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functor [2016/01/23 14:00] nikolaj |
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| === Discussion === | === Discussion === | ||
| - | * Given the functions $\mathrm{dom}$ and $\mathrm{codom}$ (at least in the meta-theory) a functor is essentially just a function in $\mathrm{Mor}_{\bf C}\to\mathrm{Mor}_{\bf D}$. This is because the object map on any $A$ is already determined by the arrow map: $FA=\mathrm{dom}\ F(1_A)$. | + | A function $f:C\to D$ maps a set of things $C=\{a,b,c,\dots\}$ into another set of things $D=\{f(a),f(b),f(c),\dots,\dots\}$ (remark: some of the listed elements in $D$ might be equal and $D$ might also be larger as the range of $f$). |
| - | * Functors map one category to another, while respecting aspects of the arrow-structure. | + | Let's write $C$ as $\{1_a,1_b,1_c,\dots\}$, which is just a formal relabeling. |
| - | * If the graph given by the objects and arrows in the domain category is thought of as "a concept", then the image of a functor is the realization of that concepts within the codomain category. | + | |
| + | A category ${\bf C}$ is richer than a set $C$: | ||
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| + | 1. There is not only a collection of special elements $1_a,1_b,1_c,\dots$, but also, for each ordered pair of those (such as $\langle 1_a,1_c\rangle$) there is a whole new collection of elements that's also in ${\bf C}$. | ||
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| + | 2. Each element knows of two other elements. (I.e. there is a "domain" and "codomain" function and these assignments should be obvious form the construction above.) | ||
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| + | 3. There is a "non-total monoid" $\circ$, with the special elements as it's units. (It's like a monoid, except it's only partially defined, e.g. $1_a\circ 1_b$ only has a value if $a=b$.) | ||
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| + | A functor is a function that respects $\circ$ in the sense of a monoid-homomorphism. | ||
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| + | (Point of view of universal constructions: If the graph given by the objects and arrows in the domain category is thought of as "a concept", then the image of a functor is the realization of that concepts within the codomain category.) | ||
| === Definitions === | === Definitions === | ||
