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Stream: learning: questions

Topic: "Interface" of a functor, what is it?

fosco (Nov 25 2025 at 21:24):

I am trying to read a rather impenetrable paper (here) that calls "interface" of a functor $M$ with small domain C and complete codomain D the following construction.
image.png
If this construction makes sense, there is certainly a way to express it in terms of a universal construction; one would be tempted to write that (in this horrible notation) $Int_M : c \mapsto \int_x Mx^{C(x,c)}$; so the interface of $M$ is the limit of M weighted by $y(c)$: too bad that the integration variable $x$ is now covariant in both positions, so the end doesn't really make sense.

fosco (Nov 25 2025 at 21:30):

This seems like a general trick to change the variance of a functor; note that instead, by the usual Togakure-ryū trick, $\int_x Mx^{C(-,x)} \cong M$. Can it be the case that what I'm using as the weight is the contravariant yoneda embedding $\dot y : C \to [C,Set]^{op}$, so the limit of $M$ (covariant) weighted by $\dot y$ yields the functor in question?

James Deikun (Nov 25 2025 at 22:45):

It seems like it's $c \mapsto \mathop{\mathrm{lim}} \mathcal C/c \xrightarrow{\pi_\mathcal{C}} \mathcal C \xrightarrow{M} \mathcal D$ if it's the way it's written; it seems like it's using the fact that the codomain fibration is a fibration, and hence gives rise to an indexed category, not just a coindexed category. Of course this actually only works if $\mathcal C$ has pullbacks. No idea why it's called an "interface".

fosco (Nov 26 2025 at 10:22):

it seems like it's using the fact that the codomain fibration is a fibration

I don't think any additional assumption is needed for the construction to make sense; the problem is: what is it, really? It's clearly some sort of weighted limit, but a trick is needed to circumvent the covariance problem.

fosco (Nov 26 2025 at 10:23):

I believe my suspicion is correct: the diagram is M, the weight is in fact a family of weights, $C \to [C,Set]^o$

fosco (Nov 26 2025 at 10:26):

thus the result is not a single object of D, but an entire functor C to D, contravariant because of how I chose coYoneda as "system of weights"; a similar (dual) trick is in Street-Walters Yoneda structures, defining the weighted colimit $w\otimes f$ :

image.png

where (in Cat) the weight is a functor $X \to [Y^o,Set]$

fosco (Nov 26 2025 at 10:27):

well, I don't want to outsource the grunt work on you to check that this indeed is the construction at hand; instead... what intuition is there behind this construction? Have you seen instances of it "in nature"? It's the first time I see this

Amar Hadzihasanovic (Nov 26 2025 at 10:39):

Just noting that the poset case seems like quite a simple thing: given an order-preserving map $f: P \to Q$ where $Q$ has all meets, the interface of $f$ is the order-reversing map sending $x$ to $\bigwedge_{y \leq x} f(y)$.

Amar Hadzihasanovic (Nov 26 2025 at 10:43):

If $P$ is something like an "information order", domain-theory style, then it would seem to me like, as we gain more information, we "see" more and more of $Q$ through $f$, and then we need to cast a wider and wider "net" (=the limit cone) to capture the whole image of $f$.

Amar Hadzihasanovic (Nov 26 2025 at 10:43):

The net being wider and wider corresponds to the tip of the cone regressing further and further back.

Amar Hadzihasanovic (Nov 26 2025 at 10:49):

Perhaps this is also the intuition that the word "interface" is trying to convey: if there is some notion of direction in the domain of $M$---perhaps it is a poset, or a directed category---so that it makes sense to look at how the image of $M$ grows along the direction, then the limit indexed by $c$ is a way of "packaging" the "image of $M$ up to object $c$" into a single object (from which it can be accessed through the cone projections), and so acts as a single-entry interface with that whole image?

fosco (Nov 26 2025 at 11:00):

I think it's something like that, just the limit projections are indexed over pairs of the form $(x, u : x\to c)$, so that you project onto the function $Mu : Mx \to Mc$, instead of just "M at level c"

fosco (Nov 26 2025 at 11:01):

("function", if M has codomain Set)

fosco (Nov 26 2025 at 11:06):

Say $M : \omega \to Set$ is a chain of sets. Clearly, the category $\omega/[n]$ is just the total order $\{0<1<\dots<n\}$.

fosco (Nov 26 2025 at 11:07):

Hence the projection $\pi_n$ is just the inclusion, and the limit of $[n] \hookrightarrow \omega \to Set$ is... $M_0$ for every n (the domain of the diagram has an initial object).

Amar Hadzihasanovic (Nov 26 2025 at 13:26):

Here's a more fun example, maybe. Let $X$ be a topological space, let $P$ be a presheaf on $X$, and let $\mathcal{U}$ be a filtered open cover of $X$ (seen as a subposet of the lattice of opens of $X$).

Then, restricting $P$ to the opens in $\mathcal{U}$, we have a functor $P_{\mathcal{U}}: \mathcal{U}^\mathrm{op} \to \mathrm{Set}$.
The interface of this functor is the functor $\mathrm{Int}_{P_\mathcal{U}}: \mathcal{U} \to \mathrm{Set}$ which sends an open $U \in \mathcal{U}$ to $\mathrm{lim}_{U \subseteq V \in \mathcal{U}} PV$.

Assuming wlog that $\mathcal{U}$ contains $\emptyset$, we have that $\mathrm{Int}_{P_\mathcal{U}} \emptyset$ is exactly the set of "compatible families" which show up in the sheaf condition for $P$ with respect to $\mathcal{U}$. The general $\mathrm{Int}_{P_\mathcal{U}} U$ seem to be something like "compatible families relative to $U$" in the sense that we are only allowed to look at the image of $P$ on opens in $\mathcal{U}$ that contain $U$.

fosco (Nov 26 2025 at 16:45):

After some more discussion and thinking, it seems that this construction is able to generate some trivial examples, and some interesting ones:

• If $M : \omega \to Set$ is a chain, $Int_M$ is the functor constant at $M0$, and if $M : \omega^o \to Set$ is an antichain, $Int_M$ is the functor constant at $\lim M$ (all unbounded subposets $\{n\ge n+1\ge \dots \}$ of $\omega^o$ are cofinal.
• If $M : O(X)^o \to Set$ is a presheaf, there is an associated copresheaf sending $U$ to $\lim_{V\supseteq U} MV$, which looks like a "costalk" (?)
• If $M : BA \to Set$ is the (left) action of a monoid $A$ on a set $X$, there is an associated right action on the same set, and this correspondence exchanges the left regular representation of A on itself, with the right regular representation.

Especially the last item of the list feels Isbell-y (Isbell duality behaves like that on the regular representations of the monoid A), which is the initial suspicion I had reading the definition...

Amar Hadzihasanovic (Nov 26 2025 at 17:03):

The second example is also constant at $MX$ if applied to the "whole" presheaf since $O(X)$ has a greatest element, that's why I suggested relativising to a filtered cover, where it does seem to become more interesting.

Morgan Rogers (he/him) (Nov 27 2025 at 08:20):

Oh when D is Set is this one of the functors in the Isbell adjunction?

fosco (Nov 27 2025 at 15:49):

I am still struggling to understand what is the universal property of this object; sending $M$ to $Int_M$ is certainly a covariant functor $[C,D] \to [C^o,D]$. Yet, look what happens when I try to check whether it is a right adjoint "for formal reasons". $Int_M(c) = \int_x Mx^{\hom(x,c)}$ (this is the only variance that makes $Int_M$ contravariant in $c$,a s it is defined to be).
image.png

fosco (Nov 27 2025 at 15:49):

which is... weird, and certainly wrong; but where is the mistake?

fosco (Nov 27 2025 at 16:28):

It also seems like "Int" is an idempotent construction:

$\begin{array}{rcl}Int_M(Int_M(c)) &=& \displaystyle \int_x \hom(x,c)\pitchfork Int_M(x)\\&=& \displaystyle \int_x \hom(x,c)\pitchfork \Big(\int_y\hom(y,x)\pitchfork My\Big)\\&=& \displaystyle \int_{xy} \hom(x,c)\pitchfork (\hom(y,x)\pitchfork My)\\&=& \displaystyle \int_{xy} (\hom(x,c)\times \hom(y,x))\pitchfork My\\&=& \displaystyle \int_{y} \Big(\int^x\hom(x,c)\times \hom(y,x)\Big)\pitchfork My\\&=& \displaystyle \int_{y} \hom(y,c)\pitchfork My\\&=& Int_M(c)\end{array}$

fosco (Nov 27 2025 at 16:34):

Furthermore, there is a universal dinatural copointing $Int_M \overset{..}\Rightarrow M$ given on components by $\pi_{c,\text{id}_c}$...

fosco (Nov 27 2025 at 16:34):

I am not particularly happy to have discovered an example of idempotent contracomonad...

Amar Hadzihasanovic (Nov 27 2025 at 16:45):

In $Mx^{C(x, c)}$, $x$ is covariant in both positions, so that's not a well-defined end...

fosco (Nov 27 2025 at 17:05):

I know, at the same time the definition as $\lim(Elts(\hom(-,c)) \to C \overset M\to D)$ plus the fact that the latter is the limit of $M$ weighted by $\hom(-,c)$, say that something like that should be true. Again, that's the only variance that makes the result contravariant in c; $\int_x Mx^{\hom(c,x)}$ makes sense, but it's just $Mc$.

fosco (Nov 27 2025 at 17:31):

Together with @Emily (she/her) I developed a language that accounts for the fact that these situations exist "in nature", but I would like to be sure there isn't an easier explanation for this construction, before I resort to "(2,0)-ends"...

Amar Hadzihasanovic (Nov 28 2025 at 08:25):

Just a simple example to show that the interface is not "idempotent" in any meaningful way: let $P$ be the poset $x, y < \top$ and $Q$ be the poset $\bot < x, y < \top$, and let $j: P \to Q$ be the obvious inclusion.

Then

• $\mathrm{Int} j: P^\mathrm{op} \to Q$ is defined by $x \mapsto x$, $y \mapsto y$, and $\top \mapsto \bot$, while
• $\mathrm{Int}(\mathrm{Int} j): P \to Q$ is constant at $\bot$.

Amar Hadzihasanovic (Nov 28 2025 at 09:19):

This reminded me of something that I worked out a few years ago, and indeed I think it's an instance of it.

Suppose we are given an indexed category $\mathcal{S}: C \to \mathrm{Cat}$.
Then, there are

1. a category $\mathrm{OplaxCone}_\mathcal{S}$ of oplax cones under $\mathcal{S}$, which are pairs of a category $D$ and an oplax natural transformation $F$ from $\mathcal{S}$ to the constant $D$ --- a component of this at $c$ is a functor $Fc: \mathcal{S}c \to D$;
2. a functor $\mathsf{Cone}_\mathcal{S}: C^\mathrm{op} \times \mathrm{OplaxCone}_\mathcal{S} \to \mathrm{Cat}$ which, on objects, is defined by sending $(c, D, F)$ to the category of cones in $D$ over $Fc: \mathcal{S}c \to D$.

Moreover, if $D$ is complete, this functor is naturally isomorphic to a functor $\mathsf{Lim}_\mathcal{S}$ which sends $(c, D, F)$ to the slice $D/{\mathrm{lim} Fc}$...

Amar Hadzihasanovic (Nov 28 2025 at 09:21):

So in particular if we fix a oplax cone $(D, F)$ under $\mathcal{S}$, this gives us a functor $C^\mathrm{op} \to \mathrm{Cat}$; actually two equivalent descriptions of it, when $D$ is complete.

Amar Hadzihasanovic (Nov 28 2025 at 09:32):

I think we can see this "interface" as arising from

• choosing $\mathcal{S}\colon C \to \mathrm{Cat}$ to be $c \mapsto C/c$,
• choosing the oplax cone under $\mathcal{S}$ to have components $F \circ \mathrm{dom}_c\colon C/c \to D$ (this is actually a "strict" cone, both lax and oplax)

Then even without any condition on $D$ we get a contravariant functor $C^\mathrm{op} \to \mathrm{Cat}$, sending $c$ to the category of cones in $D$ over $F \circ \mathrm{dom}_c$; when $D$ is complete, this functor becomes "representable" as $c \mapsto D/\mathrm{Int} Fc$

Amar Hadzihasanovic (Nov 28 2025 at 09:35):

It is interesting that the Yoneda lemma can be recovered as a very special case of the natural isomorphism between $\mathsf{Cone}_\mathcal{S}$ and $\mathsf{Lim}_\mathcal{S}$ when $\mathcal{S}: C^\mathrm{op} \to \mathrm{Cat}$ is the contravariant slice functor, $c \mapsto c/C$.

Amar Hadzihasanovic (Nov 28 2025 at 09:41):

In any case, I think the fundamental duality at play here, independent of any conditions on $D$, is the one that given a covariantly $C$-indexed category & an oplax cone under it, produces a contravariantly $C$-indexed category.

Amar Hadzihasanovic (Nov 28 2025 at 09:42):

Somehow the oplax cone acts as a dualiser...

Amar Hadzihasanovic (Nov 28 2025 at 09:54):

I had not tried to work out the functoriality of $\mathrm{OplaxCone}$, but presumably there is some sort of fibration

$\int \mathrm{OplaxCone}^C \to [C, \mathrm{Cat}]$

where the objects of the "Grothendieck construction" above are pairs of $\mathcal{S}: C \to \mathrm{Cat}$ and an oplax cone under $\mathcal{S}$; then $\mathsf{Cone}$ could assemble into a functor $\int \mathrm{OplaxCone}^C \to [C^\mathrm{op}, \mathrm{Cat}]$, and the whole duality would then be a span between $[C, \mathrm{Cat}]$ and $[C^\mathrm{op}, \mathrm{Cat}]$ with tip $\int \mathrm{OplaxCone}^C$...

fosco (Nov 28 2025 at 09:56):

I agree that whatever is going on here is formal; hence my obsession!

Amar Hadzihasanovic (Nov 28 2025 at 10:48):

I think my last message works if $[C, \mathrm{Cat}]$ and $[C^\mathrm{op}, \mathrm{Cat}]$ are categories of functors and oplax natural transformations, and the tip of the span is an (oplax, again?) comma category of $[C, \mathrm{Cat}]$ over the functor $\mathrm{Cat} \to [C, \mathrm{Cat}]$ which sends $D$ to the constant at $D$...

Amar Hadzihasanovic (Nov 28 2025 at 10:50):

Of course the projection from the comma should be some sort of fibration, but I don't know if the other leg valued in $[C^\mathrm{op}, \mathrm{Cat}]$ has some nice properties

Ruby Khondaker (she/her) (Nov 28 2025 at 12:09):

Amar Hadzihasanovic said:

In $Mx^{C(x, c)}$, $x$ is covariant in both positions, so that's not a well-defined end...

This should still work as an end though right? You have a functor $C \times C^{op} \times C \to D$ sending $(x, c, y) \mapsto My^{C(x, c)}$. So if you hold $c$ fixed, this gets you a functor $C \times C \mapsto D$ sending $(x, y) \mapsto My^{C(x, c)}$. Precomposing with the diagonal $C \to C \times C$ gets you a functor $C \to D$ sending $x \mapsto Mx^{C(x, c)}$. Then you can take the end over this by precomposing with the projection $C^\text{op} \times C \to C$, no?

I guess this is just the general phenomenon of expressing the limit of a functor $H : C \to D$ as $\int_c H(c)$, by first precomposing with the projection $C^{op} \times C \to C$.

fosco (Nov 28 2025 at 12:33):

Ruby Khondaker (she/her) said:

Amar Hadzihasanovic said:

In $Mx^{C(x, c)}$, $x$ is covariant in both positions, so that's not a well-defined end...

This should still work as an end though right? You have a functor $C \times C^{op} \times C \to D$ sending $(x, c, y) \mapsto My^{C(x, c)}$. So if you hold $c$ fixed, this gets you a functor $C \times C \mapsto D$ sending $(x, y) \mapsto My^{C(x, c)}$. Precomposing with the diagonal $C \to C \times C$ gets you a functor $C \to D$ sending $x \mapsto Mx^{C(x, c)}$. Then you can take the end over this by precomposing with the projection $C^\text{op} \times C \to C$, no?

I guess this is just the general phenomenon of expressing the limit of a functor $H : C \to D$ as $\int_c H(c)$, by first precomposing with the projection $C^{op} \times C \to C$. Perhaps I misunderstood what you meant when you said "not well-defined"?

What you are describing is a "(0,2)-end" in the language of https://arxiv.org/abs/2011.13881

Ruby Khondaker (she/her) (Nov 28 2025 at 12:42):

Oh cool! I had no idea there was a name for these sorts of things. Though, since we take two covariant arguments, would that be a $(0, 2)$-end in the language of that paper?
image.png

I suppose what I outlined in the second paragraph illustrates that $(0, 1)$-ends are equivalent to limits.

fosco (Nov 28 2025 at 12:59):

yeah, sorry I meant (0,2) :-)