Nuprl Definition : groupoid
Groupoid ==
C:SmallCategory × {inv:x:cat-ob(C) ⟶ y:cat-ob(C) ⟶ (cat-arrow(C) x y) ⟶ (cat-arrow(C) y x)|
∀x,y:cat-ob(C). ∀f:cat-arrow(C) x y.
(((cat-comp(C) x y x f (inv x y f)) = (cat-id(C) x) ∈ (cat-arrow(C) x x))
∧ ((cat-comp(C) y x y (inv x y f) f) = (cat-id(C) y) ∈ (cat-arrow(C) y y)))}
Definitions occuring in Statement :
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
equal: s = t ∈ T
Definitions occuring in definition :
product: x:A × B[x],
small-category: SmallCategory,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
cat-ob: cat-ob(C),
all: ∀x:A. B[x],
and: P ∧ Q,
equal: s = t ∈ T,
cat-arrow: cat-arrow(C),
cat-comp: cat-comp(C),
apply: f a,
cat-id: cat-id(C)
FDL editor aliases :
groupoid
Latex:
Groupoid ==
C:SmallCategory \mtimes{} \{inv:x:cat-ob(C) {}\mrightarrow{} y:cat-ob(C) {}\mrightarrow{} (cat-arrow(C) x y) {}\mrightarrow{} (cat-arrow(C) y x)|
\mforall{}x,y:cat-ob(C). \mforall{}f:cat-arrow(C) x y.
(((cat-comp(C) x y x f (inv x y f)) = (cat-id(C) x))
\mwedge{} ((cat-comp(C) y x y (inv x y f) f) = (cat-id(C) y)))\}
Date html generated:
2020_05_20-AM-07_55_02
Last ObjectModification:
2015_09_23-AM-09_29_18
Theory : small!categories
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