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

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?

• Just a couple of quick observations: in your main example, shouldn't $\alpha$ be an isomorphism, and really the inverse of a counit? Here $\eta P_a: Pa \to PPa$ and $P\eta \circ \eta P \cong 1_P$ by a (2-)monad unit constraint. So it's really $\beta$ that's the unit. Second, to elide over "foundational" problems, you could just take your main example to be categories enriched in a quantale, like $2$ or the base $([0, \infty], +)$ of Lawvere metric spaces. I happen to like this topic a lot but have slightly different ways of considering it, which I may get to later. – Todd Trimble Oct 20 '18 at 10:32
• Please, do not hesitate to share any thought on this, were it only a pointer to the literature. I prefer to avoid foundational problem taking my monads not to be endofunctors, but instead internal monoids in the skew-monoidal category $([X,Y],\lhd,J)$, where $F \lhd G\cong Lan_JF\circ G$ for a fixed $J : X \to Y$ (more than often a fully faithful one which is therefore a strong monoidal unit). – fosco Oct 20 '18 at 10:43
• There won't be any pointers to the literature, unless you count any sparse unpublished scribblings of mine on "epistemologies" as "literature". – Todd Trimble Oct 20 '18 at 10:45
• But I'd better consider more carefully your skew-monoidal remark, which sounds interesting... – Todd Trimble Oct 20 '18 at 10:46
• I do consider them literature, and I like the name. I'll have a look at your Lab. :-) so let me say it clearly, you claim that in fact $\mu \dashv \eta P$; moreover, every $P$-algebra works as left adjoint to $\eta$, so $P$ is a lax-idempotent monad. – fosco Oct 20 '18 at 10:48

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.

• I hope my yesterday's email fits my question in a sufficiently big framework to grasp where this is going :-) there's another side of the story, pretentiously aiming to give a precise connection between "all" the paradigms to do formal CT. It would be nice to fit your epistemologies in the picture! – fosco Oct 21 '18 at 17:24
• I'm still thinking over your email (thanks! it's very interesting). I hope the answer wasn't overly harsh or unfair. There are probably lots of different frameworks that could be considered. – Todd Trimble Oct 21 '18 at 17:28
• As you maybe noticed a few years before me, working with these gadgets is computationally extremely difficult (you lose an "op", you lose a statement). Your answer is a nice birdseye on what I'm trying to axiomatize. :-) – fosco Oct 21 '18 at 17:39

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).

• If $F : A \to B$ and $G : A^o \to B$, you can regard them as parallel functors $F,G : A^o \times A\to B$, and now a dinatural transformation $F \overset{..}\Rightarrow G$ amounts to what you defined, right? – fosco Oct 23 '18 at 18:39