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functor [2016/04/09 14:21] nikolaj |
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| === Discussion === | === Discussion === | ||
| 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 $f$). | 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 $f$). | ||
| - | Let write $C$ as $\{1_a,1_b,1_c,\dots\}$, which is just a formal relabeling. | + | Let's write $C$ as $\{1_a,1_b,1_c,\dots\}$, which is just a formal relabeling. |
| A category ${\bf C}$ is richer than a set $C$: | A category ${\bf C}$ is richer than a set $C$: | ||
| - | 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 ${\bf C}$. | + | |
| - | 2. 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 $a=b$.) | + | 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 generally only partially defined, where the domain and codomain function tell you which concatenations make sense (e.g. $1_a\circ 1_b$ only has a value if $a=b$). | ||
| A functor is a function that respects $\circ$ in the sense of a monoid-homomorphism. | A functor is a function that respects $\circ$ in the sense of a monoid-homomorphism. | ||
