Nuprl Definition : cat-monad
Monad(C) ==
{M:T:Functor(C;C) × nat-trans(C;C;1;T) × nat-trans(C;C;functor-comp(T;T);T)|
let T,u,m = M in
(∀X:cat-ob(C)
((cat-comp(C) (T X) (T (T X)) (T X) (u (T X)) (m X)) = (cat-id(C) (T X)) ∈ (cat-arrow(C) (T X) (T X))))
∧ (∀X:cat-ob(C)
((cat-comp(C) (T X) (T (T X)) (T X) (T X (T X) (u X)) (m X)) = (cat-id(C) (T X)) ∈ (cat-arrow(C) (T X) (T X))))
∧ (∀X:cat-ob(C)
((cat-comp(C) (T (T (T X))) (T (T X)) (T X) (m (T X)) (m X))
= (cat-comp(C) (T (T (T X))) (T (T X)) (T X) (T (T (T X)) (T X) (m X)) (m X))
∈ (cat-arrow(C) (T (T (T X))) (T X))))}
Definitions occuring in Statement :
id_functor: 1,
functor-comp: functor-comp(F;G),
nat-trans: nat-trans(C;D;F;G),
functor-arrow: arrow(F),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
spreadn: spread3,
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
product: x:A × B[x],
equal: s = t ∈ T
Definitions occuring in definition :
functor-ob: ob(F),
apply: f a,
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
equal: s = t ∈ T,
cat-ob: cat-ob(C),
all: ∀x:A. B[x],
cat-id: cat-id(C),
functor-arrow: arrow(F),
and: P ∧ Q,
spreadn: spread3,
functor-comp: functor-comp(F;G),
nat-trans: nat-trans(C;D;F;G),
id_functor: 1,
product: x:A × B[x],
cat-functor: Functor(C1;C2),
set: {x:A| B[x]}
FDL editor aliases :
cat-monad
Latex:
Monad(C) ==
\{M:T:Functor(C;C) \mtimes{} nat-trans(C;C;1;T) \mtimes{} nat-trans(C;C;functor-comp(T;T);T)|
let T,u,m = M in
(\mforall{}X:cat-ob(C). ((cat-comp(C) (T X) (T (T X)) (T X) (u (T X)) (m X)) = (cat-id(C) (T X))))
\mwedge{} (\mforall{}X:cat-ob(C)
((cat-comp(C) (T X) (T (T X)) (T X) (T X (T X) (u X)) (m X)) = (cat-id(C) (T X))))
\mwedge{} (\mforall{}X:cat-ob(C)
((cat-comp(C) (T (T (T X))) (T (T X)) (T X) (m (T X)) (m X))
= (cat-comp(C) (T (T (T X))) (T (T X)) (T X) (T (T (T X)) (T X) (m X)) (m X))))\}
Date html generated:
2020_05_20-AM-07_58_24
Last ObjectModification:
2017_01_16-PM-07_38_20
Theory : small!categories
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