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Stream: learning: questions

Topic: Pullback-stable coproducts: morphisms?

Joey Eremondi (Nov 28 2023 at 12:14):

One of Giraud's axioms is that a topos has coproducts stable under pullback, e.g. f(A+B) is isomorphic to fA + f*B. But does this extend to morphisms?

e.g. for f : D -> C, a : A->C and b : B -> C, can we relate (fa,fb) and f*(a,b), where (_,_) is the universal arrow from the coproduct?

What I'm really looking for is to relate f* id = f* (inj1, inj2) and (f* inj1, f* inj2), but I'm curious if the relationship holds generally.

Do any references spell this out explicitly? Everything I've seen just says f* colim = colim f* or "base change functor preserves coproducts" but I've never seen it written out explicitly, and while I'm building the intuition for the abstract descriptions it's just not quite there yet.

Damiano Mazza (Nov 28 2023 at 14:16):

I don't know of any reference spelling it out explicitly, but it is true that in any extensive category (in particular, a topos) $f^\ast[a,b]$ is the same as $f^\ast a+f^\ast b$, where I denote by $h+h'$ the arrow $[\iota_1 h,\iota_2 h']:X+X'\to Y+Y'$ for any $h:X\to Y$, $h':X'\to Y'$.

(It is not true that $f^\ast[a,b]$ is equal to $[f^\ast a,f^\ast b]$ as the latter expression is not even well-typed: $f^\ast a$ has codomain $A$, $f^\ast b$ has codomain $B$).

Damiano Mazza (Nov 28 2023 at 14:23):

The key point is that, in an extensive category, any arrow $h:E\to A+B$ is necessarily of the form $h_1+h_2$ with $h_1:E_1\to A$, $h_2:E_2\to B$ and $X=X_1+X_2$. You use this to show that the commuting square $[a,b](f^\ast a+f^\ast b)=f[a^\ast f,b^\ast f]$ is in fact a pullback.

Damiano Mazza (Nov 28 2023 at 14:23):

In the proof, you have two arrows $h:E\to A+B$ and $k:E\to D$ such that $[a,b]h=fk$, you decompose $h$ into $h_1+h_2$ as above, observe that $ah_1=fk\iota_1$ and $bh_2=fk\iota_2$, which gives you canonical arrows $u_1$ and $u_2$, and your canonical arrow for the first diagram is $u_1+u_2$.

Damiano Mazza (Nov 28 2023 at 14:31):

(By the way, when you want to write math expressions, you may write LaTeX code by using double-dollar signs (rather than a single dollar sign as one would normally do in LaTeX). Zulip uses Markdown in messages, in which "asterisk" (or "star") signs have a special meaning (they are used to do italic and bold), so your original post does not render as expected: some "stars" disappear and you see italic text starting/ending instead).

Mike Shulman (Nov 28 2023 at 15:41):

More generally, when we say that some functor (like pullback) preserves a limit or colimit (like coproduct), we mean not just that there is some isomorphism, but that the canonical comparison map defined using the universal properties is an isomorphism. The definition of that comparison map then ensures that it commutes with the action on morphisms.