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Stream: learning: questions

Topic: ✔ Subobject classifier question

Mike Stay (Jan 27 2023 at 22:45):

It seems like there's not a unique "character" morphism $\chi$ that turns this diagram into a pullback square in Set, since $\chi$ could just as well pick out $\bot$ as $\top$:
$\begin{array}{ccc}0&\xrightarrow{!}&1\\ !\downarrow & & ⇣\chi \\ 1 & \xrightarrow[\top]{} & 2 = \{\top, \bot\}\end{array}$

What am I missing?

P.S. What's the latex command for the "subobject" arrow that looks like ⧽—→?

John Baez (Jan 27 2023 at 22:50):

If $\chi$ maps the one element of $1$ to $\top$, this square would not be a pullback square.

John Baez (Jan 27 2023 at 22:51):

The pullback of functions $f: X \to A$ and $g: Y \to A$ is the set of pairs $(x,y) \in X \times Y$ such that $f(x) = g(y)$.

John Baez (Jan 27 2023 at 22:53):

I use \hookrightarrow for monics, giving $\hookrightarrow$. So I'm tempted to answer the way people often do on StackExchange when people ask "how do you do X?"

Namely, "Don't do X! Do Y!" :laughing:

John Baez (Jan 27 2023 at 22:55):

But someone here will know.

David Egolf (Jan 27 2023 at 23:13):

Looking at this, I think it might be \rightarrowtail which gives: $\rightarrowtail$

Mike Stay (Jan 27 2023 at 23:18):

If $\chi$ maps the one element of $1$ to $\top$, this square would not be a pullback square.

Thanks, I knew it was something simple!

I presume that something similar is true when working in a topos with more truth values (e.g. multisets) and the mono on the top is $\bot\colon 1 \rightarrowtail \Omega$ so that $\chi_\bot\colon \Omega \to \Omega$ is negation. At first glance it looked to me like the third truth value could go either to top or to bottom, but I guess one of those won't make 1 the pullback.

\rightarrowtail

Thanks!

Notification Bot (Jan 28 2023 at 17:04):

Mike Stay has marked this topic as resolved.

Matteo Capucci (he/him) (Jan 29 2023 at 09:52):

Bear in mind people usually call $\chi$ the morphism at the bottom of the pullback square, whereas the morphism on the right side is the subobject classifier itself