$P = [-°,Set]$ is a contravariant co/lax idempotent monad, whose multiplication is determined by the unit

Question

A unidetermined contramonad is a 2-monad $T : {\cal C}\to \cal C$ such that

1. $T$ is contravariant, i.e. a contravariant endofunctor;
2. the multiplication $\mu_A : TTA \to TA$ is determined as $T\eta_A = T(A\to TA)$.

Q(-1): Is this even a thing? Does this definition already exist under another name?

An example of such monad is the presheaf construction $P : A\mapsto [A°,Set]$; it has the Yoneda embeddings as units, and it is in fact possible to show that $T\eta_A$ acts as a multiplication, in view of the general fact that $$ [[PA°,Set]°,Set] \underset{PP\eta}{\overset{P\eta P}\leftrightarrows} [PA°,Set] $$ is an adjunction ($PP\eta\dashv P\eta P$). (notation: whenever $X$ is a large category, $[X°,Set]$ is the category of small functors).

Q0: Or is it (coming from the adjunction $P\eta\dashv \eta P$)? I'm able to find two natural transformations:

• $\alpha : F \Rightarrow P\eta(\eta P(F))$, induced by the cowedge $$ Fa\times A(-,a)\to F$$ (or, rather, induced by the action of $F$ on arrows): $\alpha$ mates to $$ \tilde\alpha : Fa \to Set(A(-,a),F) $$ (and $\tilde\alpha$ is invertible, for that matter: Yoneda lemma).
• $\beta : \Theta \Rightarrow \eta P(P\eta(\Theta))$, induced by the action of $\Theta$ on morphisms: its components are $$ PA(\hom(-,a),F)\to PA(\Theta F,\Theta(\hom(-a,)))$$ who mate to a family of maps $$ \tilde\beta : \Theta(F) \to PA(F, \Theta(\hom(-,a)))$$

Now... who's the unit? Who's the counit?

$P$, being a free cocompletion, is a KZ-monad. As soon as one wants to write down explicitly what this structure is, however, they have to face a few slight inaccuracies in how $P$ was defined;

• first of all it is not only contravariant, but also partially defined; it is a so-called relative monad, like a monad but not an endofunctor. In this particular case, $P : cat^\text{coop}\to Cat$ is a monad relative to $i^\text{coop} : cat^\text{coop} \to Cat^\text{coop}$, the inclusion of small into locally small categories.

  Q1: Am I wrong if I define a contravariant (total) monad to be a contravariant endofunctor $T : {\cal C}\to \cal C$ which is relative to $1^\text{op} : {\cal C}^\text{op} \to {\cal C}^\text{op}$?

• second, it seems to satisfy mixed properties of a lax and a colax idempotent 2-monad: in particular, it seems to me that every $P$-algebra $a :PA \to A$ is a left adjoint to the unit (so lax), and yet $\mu \dashv \eta P$ (so colax).

  Q2: Am I committing a mistake? If not, does these mixed properties have to do with the fact that $P$ is contravariant?

• third it is "unidetermined", i.e. $\mu$ is determined by $\eta$.

  Q3: How does this affect the equations defining a KZ-monad, if at all?

Answer 1

It's a little weird I guess to consider "contravariant monads"; I don't think I've seen them defined before. If one considers such a notion at just the 1-categorical level, then a question is: what should an algebra map $f: A \to B$ mean? The obvious diagram one writes down would be

$$\begin{array}{l} TA & \stackrel{a}{\to} & A \\ \uparrow \; Tf & & \downarrow\; f \\ TB & \stackrel{b}{\to} & B \end{array}$$

but commutativity of such doesn't really fit the examples you have in mind, not even up to isomorphism when we consider 2-categorical structure.

Much more telling is that morally there's both a covariant $T$ and a contravariant $T$ (which I'll denote as $T^\ast$), and what we have is $Tf \dashv T^\ast f$ for all morphisms $f: A \to B$. You are quite right that in your examples we have that $\mu_A = T^\ast \eta_A: TTA \to TA$. So in fact in the examples you are considering, we have an adjoint string

$$T\eta \dashv T^\ast \eta \dashv \eta T$$

and I think that helps to keep things straight. It fits in well with the basic theory of KZ (= lax idempotent) monads (going back to early 70's work of K = Anders Kock) where we have an adjoint string $T\eta \dashv \mu \dashv \eta T$.

To stir the pot even more, there is in your examples a second (covariant) monad which might be denoted $T^o$. By duality this carries a colax idempotent monad structure, and then there are relations between the two such as an "Isbell conjugation" map

$$T \to T^o$$

that is contravariantly adjoint to itself (this time the contravariance is at the 2-cell level). Now I think $T^o(A)$ should be a $T$-algebra and that the Isbell conjugation displayed above will be a $T$-algebra map, although I suspect that in the relative monad setting you are considering, you will not get that all $T^o$-algebras carry $T$-algebra structures (i.e., not all cototal categories are total, a known fact for ordinary category theory with classical foundations) -- although IIRC such a result did hold in the "epistemology" setting (that I was considering many years ago but which begins to fade from memory), where the key 2-monads involved are actual endofunctors.

I'm sorry if this is not completely responsive to all your questions -- the main takeaway is that there are deeper levels to your examples than I think can be reasonably accounted for with only a contramonad formalism, for example the level $T \dashv T^\ast$. Just to say one more thing in anticipation of future discussion: if it helps, I believe you can think of an epistemology as basically equivalent to a Yoneda structure in which all 1-cells are admissible, but that in my brief nLab note I was considering more of an interplay between the Kleisli bicategory $\mathbf{B}$ of $T$ (think profunctors), and its associated bicategory of maps $\text{Map}(\mathbf{B})$ (same objects as $\mathbf{B}$, but the 1-cells are left adjoints. Here $\text{Map}(\mathbf{B})$ gives back the original 2-category of the Yoneda structure, up to a notion of Morita equivalence anyway, and the inclusion $i: \text{Map}(\mathbf{B}) \hookrightarrow \mathbf{B}$ has a KZ right bi-adjoint $P$ which governs essentially all the structure.

Answer 2

This is just a (too long) comment. As Todd Trimbe observed, there is a slightly incongruence: If $T: \mathcal{C}\to \mathcal{C}^{op}$ it isn't a proper endomorphism, then we have to interpret $T\circ T$ as $T^{op}\circ T: \mathcal{C}\to \mathcal{C}^{op}$ then how we fill the question mark in below?:

$\mu: TT\Rightarrow T: \mathcal{C}\to (?)$.

I only propose a way for make the problem formally coherent.

If $F: \mathcal{A}\to \mathcal{B}, G: \mathcal{A}\to \mathcal{B}^{op}$ in $Cat$ we define a anti-transformation $\alpha: F\Rightarrow^* G $ (marking it by "*") as a family of maps $\alpha_A: F(A) \to G(A)$ such that $\alpha_A= G(f)\circ \alpha_B\circ F(f)\ f: A\to B$, a anti-transformation $\beta: G\Rightarrow F$ has quite similar definition

(exactly is one $\beta: G^{op}\Rightarrow F^{op}$ as above, a anti-transformation is a particular case of dinatural transformation like $F\xrightarrow{ditransf.} G^{op}$)

Then I TRY to define a 2-category $Cat^\pm$ whit objects the small categories, with arrow's $\mathcal{A}\to \mathcal{B}$ of type $(F, +),\ (G, -)$ for $F$ (resp. $G$) a covariant (contravariant) functor from $\mathcal{A}$ to $\mathcal{B}$, with cell's like $(\sigma, s\cdot t): (H, s)\Rightarrow (K, t)$ that is a (usual) transformation $F\Rightarrow G$ if $s\cdot t=1$ or anti-transformation $F\Rightarrow G$ if $s\cdot t=-1$. Now the horizontal composition is defined as $(G, s)\circ (F, 1)= (G\circ F, s),\ (G, s)\circ (F, -1)= (G^{op}\circ F, -s)$, this well defined on arrows, on cell is quite similar (observing that a dual of a transformation (resp. anti-transformation) is still a transformation (resp. anti-transformation)), the vertical composition of cells is defined on components and we have that $(\tau, t)\ast (\sigma, s)= (\tau\ast \sigma, t\cdot s)$.

Edit: I see now that Godement rule failed, this structure neither is a sesquilinear category, but may be enough for working by triple.

Edit: (continue from my previous comment):

If $M$ in a monoid viewing as a one object ($\bullet$) category, let $\Gamma(M):= \bullet\downarrow M$ where a morphism is represented as $x: a\to b$ where $a, b, x\in M, x\circ a=b$, then $M$ is a group iff $M$ is a group. Let $\{1-, 1,\cdot\}$ the multiplicative couple to the additive group $Z_{(2)}$ (module $2$ numbers), and let $I^\pm:= \Gamma(\{+1, -1\})$.

I define $\pm$.category as a small category $\mathcal{C}$ with a isomorphism $\mathcal{C}\cong ||\mathcal{C}||\times I^\pm$, where I call $||\mathcal{C}||$ the category of "essential" object and morphisms of $\mathcal{C}$, then each object of $\mathcal{C}$ is representable as $(X, a)$ where $X\in ||\mathcal{C}||$, $a=\pm 1$, and a morphism like $(f, c): (X, a)\to (Y, b)$ where $f:X\to Y$ in $||\mathcal{C}||$, and $c=a\cdot b$, with composition on $||\mathcal{C}||$ and multiplication of signs, observe that given $(f, c)$ as before we have also $(f, c): (X, -a)\to (Y, -b)$.

There is the natural projection $\pi:\mathcal{C}\to I^\pm$, a $\pm$ functor between $\pm$ categories $F: \mathcal{A}\to \mathcal{B}$ is a "sign-preserving" i.e. a functor $F: ||\mathcal{A}||\to ||\mathcal{B}||$ and we write also $F: \mathcal{A}\to \mathcal{B}$ as the functor defined as $F(X, a)= (F(X), a),\ F(f, c)=(F(f), c)$.

In $Cat\downarrow I^\pm$ we have a product, this is a monoidal product of the category of signed categories and signed functors, exactly $\mathcal{A}\otimes \mathcal{B}:= \mathcal{A}\times_{I^\pm} \mathcal{B}$ as object as $(A, B, a), (A, B)\in ||\mathcal{A}\times \mathcal{B}||,\ a=\pm1$, the unitary object is just $I^\pm$ (consider $\underline{1}\times I^\pm\cong I^\pm$). Then we have the monoidal category $Cat_{\pm}$ of signed categories and signer functors. I define a $Cat_{\pm}$-category $Cat^{\pm}$ with object the some of $Cat$, where objects of $[\mathcal{A},\ \mathcal{B}]$ write as $(F, a): \mathcal{A}\to \mathcal{B}$ are functors $F: \mathcal{A}\to \mathcal{B}$ if $a=1$, or functors $F: \mathcal{A}\to \mathcal{B}^{op}$ if $a=-1$, morphism $(\sigma, s) : (F, a)\to (G, b)$ are usual transformation $\sigma: F\to G$ is $s=1$, and anti-transformation (see mine precedent comment) if $s=-1$. About the composition $[\mathcal{B},\ \mathcal{C}]\otimes [\mathcal{A},\ \mathcal{B}]\to [\mathcal{A},\ \mathcal{C}]$ define $(\tau, t)\circ (\sigma, s)$ where $(\tau, t): (G_1, b_1)\Rightarrow (G_2, b_2)$, $(\sigma, a): (F_1, a_1)\Rightarrow (F_2, a_2)$ as follow:

firs of all observing that $a_1=b_1,\ a_2=b_2$. If $s=t=1$ the definition is just the usual. If $s=-1, t=-1$ then:

if $a_1=b_1=1$ (then $b_2=a_2=-1$) let $(\tau\circ \sigma, -1)$ with components $G_1F_1(A)\xrightarrow{G_1\sigma}G_1F_2(A)\xrightarrow{\tau_{F_2A}} G_2F_2(A)$.

if $a_1=b_1=-1$ (then $a_2=b_2=-1$) let $(\tau\circ \sigma, -1)$ with components $G_1F_1(A)\xrightarrow{\tau_{F_1A}}G_2F_1(A) \xrightarrow{G_2\sigma}G_2F_2(A)$. (I seem that it work well).

The Yoneda $Y_\mathcal{C}:=[(-)^{op}, Set]: \mathcal{C}\mapsto \mathcal{C}^>$ (being natural on morhpism and cell) is a ($Cat_{\pm}$)-"endo"-functor (with obvious consideration about set's universe expansion).