Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: Dinatural category of elements

Patrick Nicodemus (Mar 26 2025 at 14:28):

A colleague of mine (who I'm not sure is on this server, so I'm asking on their behalf) came up with/discovered an interesting construction.

Let $F : \mathcal{C}\times\mathcal{C}^{\rm op}\to\mathbf{Sets}$ be a functor. The "dinatural category of elements" of $F$ is the category whose objects are pairs $(c, x)$ with $x\in F(c,c)$. A morphism $(c,x)\to (d,y)$ is $f : c\to d$ such that $F(f,1)(x) = F(1,f)(y)$. Composition and identities are clear. This is equipped with a projection functor $\pi : \mathrm{el}(F)\to C$, but it is apparently not a discrete fibration like the usual category of elements. The connected components of $\mathrm{el}(F)$ correspond to elements of the coend of $F$, similarly to how the colimit of a presheaf is the set of connected components of its category of elements. The set of sections of $\pi$ in $\mathbf{Cat}$ correpond to elements of the end of $F$, similar to how sections of the discrete fibration of the category of elements of a presheaf correspond to elements of the limit of $F$.

Can others elaborate on this? Does it have a name?

Patrick Nicodemus (Mar 26 2025 at 14:36):

I also observed that for any functor $G : \mathcal{A}\times\mathcal{A}^{\rm op}\to \mathcal{B}$ and $b \in B$ one can form a "dinatural comma category" whose objects are pairs $(a, f : G(a,a)\to b)$, a morphism $(a, f)\to (a',f')$ is a morphism $g : a \to a'$ such that $f\circ G(g, 1)=f'\circ G(1,g)$. This "dinatural category of elements" is a particular example of this construction applied to the (ordinary, usual) Yoneda embedding $y : \mathcal{C}^{\rm op}\times\mathcal{C}\to [\mathcal{C}\times\mathcal{C}^{\rm op};\mathbf{Sets}]$. So, this leads me to ask whether there is some specialization of the theory of 2-categories which allows for such "dinatural comma categories" to be expressed as a 2-limit in some way.

Patrick Nicodemus (Mar 26 2025 at 14:41):

Taking the discussion in a somewhat orthogonal direction, just as the "category of elements" construction for a presheaf can be generalized to the Grothendieck construction for indexed categories, it is clear how to generalize this construction to give a "dinatural Grothendieck" construction for functors $F : \mathcal{C}\times\mathcal{C}^{\rm op}\to\mathbf{Cat}$, so if anyone wants to comment on that, feel free

Andrea Laretto (Mar 26 2025 at 15:07):

I also very recently stumbled upon the very same definition! The main problem that left me a bit dissatisfied with this definition is that you do not ever talk about morphisms between $x$ and $y$ in any fiber, and this is obviously connected to the problem that given a morphism $f : c \to d$ one cannot directly relate $P(c,c)$ and $P(d,d)$... so possibly a more enlightened 2-categorical definition might include some sort of laxity instead of the "wedge-like" equation.

Amar Hadzihasanovic (Mar 26 2025 at 15:11):

Just another way you can see this category.
In general, given a functor $\pi: \mathcal{C} \to \mathcal{D}$, one can consider a category of sections of $\pi$, whose

• objects are sections of $\pi$, that is, functors $s: \mathcal{D} \to \mathcal{C}$ such that $\pi \circ s = \mathrm{id}_\mathcal{D}$,
• morphisms from $s$ to $t$ are natural transformations $\alpha: s \Rightarrow t$ such that, for all objects $d$ of $\mathcal{D}$, $\pi(\alpha_d) = \mathrm{id}_d$.

Let $\mathbb{N}$ denote the monoid of natural numbers with addition, seen as a category with a single object.
An endo-profunctor $F$ on $\mathcal{C}$ determines and can be reconstructed from a functor $\pi_F: \mathcal{C}_F \to \mathbb{N}$, where

• $\mathcal{C}_F$ is obtained by adjoining a morphism $x: c \to d$ to $\mathcal{C}$ for every element $x \in F(c, d)$, and quotienting by the equations $gxf = y$ whenever $F(g, f)(x) = y$,
• $\pi_F$ is defined by $\pi_F(f) = 0$ if $f$ is a morphism in $\mathcal{C}$ and $\pi_F(x) = 1$ if $x$ is an element of the profunctor.

Then an equivalent description of your "dinatural category of elements" is the category of sections of $\pi_F$.

Amar Hadzihasanovic (Mar 26 2025 at 15:19):

Notice that “endo-profunctors” are essentially equivalent to “categories over $\mathbb{N}$”; every endo-profunctor determines a category over $\mathbb{N}$ the way that I described, and, vice versa, every functor $\pi: \mathcal{D} \to \mathbb{N}$ determines a profunctor $F_\pi$ on the subcategory $\pi^{-1}(0)$ of $\mathcal{D}$, by letting $F_\pi(c, d) \coloneqq \{ x \in \mathcal{D}(c, d) \mid \pi(x) = 1 \}$, with the action determined by morphism composition in $\mathcal{D}$.
The two constructions are inverse to each other up to natural isomorphism.

Amar Hadzihasanovic (Mar 26 2025 at 15:23):

This is sometimes a nice perspective because it shows you that a “category together with an endo-profunctor on it” can be faithfully encoded as the data of a “category [but a different one!] together with an $\mathbb{N}$-valued, additive measure on its morphisms”.

Amar Hadzihasanovic (Mar 26 2025 at 15:28):

Then your category of sections has, as objects, objects of this category together with a measure-1 endomorphism on them; this may sometimes be interpreted as a category of dynamical systems...

fosco (Mar 26 2025 at 15:32):

I see this story from a slightly different point:

• A monad in profunctors is a category
• A mere endo-1-cell in profunctors is a (directed) graph

In such directed graph, you can study iterated compositions of the profunctors with itself, studying "flows" of the discrete dynamical system prescribed by the endo-profunctors.

Amar Hadzihasanovic (Mar 26 2025 at 15:33):

Amar Hadzihasanovic said:

Notice that “endo-profunctors” are essentially equivalent to “categories over $\mathbb{N}$”; every endo-profunctor determines a category over $\mathbb{N}$ the way that I described, and, vice versa, every functor $\pi: \mathcal{D} \to \mathbb{N}$ determines a profunctor $F_\pi$ on the subcategory $\pi^{-1}(0)$ of $\mathcal{D}$, by letting $F_\pi(c, d) \coloneqq \{ x \in \mathcal{D}(c, d) \mid \pi(x) = 1 \}$, with the action determined by morphism composition in $\mathcal{D}$.
The two constructions are inverse to each other up to natural isomorphism.

Sorry, this is not true, lol --- it is true that $\pi \mapsto F_\pi$ is a left inverse of $F \mapsto \pi_F$, but not the other way around

fosco (Mar 26 2025 at 15:34):

Indeed, I was starting to express doubt about it. Not categories over NN...

Amar Hadzihasanovic (Mar 26 2025 at 15:36):

Yes I was thinking about the (true) fact that profunctors = categories over the arrow and I turned it too hastily into endo-profunctors = categories over the loop, but it's only an “inclusion” in the latter case.

Amar Hadzihasanovic (Mar 26 2025 at 15:48):

I'd hasard the conjecture that endo-profunctors = categories with a strict Conduché functor to $\mathbb{N}$...

Amar Hadzihasanovic (Mar 26 2025 at 15:55):

Thus Conduché functors into B correspond to pseudofunctors from B, regarded as a locally discrete bicategory, to the bicategory Prof.

Well this seems to confirm it, since pseudofunctors from $\mathbb{N}$ to $\mathrm{Prof}$ classify endo-profunctors.

Amar Hadzihasanovic (Mar 26 2025 at 16:15):

So my intuition is that a Conduché functor $\mu: \mathcal{C} \to \mathbb{N}$ is like a measure on morphisms of $\mathcal{C}$ that is additive wrt composition, and with the property that every morphism of length $n$ can be decomposed into $n$ morphisms of length $1$; this is the sort of thing that seems quite natural in modelling some categories of discrete-time processes

Mike Shulman (Mar 26 2025 at 16:16):

I think the "standard" construction that this is most closely related to is the "two-sided category of elements", which associates to any profunctor $C\nrightarrow D$ a [[two-sided discrete fibration]] $C \leftarrow E \to D$, which is a fibration over $D$ and an opfibration over $C$, compatibly, and discrete over $C\times D$ (but not over $C$ or $D$ individually). Then if it should happen that $C=D$, you can pull back along the diagonal $C\to C\times C$ and I think you get your construction.