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Stream: learning: questions

Topic: How do I call this equivalence of arrows?

Ignat Insarov (Apr 16 2025 at 19:31):

Consider the equation $g = f ∘ a$. In some sense it tells us that $g$ is longer than $f$, or at least as long as $f$. This can be formalized like so:

1. Take the category of arrows and commutative squares over an ambient category $\mathcal C$.
2. Given two arrows $g$ and $f$ in $\mathcal C$, squish all commutative squares of form $g ∘ \operatorname {id} = b ∘ f$ and of form $\operatorname {id} ∘ g = f ∘ a$ into the symbol $g ≤ f$.
3. Take the transitive closure of the relation defined by $≤$, for good measure. It is automatically reflexive because $f ∘ \operatorname {id} = \operatorname {id} ∘ f$.

Equivalently, $g ≤ f ≔ ∃ab. g = b ∘ f ∘ a$.

Thus we attain a preorder on arrows of $\mathcal C$. This preorder has the equivalence fragment $≡$ where $g ≡ f ≔ g ≤ f ∧ f ≤ g$. This is indeed an equivalence. It is symmetric by symmetry of $∧$, reflexive because $≤$ is, and it is transitive like so:

$h ≡ g ∧ g ≡ f \\ h ≤ g ∧ g ≤ h ∧ g ≤ f ∧ f ≤ g \\ h ≤ f ∧ f ≤ h \\ h ≡ f$

For a concrete category, this notion is weaker than set theoretic pointwise equality of functions, and yet it captures what we care about. Does this equivalence have a name? And is the relation $≤$ good for anything?

Ignat Insarov (Apr 16 2025 at 20:41):

And also the arrows of the category $\mathcal C ^→$ — the commutative squares — give us the notion of one arrow being «before» the other. The equivalence we get from this partial order coincides with the equivalence defined above.

John Baez (Apr 16 2025 at 20:42):

This is interesting because people much more often study the obvious preorder on arrows to a fixed object.

Morgan Rogers (he/him) (Apr 17 2025 at 08:19):

A concise-ish way to describe this with standard constructions is as the (dual of the) poset reflection of the [[twisted arrow category]]

Ignat Insarov (Apr 17 2025 at 12:49):

«Twisted arrow category» is exactly what I needed to be in my vocabulary!

Matteo Capucci (he/him) (Apr 17 2025 at 13:35):

probably one of my least fav terminologies. it's just:tm: the graph of the hom profunctor!

Ignat Insarov (Apr 17 2025 at 14:08):

What is a graph of a profunctor?

Matteo Capucci (he/him) (Apr 17 2025 at 14:38):

every profunctor $P:\cal A \to B$ corresponds to a [[two-sided fibration]] ${\cal A} \leftarrow \int P \to \cal B$ where $\int P$ has objects triples $(a,b,p \in P(a,b))$ and as maps $(a,b,p) \to (a',b',p')$ pairs $(f:a \to a',g:b' \to b)$ such that $P(f,g)(p') = p$.

Mike Shulman (Apr 17 2025 at 16:11):

I think the twisted arrow category is important enough to deserve its own name in addition to "the graph of the hom profunctor".

Evan Patterson (Apr 17 2025 at 18:35):

Another reason is that there are four different graphs of a profunctor, depending on how you handle the variances!

Nathanael Arkor (Apr 17 2025 at 19:40):

I think this is a slightly misleading perspective. Given a chosen definition of distributor, there is a single notion of graph (which corresponds to the tabulator in the double category of distributors). You can then rearrange it into three other distributors and take their graphs.

Evan Patterson (Apr 17 2025 at 21:19):

Right, that's what made me want to comment here! For me, the graph of the hom-profunctor is the (ordinary) arrow category, not the twisted arrow category, because it's the former that is the tabulator of hom-profunctor viewed as a proarrow in the double category of profunctors.

Evan Patterson (Apr 17 2025 at 21:25):

I got bit by this once because I read somewhere on the nLab that the category of elements of the hom-profunctor is the twisted arrow category and only later realized that the tabulator is something else.

Mike Shulman (Apr 17 2025 at 23:03):

That's right. And I think the graph that @Matteo Capucci (he/him) described is the one that's the tabulator, not the one that's the twisted arrow category.

Matteo Capucci (he/him) (Apr 18 2025 at 05:25):

Mike Shulman said:

I think the twisted arrow category is important enough to deserve its own name in addition to "the graph of the hom profunctor".

I agree, I just dislike the name twisted arrow :man_shrugging:

Matteo Capucci (he/him) (Apr 18 2025 at 05:27):

Mike Shulman said:

That's right. And I think the graph that Matteo Capucci (he/him) described is the one that's the tabulator, not the one that's the twisted arrow category.

:thinking: isn't $C \leftarrow Tw(C) \to C$ the result of applying the construction above to the hom profunctor? or am I missing an op?

John Baez (Apr 18 2025 at 05:32):

I like the name twisted arrow category, both because it's descriptive and because it reminds of the Buffalo Springfield song "Broken Arrow".

Mike Shulman (Apr 18 2025 at 06:01):

I think you're missing an op. The (domain, codomain) projection from the twisted arrow category lands in $C \times C^{\rm op}$, precisely because it's twisted. The non-twisted case is when the arrows go the same direction for both domain and codomain.

Oscar Cunningham (Apr 18 2025 at 18:25):

(deleted)

Oscar Cunningham (Apr 18 2025 at 18:26):

Matteo Capucci (he/him) said:

Mike Shulman said:

I think the twisted arrow category is important enough to deserve its own name in addition to "the graph of the hom profunctor".

I agree, I just dislike the name twisted arrow :man_shrugging:

I assumed it was the twisted (arrow category), not the (twisted arrow) category!