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over_category [2015/03/18 17:11] nikolaj |
over_category [2015/03/18 17:16] nikolaj |
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| | @#55CCEE: context | @#55CCEE: $T\in{\bf C}$ | | | @#55CCEE: context | @#55CCEE: $T\in{\bf C}$ | | ||
| | @#FFBB00: definiendum | @#FFBB00: ${\bf C}/T$ | | | @#FFBB00: definiendum | @#FFBB00: ${\bf C}/T$ | | ||
| - | | @#FF9944: definition | @#FF9944: $\mathrm{Ob}_{{\bf C}/T}:=$ all arrows $f$, such that there is an $S\in{\bf C}$, such that $f:{\bf C}[S,T]$ | | + | | @#FF9944: definition | @#FF9944: $\mathrm{Ob}_{{\bf C}/T}:=$ all arrows $f$, such that there is an object $S\in{\bf C}$, such that $f:{\bf C}[S,T]$ | |
| | @#FF9944: definition | @#FF9944: ${\bf C}/T[f,g]:=$ all arrows $h$, such that $h\circ g = f$ | | | @#FF9944: definition | @#FF9944: ${\bf C}/T[f,g]:=$ all arrows $h$, such that $h\circ g = f$ | | ||
| ----- | ----- | ||
| + | === Elaboration === | ||
| + | Given a (target) object $T$ in any category ${\bf C}$, the over category ${\bf C}/T$ has objects all arrows into $T$, and arrows all forwardings of their domains (there is one for each commutative triangle). | ||
| + | |||
| + | === Discussion === | ||
| + | |||
| + | |||
| === Reference === | === Reference === | ||
| nLab: [[http://ncatlab.org/nlab/show/overcategory|Overcategory]] | nLab: [[http://ncatlab.org/nlab/show/overcategory|Overcategory]] | ||
