The n-Category Café

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.