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Stream: learning: questions

Topic: ✔ Question about a proof from 'Non-Canonical Isomorphisms'

Nayan Rajesh (Jun 02 2025 at 13:33):

The square

image.png

need not always commute, even when $0$ is both initial and terminal. For example in the category of abelian groups where finite products and finite coproducts coincide, consider the natural isomorphism,

$G\times H \xrightarrow{\psi_{G,H}} G\times H$

defined by $\psi_{G,H}((g,h)) = (-g,-h)$. Then, the composite $G \xrightarrow{\iota} G\times 1 \xrightarrow{\psi_{G,1}} G\times 1$ sends $g$ to $(-g,*)$, while $G \xrightarrow{\langle 1,0\rangle} G\times 1$ sends $g$ to $(g,*)$

Morgan Rogers (he/him) (Jun 02 2025 at 13:36):

What are the constraints on $\psi$ in your diagram? The notation suggests it's a component of a natural transformation?

Morgan Rogers (he/him) (Jun 02 2025 at 13:36):

Oh I suppose it's a "non-natural transformation" given the title of the topic

Nayan Rajesh (Jun 02 2025 at 13:45):

$\psi$ is still natural; the non-canonicity of it refers to the fact that it is not required to be the canonical map

$X +Y \xrightarrow{\langle [1_X, 0_{Y,X}], [0_{X,Y}, 1_Y] \rangle} X\times Y$

that exists in a category with zero morphisms, $0_{X,Y}: X \to Y$

Morgan Rogers (he/him) (Jun 02 2025 at 14:12):

My apologies, my mind mistakenly filled in the middle word. Looking at the screenshot you took, their diagram should instead be expanded as follows:
image.png

In the upper row, $\iota_1$ and $\pi_1$ are isomorphisms since $0$ is initial and terminal and Lack "identifies" $Y$ with $Y+0$ along $\iota_1$ and similarly for $Y \times 0$. I agree that this is somewhat confusing, but the important thing is that we don't require $\psi_{Y,0}$ to be the identity as your diagram would suggest.

Nayan Rajesh (Jun 02 2025 at 14:44):

Thank you! This makes so much more sense. I had my arrows pointing in the other direction so that I could accommodate the pairing $Y \xrightarrow{\langle 1, 0\rangle} Y\times Z$, but your diagram tells me that I don't need to.

Notification Bot (Jun 02 2025 at 14:44):

Nayan Rajesh has marked this topic as resolved.