Finn Lawler product-absolute pullback (changes)

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In category with finite products, the following squares must be pullbacks, for any morphisms f,gf,g:

X → ⟨ (1,f ⟩ ) X×Y f↓ (A) ↓f×Y Y → Δ d Y×YX →1 X 1↓ (B) ↓ Δ d X → Δ d X×XX×X′ →1×g X×Y′ f×1↓ (C) ↓f×1 Y×X′ →1×g Y×Y′ \array{ X & \xrightarrow{\langle \xrightarrow{(1, 1, f)} f \rangle} & X \times Y \\ \mathllap{f} \downarrow & (A) & \downarrow \mathrlap{f \times Y} \\ Y & \xrightarrow{\Delta} \xrightarrow{d} & Y \times Y } \qquad \qquad \qquad \array{ X & \xrightarrow{1} & X \\ \mathllap{1} \downarrow & (B) & \downarrow \mathrlap{\Delta} \mathrlap{d} \\ X & \xrightarrow{\Delta} \xrightarrow{d} & X \times X } \qquad \qquad \qquad \array{ X \times X' & \xrightarrow{1 \times g} & X \times Y' \\ \mathllap{f \times 1} \downarrow & (C) & \downarrow \mathrlap{f \times 1} \\ Y \times X' & \xrightarrow{1 \times g} & Y \times Y' }

as must the naturality square for the symmetry σ:X×Y≅Y×X\sigma \colon X \times Y \cong Y \times X , the product on one side or the other with an identity morphism of a pullback square, square and (which the pasting of two pullback squares side by side. These pullbacks must be preserved by any product-preserving functor, so we will call themtype D), and the pasting of two pullback squares side by side. These pullbacks must be preserved by any product-preserving functor, so we call them product-absolute.

This fact was already noted for squares of types (A) A and (C) C byLawvere; the others are given by Seely. See also Todd Trimble’s exposition, noting in particular that the squares expressing coassociativity of diagonal maps d:X→X×Xd \colon X \to X \times X are product-absolute pullbacks, which we call type E.

Last revised on December 8, 2015 at 17:41:17. See the history of this page for a list of all contributions to it.

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