Spahn couniversal mapping property 1 (Rev #3, changes)

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Let

category: universal construction
F:X→CF:X\to C be a functor between categories. A CC morphism f:a→F(x)f:a\to F(x) is called to satisfy the couniversal mapping property with respect to FF, if f=0∈C/Ff=0\in C/F is initial in the comma category C/FC/F.

This means: for anyy∈Xy\in X and any morphism ϕ:a→G(y)\phi:a\to G(y) there is a morphism ψ:x→y\psi:x\to y such that

a ↦f F(x) ↘ ϕ ↓ G(ψ) G(y)\array{ a&\stackrel{f}{\mapsto}&F(x) \\ &\searrow^\phi&\downarrow^{G(\psi)} \\ &&G(y) }

Let F:X→CF:X\to C be a functor between categories. A CC morphism f:a→F(x)f:a\to F(x) is called to satisfy the couniversal mapping property with respect to FF, if f=0∈C/Ff=0\in C/F is initial in the comma category C/FC/F.

This means: for any y∈Xy\in X and any morphism ϕ:a→G(y)\phi:a\to G(y) there is a morphism ψ:x→y\psi:x\to y such that
a ↦f F(x) ↘ ϕ ↓ G(ψ) G(y)\array{ a&\stackrel{f}{\mapsto}&F(x) \\ &\searrow^\phi&\downarrow^{G(\psi)} \\ &&G(y) }

commutes.

For c∈Cc\in C the initial object f:c→F(x)f:c\to F(x) in c/Fc/F is called couniversal arrow from cc to RR.

Revision on July 15, 2012 at 13:32:48 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

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