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limit_._category_theory [2014/09/30 17:01] nikolaj |
limit_._category_theory [2015/03/16 18:13] nikolaj |
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| | @#FFBB00: definiendum | @#FFBB00: $\mathrm{lim}\,F$ ... terminal morphism from $\Delta$ to $F$ | | | @#FFBB00: definiendum | @#FFBB00: $\mathrm{lim}\,F$ ... terminal morphism from $\Delta$ to $F$ | | ||
| - | ==== Discussion ==== | + | ----- |
| === Elaboration === | === Elaboration === | ||
| A limit of a functor with image in ${\bf C}$, if it exists, is a particular terminal morphism in the functor category ${\bf C}^{\bf D}$. That definition emphasizes the underlying universal property, but it's a little hard to understand what's really going on. The reason is that a morphisms in a functor category is a natural transformation and this means there are a couple algebraic identities which don't get explicitly mentioned in the definition above. If you don't know limits, skip the next section and read the ones using cones first. | A limit of a functor with image in ${\bf C}$, if it exists, is a particular terminal morphism in the functor category ${\bf C}^{\bf D}$. That definition emphasizes the underlying universal property, but it's a little hard to understand what's really going on. The reason is that a morphisms in a functor category is a natural transformation and this means there are a couple algebraic identities which don't get explicitly mentioned in the definition above. If you don't know limits, skip the next section and read the ones using cones first. | ||
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| In the more abstract definition using the functor category, these several morphisms are encoded into a natural transformation (=a single morphism, but in another category). The trick is that a natural transformation is an arrow in a functor category and this way one can specify the limit as terminal morphism. | In the more abstract definition using the functor category, these several morphisms are encoded into a natural transformation (=a single morphism, but in another category). The trick is that a natural transformation is an arrow in a functor category and this way one can specify the limit as terminal morphism. | ||
| + | |||
| + | === Discussion === | ||
| + | The object of the limit is like the full solution to a problem posed by the concept captured by the functor. | ||
| === Examples === | === Examples === | ||
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| Wikipedia: [[http://en.wikipedia.org/wiki/Limit_%28category_theory%29|Limit (category theory)]] | Wikipedia: [[http://en.wikipedia.org/wiki/Limit_%28category_theory%29|Limit (category theory)]] | ||
| - | ==== Parents ==== | + | ----- |
| === Context === | === Context === | ||
| [[Functor category]], [[Diagonal functor]] | [[Functor category]], [[Diagonal functor]] | ||
| === Subset of === | === Subset of === | ||
| [[Terminal morphism]] | [[Terminal morphism]] | ||
