# Notation for base change over multiple bases

In a paper that I'm writing, the following comes up:

Construction. Let $$\mathscr C$$ be a finitely complete category, and let $$A_1 \to B_1, \ldots, A_n \to B_n$$ be morphisms in $$\mathscr C$$. Let $$B$$ be an object with maps $$\pi_i \colon B \to B_i$$ for all $$i \in \{1,\ldots, n\}$$. Then construct the base change $$A = \bigg(\ldots\bigg(\bigg( B \underset{B_1}\times A_1\bigg) \underset{B_2}\times A_2 \bigg) \ldots \bigg) \underset{B_n}\times A_n$$ of $$B$$ along all morphisms $$A_1 \to B_1, \ldots, A_n \to B_n$$.

Picture. In the case $$n = 2$$, the picture looks as follows:

Neither square is a pullback, but $$A$$ is the limit of the rest of the diagram.

Question. Is there any alternative notation for the limit $$A$$ defined above?

## 1 Answer

$$A$$ can be identified with the pullback $$B\times_{\prod_i B_i}\prod_i A_i.$$ (Which I think is nicer notation)

Proof:

Let $$h : B\to \prod_i B_i$$ be the canonical map. Let $$k = \prod_i (A_i\to B_i)$$ be the product of the individual maps, with the individual maps being $$\alpha_i : A_i\to B_i$$.


Then $$hf=kg$$ if and only if $$\pi_ihf = \pi_ikg$$ for all $$i$$, but $$\pi_i kg$$ is equivalently the map $$X\xrightarrow{g_i} A_i \xrightarrow{\alpha_i} B_i.$$ Thus $$hf = kg$$ if and only if $$\pi_ihf = \alpha_i g_i$$ for all $$i$$. Conversely, any family of maps $$(g_i)_i$$ with $$\pi_ihf = \alpha_i g_i$$ for all $$i$$ induces a map $$g: X\to \prod A_i$$ such that $$hf=kg$$. Hence

$$\{(f,g) \mid f:X\to B, g:X\to \prod_i A_i, hf = kg\} \simeq$$ $$\{(f,(g_i)_i) \mid f:X\to B, g_i:X\to A_i, \pi_ihf = \alpha_i g_i\} \simeq \C(X,A)$$ with the last natural isomorphism being by definition of the limit.

Thus by the Yoneda lemma, the claim follows. $$\blacksquare$$

• Oh, of course! That's a big improvement. This is pretty close to how one constructs arbitrary finite limits from products and equalisers. – Remy Feb 4 '20 at 19:52
• @Remy Yes, the intuition was the same, or at least similar. We replace a cone of maps with a single map to the product. – jgon Feb 5 '20 at 5:05