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Stream: practice: terminology & notation

Topic: mixed composite?

Eric M Downes (Jun 28 2024 at 00:49):

You have functors $F:{\sf C \to D}, G:{\sf D\to C}$, and $G\circ F$ fixes an object $B\in{\sf C}_0$. You also have a path $(A\overset{a}{\to}B\overset{b}{\to}A) \in {\sf C}_2$. What would you call a “mixed” path like the following?

$b\circ_{\sf C}G(\ldots\circ_{\sf D}\ldots)F\circ_{\sf C}a:A\leadsto A$

That is, a composite through multiple categories whose diagram may not commute, that is may not correspond to any endomorphism of $A$ in $\sf C_1$.

it is awkward, but this captures an important computation, and if I am to build a category whose paths are such morphisms, it would be convenient to have a term for them while doing so.

Eric M Downes (Jun 28 2024 at 02:10):

In case that's not helpful, here's a simpler question; if I call the center part "the image of $B$ under $G(\ldots\circ_{\sf D}\ldots)F$" and assign that image a name, what notation would easily communicate to you "this is not necessarily a morphism in $\sf C$"?

Kevin Carlson (Jun 28 2024 at 02:13):

I don't understand what the center part even means, I don't think? I can understand $G(\ldots\circ_D\ldots),$ it's $G$ of some morphism in $\mathsf D,$ but I don't understand how to parse $F\circ_C a$ or $G(\ldots\circ_D\ldots)F$ separately, and it doesn't look any clearer what they mean together.

Eric M Downes (Jun 28 2024 at 02:29):

Hmmmm... does this help? Everything on the bottom is in $\sf C$, we leave $\sf C$ and then we return to the same object we left, but in a way that may not correspond to any morphism within $\sf C$.

I'm thinking of $G\ldots F$ as "the action of -thing- on $B$", I don't know if others will understand that, though.

Rémy Tuyéras (Jun 28 2024 at 02:31):

@Eric M Downes Just for confirmation, do you mean that your path $A \leadsto A$ of the form:

$A \mathop{\to}\limits^{a} B \mathop{\to}\limits^{G(f_0)} G(X_1) \mathop{\to}\limits^{G(f_1)} \dots G(X_{n}) \mathop{\to}\limits^{G(f_n)} B \mathop{\to}\limits^{b} A$

such that $f_0$ is an arrow of the form $F(B) \to X_1$ in $\mathsf{D}$ and $f_n$ is an arrow of the form $X_{n} \to F(B)$ in $\mathsf{D}$?

Eric M Downes (Jun 28 2024 at 02:32):

Yes! Thanks for deciphering my ravings.

Rémy Tuyéras (Jun 28 2024 at 02:42):

If I had to formalize these paths, I would probably start looking at the category of $A$-elements over the functor $G$ (let us denote it as $A\downarrow G$) whose

• objects are pairs $(x:A \to G(X),X \in \mathsf{Obj}(\mathsf{C}))$
• and whose arrows $(x:A \to G(X),X) \to (y:A \to G(Y),Y)$ are given by arrows $f:X \to Y$ in $\mathsf{D}$ such that $G(f) \circ x = y$

Rémy Tuyéras (Jun 28 2024 at 02:47):

In fact, I would consider the category $A\downarrow G \downarrow A$ whose

• objects are triple $(x:A \to G(X), x':G(X) \to A,X \in \mathsf{Obj}(\mathsf{C}))$
• and whose arrows $(x:A \to G(X), x':G(X) \to A,X) \to (y:A \to G(Y), y':G(Y) \to A,Y)$ are given by arrows $f:X \to Y$ in $\mathsf{D}$ such that $G(f) \circ x = y$ and $y' \circ G(f)= x'$

Rémy Tuyéras (Jun 28 2024 at 02:49):

Then, your path is an endo-path on the object $(a,b,F(B))$ in the category $A \downarrow G \downarrow A$

Eric M Downes (Jun 28 2024 at 02:50):

Nice! That's very thoight-provoking... thank you!

Rémy Tuyéras (Jun 28 2024 at 03:24):

@Eric M Downes Actually, I went a bit too fast! With the information that you provided, your path is in fact a path of the form:

$(a,b',F(B)) \to \dots \to (a',b,F(B))$

in $A \downarrow G \downarrow A$. It is only an endopath if $G(f_n) \circ \dots \circ G(f_0) = \mathsf{id}_B$.

Eric M Downes (Jun 28 2024 at 04:36):

In the simplest possible case I can construct $\gamma:=G(f_n)\circ\ldots\circ G(f_0)$ satisfies $\gamma\circ\gamma=id_{B}$.

Matteo Capucci (he/him) (Jun 28 2024 at 07:26):

Another way to get such heteromorphisms is through profunctors, which make precise the idea that these paths don't compose with themselves but can be precomposed and postcomposed by arbitrary morphisms. This one should be the composite of Hom(-, F) and Hom(G, -).

Matteo Capucci (he/him) (Jun 28 2024 at 07:27):

(Kinda -- you seem to fix some objects & look at the 'endoheteromorphisms' but that's extra manipulations on top of the profunctor, ie fix A and A as arguments)