T$. We write $R\colon A \to B[T]$ for the non-identity component. Functoriality implies that $u^T R = u$.

### Monadicity

As above, the weight $W'$ for $u$-split simplicial objects in $A$ is defined to be a pushout

$\begin{matrix}
Adj_+ \times \Delta^{op} & \longrightarrow & Adj_+ \times \Delta_\infty \\
\downarrow & & \downarrow \\
Adj_-\times\Delta^{op} & \longrightarrow & W'
\end{matrix}$
Here we’re using $\Delta^{op}$ and not $\Delta_+^{op}$ because we’re defining the weight for the (simplicial object shaped) diagrams, not the (augmented simplicial object shaped) cones under these diagrams. We say that $A$ *admits colimits of* $u$-*split simplicial objects* if there is an absolute left lifting diagram

$\begin{matrix} & & A=\{Adj_-,H\} \\ & {\mathrm{colim}}\underset{\Uparrow}{\nearr} &\downarrow\mathrm{const} \\ \{W',H\} & \longrightarrow & \Delta^{op}\pitchfork A=\{Adj_-\times\Delta^{op},H\} \end{matrix}$

The weight lan $W_-$ defines a cone under the pushout defining $W'$ in the expected way. Hence, there is a map

$L\colon B[T]=\{\mathrm{lan} W_-, H\}\to\{W',H\}\to A$

under the hypothesis that $A$ admits colimits of $u$-split simplicial objects. It follows from the universal property of the absolute lifting that this is left adjoint to the comparison map $R:A \to B[T]$.

Let $\iota$ and $\nu$ denote the unit and counit of the comparison adjunction $L \dashv R$. If $u\colon A \to B$ preserves colimits of $u$-split simplicial objects, then, as in the classical proof, it follows that $u^T\iota$ is an isomorphism; as $u^T$ is conservative, $\iota$ must be an isomorphism. Now $\iota_R$ and hence $R\nu$ is an isomorphism, so $u\nu = u^TR\nu$ is also an isomorphism. Finally, if $u$ is conservative we conclude that $\nu$ is an isomorphism, in which case $L\dashv R$ is an adjoint equivalence, completing the proof.

## Re: A weighted limits proof of monadicity

In Richard’s talk at the pre-CT workshop, he mentioned that by right Kan extending along the inclusion of

MndintoAdj, you can obtain the entire monadic adjunction as a diagram. I liked that idea because working with derivators has taught me to sometimes prefer Kan extending along fully faithful functors over constructing (weighted) (co)limits. Can you reformulate your other weighted limits as fully faithful Kan extensions? If so, it might be the easiest way to do something analogous in a “2-derivator” (whatever that is).