Nuprl Definition : cat-functor
Functor(C1;C2) ==
{FM:F:cat-ob(C1) ⟶ cat-ob(C2)
× (x:cat-ob(C1) ⟶ y:cat-ob(C1) ⟶ (cat-arrow(C1) x y) ⟶ (cat-arrow(C2) (F x) (F y)))|
let F,M = FM
in (∀x:cat-ob(C1). ((M x x (cat-id(C1) x)) = (cat-id(C2) (F x)) ∈ (cat-arrow(C2) (F x) (F x))))
∧ (∀x,y,z:cat-ob(C1). ∀f:cat-arrow(C1) x y. ∀g:cat-arrow(C1) y z.
((M x z (cat-comp(C1) x y z f g))
= (cat-comp(C2) (F x) (F y) (F z) (M x y f) (M y z g))
∈ (cat-arrow(C2) (F x) (F z))))}
Definitions occuring in Statement :
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
spread: spread def,
product: x:A × B[x],
equal: s = t ∈ T
Definitions occuring in definition :
set: {x:A| B[x]} ,
product: x:A × B[x],
function: x:A ⟶ B[x],
spread: spread def,
and: P ∧ Q,
cat-id: cat-id(C),
cat-ob: cat-ob(C),
all: ∀x:A. B[x],
equal: s = t ∈ T,
cat-arrow: cat-arrow(C),
cat-comp: cat-comp(C),
apply: f a
FDL editor aliases :
cat-functor
Latex:
Functor(C1;C2) ==
\{FM:F:cat-ob(C1) {}\mrightarrow{} cat-ob(C2) \mtimes{} (x:cat-ob(C1)
{}\mrightarrow{} y:cat-ob(C1)
{}\mrightarrow{} (cat-arrow(C1) x y)
{}\mrightarrow{} (cat-arrow(C2) (F x) (F y)))|
let F,M = FM
in (\mforall{}x:cat-ob(C1). ((M x x (cat-id(C1) x)) = (cat-id(C2) (F x))))
\mwedge{} (\mforall{}x,y,z:cat-ob(C1). \mforall{}f:cat-arrow(C1) x y. \mforall{}g:cat-arrow(C1) y z.
((M x z (cat-comp(C1) x y z f g))
= (cat-comp(C2) (F x) (F y) (F z) (M x y f) (M y z g))))\}
Date html generated:
2020_05_20-AM-07_50_43
Last ObjectModification:
2015_09_23-AM-09_29_07
Theory : small!categories
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