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Dimitri Zaganidis
Dimitri Zaganidis
  • 69
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Is there an analog of adjoint functor theorem for adjunctions inof two variables?

Let $L:\mathscr A \times \mathscr B \longrightarrow \mathscr C$ and $R_1:\mathscr B^{op} \times \mathscr C \longrightarrow \mathscr A$ be two functors such that there is a bijection

$$ \mathscr C(L(A,B),C) \cong \mathscr A( A, R_1(B,C)) $$ natural in $A,B,C$.

Is there any sufficient conditions to ensure existence of a functor $ R_2: \mathscr A^{op} \times \mathscr C \longrightarrow \mathscr B$ such that $(L,R_1,R_2)$ is an adjunction inof two variables?

This mean that we would have bijections

$$ \mathscr C(L(A,B),C) \cong \mathscr A( A, R_1(B,C)) \cong \mathscr B( B, R_2(A,C)) $$ natural in $A,B,C$.

Is there an analog of adjoint functor theorem for adjunctions in two variables?

Let $L:\mathscr A \times \mathscr B \longrightarrow \mathscr C$ and $R_1:\mathscr B^{op} \times \mathscr C \longrightarrow \mathscr A$ be two functors such that there is a bijection

$$ \mathscr C(L(A,B),C) \cong \mathscr A( A, R_1(B,C)) $$ natural in $A,B,C$.

Is there any sufficient conditions to ensure existence of a functor $ R_2: \mathscr A^{op} \times \mathscr C \longrightarrow \mathscr B$ such that $(L,R_1,R_2)$ is an adjunction in two variables?

This mean that we would have bijections

$$ \mathscr C(L(A,B),C) \cong \mathscr A( A, R_1(B,C)) \cong \mathscr B( B, R_2(A,C)) $$ natural in $A,B,C$.

Is there an analog of adjoint functor theorem for adjunctions of two variables?

Let $L:\mathscr A \times \mathscr B \longrightarrow \mathscr C$ and $R_1:\mathscr B^{op} \times \mathscr C \longrightarrow \mathscr A$ be two functors such that there is a bijection

$$ \mathscr C(L(A,B),C) \cong \mathscr A( A, R_1(B,C)) $$ natural in $A,B,C$.

Is there any sufficient conditions to ensure existence of a functor $ R_2: \mathscr A^{op} \times \mathscr C \longrightarrow \mathscr B$ such that $(L,R_1,R_2)$ is an adjunction of two variables?

This mean that we would have bijections

$$ \mathscr C(L(A,B),C) \cong \mathscr A( A, R_1(B,C)) \cong \mathscr B( B, R_2(A,C)) $$ natural in $A,B,C$.

Source Link
Dimitri Zaganidis
Dimitri Zaganidis
  • 69
  • 5

Is there an analog of adjoint functor theorem for adjunctions in two variables?

Let $L:\mathscr A \times \mathscr B \longrightarrow \mathscr C$ and $R_1:\mathscr B^{op} \times \mathscr C \longrightarrow \mathscr A$ be two functors such that there is a bijection

$$ \mathscr C(L(A,B),C) \cong \mathscr A( A, R_1(B,C)) $$ natural in $A,B,C$.

Is there any sufficient conditions to ensure existence of a functor $ R_2: \mathscr A^{op} \times \mathscr C \longrightarrow \mathscr B$ such that $(L,R_1,R_2)$ is an adjunction in two variables?

This mean that we would have bijections

$$ \mathscr C(L(A,B),C) \cong \mathscr A( A, R_1(B,C)) \cong \mathscr B( B, R_2(A,C)) $$ natural in $A,B,C$.