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A blog mostly on math, physics, and computer science.
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Derek Elkins
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2025-12-22 08:59:07
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Home Home About Contact Reading List Archive RSS Atom Hedonistic Learning Learning for the fun of it Preserving, Reflecting, and Creating Limits Introduction This is a brief article about the notions of preserving, reflecting, and creating limits and, by duality, colimits. Preservation is relatively intuitive, but the distinction between reflection and creation is subtle. Preservation of Limits A functor, |F|, preserves limits when it takes limiting cones to limiting cones. As often happens in category theory texts, the notation focuses on the objects. You’ll often see things like |F(X \times Y) \cong FX \times FY|, but implied is that one direction of this isomorphism is the canonical morphism |\langle F\pi_1, F\pi_2\rangle|. To put it yet another way, in this example we require |F(X \times Y)| to satisfy the universal property of a product with the projections |F\pi_1| and |F\pi_2|. Other than that subtlety, preservation is fairly intuitive. Reflecti...
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