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Stream: learning: reading & references

Topic: Naming conventions for universal arrows in 2-categories

Patrick Nicodemus (Apr 16 2025 at 22:26):

Let $G : D\to C$ be a functor between categories. Let $c$ be an object in $C$. The notion of a "universal arrow" $(d, \eta : c \to G d)$ from $c$ into $D$ is standard.

Is there a special term for the case where we require that $G(d)=c$ and $\eta$ is the identity homomorphism? Such a thing would be useful in the case that we view $D$ as fibered over $C$ by $G$, and are looking for $d$ "over" $c$.

If $C, D$ are 2-categories or bicategories, one can consider other notions of universal arrow from $c$ into $G$:

• $(d, \eta : c \to G d)$ such that for every $(d', f : c\to G d')$ there is a pair $(s : d\to d', \alpha : f\Rightarrow G(s)\circ \eta)$ which is initial among such pairs (we could call this a "lax universal arrow" as it seems to correspond to this https://ncatlab.org/nlab/show/lax+2-adjunction)
• $(d, \eta : c \to G d)$ such that for every $(d', f : c\to G d')$ there is an essentially unique pair $(s : d\to d', \alpha : f\Rightarrow G(s)\circ \eta)$ up to isomorphism (would seem to correspond to a biadjunction)
  What do we call these notions of universal arrow?

Nathanael Arkor (Apr 16 2025 at 22:45):

Is there a special term for the case where we require that $G(d)=c$ and $\eta$ is the identity homomorphism?

A universal arrow in this sense exhibits a reflective subcategory. If you ask for the unit to be an identity, you're expressing that $D$ and $C$ are equivalent. Perhaps I'm misinterpreting your question?

Edit: I misread.

Patrick Nicodemus (Apr 16 2025 at 22:53):

It would be an equivalence if both the unit and counit were isomorphisms, but I only asked that one be an isomorphism. We can consider the case of the counit if you prefer to think about reflective subcategories. Let's talk about reflective subcategories because that's closer to what I'm thinking about.

So you have the left adjoint $F: C \to D$, this is known, and you're trying to construct a right adjoint right inverse to $F$, this is equivalent to showing that for each $d$ in $D$ there is $c\in C$ such that the identity homomorphism $F(d)=c$ is a universal arrow from $F$ to $c$. Motivating example, $p : E\to B$ is a functor and we want to prove it's a Grothendieck fibration, which is equivalent to proving that the functor $E^I\to B^I\times_BE$ has a right adjoint right inverse.

Patrick Nicodemus (Apr 16 2025 at 22:58):

My question is, is there a better term for these identity universal arrows from $F$ to $d$? It's not really an arrow, it's an object $c$ in $C$ with the property that for all $c'$ in $C$ and all $g: F(c') \to d$, there is a unique $f : c'\to c$ with $F(f)=g$. I'd rather not mention composition with the identity because in my application I want to talk about bicategories as well, and then $F(f)\circ 1 \neq F(f)$ in general and I have to account for the right unitor everywhere.