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: learning: questions

Topic: a simple 2-categorical exercise

Ambroise (Oct 11 2023 at 16:53):

Hello, I am trying to improve my 2-categorical skills with the following exercise: prove that if a category B has an initial object b, then a natural transformation $\alpha$ between a constant functor from B and another one is uniquely determined by its component $\alpha_b$. This is easy to show directly, but is there a 2-categorical proof of that relying on some adequate notions of an object of a 2-category with an initial object and similar stuff?

Patrick Nicodemus (Oct 11 2023 at 19:30):

I don't know how standard this is, but I have recently been working with 2-categories a lot, and a useful definition for my purposes is
that if $\mathcal{A}$ is a given 2-category and $X$ is an object in $\mathcal{A}$, $X$ has an initial object if the category $\mathcal{A}(Y,X)$ has an initial object for every $Y$, and these initial objects are preserved under the functors $\mathcal{A}(Y,X)\to\mathcal{A}(Z,X)$ given by reindexing. In some cases it might be useful to use the stronger definition that there is a specific choice of terminal object in each category $\mathcal{A}(Y,X)$ and these specific choices are preserved under reindexing. Such a specific choice of section would correspondto the 2-presheaf $\mathcal{A}(-,X)$ having a "global section" from the terminal presheaf $\mathbf{1}:\mathcal{A}\to\mathbf{Cat}$ which sends everything to the terminal category.

Patrick Nicodemus (Oct 11 2023 at 19:33):

I'm not sure that clarifies the proof so much as it gives a way to extend your proof to other 2-categories by bootstrapping it.

Ambroise (Oct 12 2023 at 07:33):

Thanks! I am not sure how to complete the proof, knowing this property. How would you proceed?

Kengo Hirata (Oct 12 2023 at 15:09):

I have a 2-categorical proof with a 2-terminal object 1.
First, observe that a category with an initial object is an object $B$ in $\mathbf{Cat}$ with an adjunction $b\dashv ! : 1 \rightarrow B$.
Then, from the mate correspondence, a 2-cell $b \circ ! \Rightarrow f$ is bijective to $! \Rightarrow !\circ f$ which is unique by the definition of terminal object.

Patrick Nicodemus (Oct 12 2023 at 15:15):

What I meant is the following. I don't insist this is the unique or most natural way to adapt this to 2-categories, just what came to mind.

Let $Y,X$ be objects in our 2-category $\mathcal{A}$. Let $a, b : Y\to X$ and $\tau :a\Rightarrow b$. Assume $\mathcal{A}$ has a 2-terminal object $\mathbf{1}$ and that $a$ and $b$ both factor through the 2-categorical object by 1-cells $a', b' : \mathbf{1}\to X$.

Now let $Z$ be arbitrary and $j : Z\to Y$. We have an initial object $0 \in \mathcal{A}(Z,Y)$ and an initial map $e : 0\Rightarrow j$.
By some basic theorems about the naturality of whiskering, we have that $b\cdot e\circ \tau\cdot 0 : a\cdot 0\Rightarrow b\cdot 0\Rightarrow bj = \tau\cdot j\circ a\cdot e : a\cdot 0\Rightarrow a\cdot j\Rightarrow b\cdot j$.

Because $a,b$ factor through the terminal object, $a\cdot j =a\cdot 0$ and $a\cdot e$ is the identity 2-cell, and similarly $b\cdot 0\Rightarrow b\cdot j$ is the identity 2-cell. This forces $\tau\cdot 0 = \tau\cdot j$. Thus $\tau\cdot j$ is determined by $\tau\cdot 0$.
The intuition here is that $\tau\cdot j$ is the component of $\tau$ on the "generalized object" $j$. One can always take $j$ to be the identity 1-cell on $Y$, and then you get a particularly strong instance of this claim, that $\tau$ itself is completely determined by $\tau\cdot 0$.

Ambroise (Oct 12 2023 at 18:05):

Thank you both! I think I got the idea.