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functor [2016/04/09 14:22] nikolaj |
functor [2016/04/09 14:31] nikolaj |
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| 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. | ||
