Leave $ Spc_S = L_ {Nis} sPre (Sm_S) $ Y $ Spc_S ^ { mathbb {A} _1} = L _ { mathbb {A} _1} L_ {Nis} sPre (Sm_S) $, the location of two Bousfield's $ sPre (Sm_S) $ To obtain the category of motivated homotopy. $ Spc_S ^ { mathbb {A} _1} $.

Here is the definition of $ mathbb {A} _1 $-homotopia

A map $ f: F a G $ is a $ mathbb {A} _1 $-difference of equivalence if it is a $ mathbb {A} _1 $– local weak equivalence in $ Spc_S ^ { mathbb {A} _1} $, which means that for any fibrant object $ X $, there is a weak equivalence of simplicial sets.

$ map (F, X) a map (G, Y) $

where $ map (X, Y) $ is the mapping space in $ sPre (Sm_S) $.

I'm not sure how to show that $ mathbb {A} _1 $equivalence -homotopy is a $ mathbb {A} _1 $– Weak equivalence What is the relationship between $ F times mathbb {A} _1 $ Y $ F times Delta_1 $ where $ Delta_1 $ is the constant simplicial presemigo in $ Delta_1 $?