{"id":95710,"date":"2022-01-27T10:51:11","date_gmt":"2022-01-27T16:51:11","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?page_id=95710"},"modified":"2025-06-09T11:47:17","modified_gmt":"2025-06-09T16:47:17","slug":"pullbacks","status":"publish","type":"page","link":"https:\/\/www.johndcook.com\/blog\/pullbacks\/","title":{"rendered":"Pullbacks"},"content":{"rendered":"

A pullback is a limit over a diagram of the following shape.<\/p>\n

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The pullback is a sort of product of A<\/em> and B<\/em> that depends on C<\/em>, and so it is sometimes written as a product with a subscript C on the product symbol: A<\/em> \u00d7C<\/em><\/sub> B<\/em>.<\/p>\n

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In all these diagrams, the givens will be in black and the pullback will be in royal blue.<\/p>\n

The limit of a diagram should have morphisms to all the objects in the diagram, so there should be a morphism from the pullback to C<\/em>. But that morphism is redundant: if the square commutes, then the morphism on the diagonal is determined by the morphisms on the side.<\/p>\n

In the previous diagram, f<\/em>‘ is called the pullback of f<\/em> along g<\/em>, and g<\/em>‘ is called the pullback of g<\/em> along f<\/em>.<\/p>\n

Products, intersections, inverse images, kernels, and equalizers are all pullbacks. We’ll go into each below.<\/p>\n

The product of two objects is the pullback of the diagram mapping both objects to a terminal object 1. So pullbacks are not just analogous to products, they are a generalization of products.<\/p>\n

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The intersection of two sets U<\/em> and V<\/em> is the pullback of the diagram of U<\/em> and V<\/em> inserting into their union. So you could also think of a pullback as a generalization of intersection.<\/p>\n

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Hooks on arrows indicate inclusion maps.<\/p>\n

Let f<\/em> be a function from a set A<\/em> to a set B<\/em>, and let C<\/em> be a subset of B<\/em>. Then the inverse image of C<\/em> is a pullback. The function f<\/em>* is the restriction of f<\/em> to the inverse image of C<\/em>.<\/p>\n

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Let f<\/em> be a monoid homomorphism f<\/em>: M<\/em> \u2192 N<\/em> and let O<\/em> be the one-element monoid. Then K<\/em>, the kernel of f<\/em>, is a pullback.<\/p>\n

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The one-element monoid O<\/em> is both initial and terminal, so there’s a unique map from K<\/em> to O<\/em> and from O<\/em> into N<\/em>.<\/p>\n

Finally, equalizers are pullbacks. Given two morphisms from A<\/em> into B<\/em>, their pullback is an equalizer.<\/p>\n

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More on category theory<\/h2>\n