Categorical Eilenberg–Watts equivalences for finite linear categories

Use the supplied finite-dimensional unital $k$-algebras $R,S$ and $k$-linear equivalences of [F4] $\mathcal{A}\simeq R\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$, $\mathcal{B}\simeq S\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$, under which finite $(\mathcal{B},\mathcal{A})$-bimodules correspond to finite $(S,R)$-bimodules; the two classification theorems [F1] and [F2] are stated for such algebras and are equivalences of categories including all natural transformations, hence after transport they identify finite $(\mathcal{B},\mathcal{A})$-bimodules with ${\mathrm{Rex}}_k(\mathcal{A},\mathcal{B})$ and with ${\mathrm{Lex}}_k(\mathcal{A},\mathcal{B})$ (Equivalence, quasi-inverse, and adjoint equivalence of categories, Natural transformation and its components, The opposite ring $R^{\mathrm{o}\mathrm{p}}$). Composing with the equivalence ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}⊠\mathcal{B}\simeq (\mathcal{B},\mathcal{A})\text{-}\mathrm{b}\mathrm{i}\mathrm{m}\mathrm{o}\mathrm{d}$ of [F3] exhibits ${\mathrm{\Phi }}^r$ and ${\mathrm{\Phi }}^l$ as equivalences onto ${\mathrm{Rex}}_k(\mathcal{A},\mathcal{B})$ and ${\mathrm{Lex}}_k(\mathcal{A},\mathcal{B})$, with quasi-inverses the transported inverse constructions of [F1] and [F2].

Work in the supplied small module models, with the scalar and size conventions of [F1] and [F2]. For a $k$-linear functor $G:R\text{-}\mathrm{m}\mathrm{o}\mathrm{d}\to S\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ put $H_G(a,b)=G(b){\otimes }_k{a}^{\ast }$, an $(S,R)$-bimodule with right action $(v\otimes \lambda )r=v\otimes (\lambda \cdot r)$. This models $\bar{a}⊠G(b)$ by [F3]. Give $G(R)$ the right action $vr=G(r_r)(v)$, where $r_r(t)=tr$, and give $G(R^{\ast })$ the right action $vr=G(t_r)(v)$, where $t_r(\lambda )=\lambda \cdot r$. Here $(r\lambda )(t)=\lambda (tr)$ and $(\lambda \cdot r)(t)=\lambda (rt)$ on $R^{\ast }$. These maps are left $R$-linear, and functoriality and $k$-linearity give commuting, scalar-compatible bimodule actions. The finite-dual-basis map $v\otimes \lambda \mapsto [x\mapsto \lambda (x)v]$ identifies $H_G(a,a)$ with ${\mathrm{Hom}}_k(a,G(a))$; it is independent of the basis, with right action $(pr)(x)=p(rx)$. We construct the universal families of [F7] in these finite bimodules.

Under the correspondence of step 1.1 the functor ${\mathrm{\Phi }}^r(M)=M{\otimes }_{\mathcal{A}}-$ is well defined, $k$-linear and right exact, and ${\mathrm{\Phi }}^l(M)={\mathrm{Hom}}_{\mathcal{A}}(M^{\ast },-)$ is well defined, $k$-linear and left exact, because these are the transported statements of F1 and [F2] for the model algebras (k-linear categories and k-linear functors, Left exact and right exact functors, $(S,R)$-bimodules and commuting left and right scalar actions).

For $x\in a$ let ${\mathrm{\ell }}_x:R\to a$ be ${\mathrm{\ell }}_x(t)=tx$. Define $i_a:G(R)\to H_G(a,a)$ by $i_a(v)(x)=G({\mathrm{\ell }}_x)(v)$ in the preceding Hom identification. It is $S$-linear and right $R$-linear because ${\mathrm{\ell }}_{rx}={\mathrm{\ell }}_x{r}_r$. For an $R$-linear $u:a\to b$, $u{\mathrm{\ell }}_x={\mathrm{\ell }}_{u(x)}$, so $G(u)i_a(v)(x)=i_b(v)(u(x))$; this is dinaturality. For any dinatural bimodule maps $j_a:Z\to H_G(a,a)$, set $h(z)=j_R(z)(1)$. Dinaturality at ${\mathrm{\ell }}_x$ gives $j_a(z)(x)=G({\mathrm{\ell }}_x)(h(z))$, so $j_a=i_{a}h$. The map $h$ is $S$-linear, and it is right $R$-linear since $h(zr)=j_R(zr)(1)=j_R(z)(r)=G(r_r)(h(z))$. Evaluation at $1$ gives $i_R(v)(1)=v$, proving uniqueness of $h$. Thus $G(R)$ with $i_a$ is ${\int }_{a}G(a){\otimes }_k{a}^{\ast }$ in the finite bimodule category.

For $\lambda \in a^{\ast }$ define the left $R$-linear map $t_{\lambda }:a\to R^{\ast }$ by $t_{\lambda }(x)(r)=\lambda (rx)$, and put $q_a(v\otimes \lambda )=G(t_{\lambda })(v)$. Bilinearity gives a map $H_G(a,a)\to G(R^{\ast })$; it is $S$-linear and right $R$-linear because $t_{\lambda \cdot r}=t_r{t}_{\lambda }$. For $u:a\to b$ and $\mu \in b^{\ast }$, $t_{\mu }u=t_{\mu u}$, so $q_b(G(u)v\otimes \mu )=q_a(v\otimes \mu u)$, proving dinaturality. Given any dinatural bimodule maps $c_a:H_G(a,a)\to Z$, let $\epsilon \in (R^{\ast }{)}^{\ast }$ be evaluation at $1$, and define $h(v)=c_{R^{\ast }}(v\otimes \epsilon )$. Since $\epsilon t_{\lambda }=\lambda$, dinaturality at $t_{\lambda }$ yields $h{q}_a=c_a$. This $h$ is $S$-linear; moreover $\epsilon t_r=\epsilon \cdot r$ (both evaluate $\lambda$ at $r$), so dinaturality at $t_r$ and right $R$-linearity of $c_{R^{\ast }}$ give $h(vr)=c_{R^{\ast }}(G(t_r)v\otimes \epsilon )=c_{R^{\ast }}(v\otimes \epsilon \cdot r)=h(v)r$. Finally $t_{\epsilon }:R^{\ast }\to R^{\ast }$ is the identity, hence $q_{R^{\ast }}(v\otimes \epsilon )=v$, proving uniqueness of $h$. Thus $G(R^{\ast })$ with $q_a$ is ${\int }^{a}G(a){\otimes }_k{a}^{\ast }$ in the finite bimodule category.

In the module models, $M=b{\otimes }_k{a}^{\ast }$ has dual $M^{\ast }\cong b^{\ast }{\otimes }_{k}a$, by the evaluation pairing and finite dual bases. Put $Q=a^{\ast }{\otimes }_{R}X$. For $f\in {\mathrm{Hom}}_R(X,a)$ define $J(f)\in Q^{\ast }$ by $J(f)(\lambda \otimes x)=\lambda (f(x))$. Balancedness follows from $R$-linearity of $f$. Conversely, for $g\in Q^{\ast }$ set $f_g(x)={\mathrm{ev}}_a^{-1}(\lambda \mapsto g(\lambda \otimes x))$. Then $f_g(rx)=r{f}_g(x)$ because $g(\lambda \otimes rx)=g((\lambda \cdot r)\otimes x)$ and $(\lambda \cdot r)(y)=\lambda (ry)$. The two constructions are inverse, so $J:{\mathrm{Hom}}_R(X,a)\to Q^{\ast }$ is a natural isomorphism. Dualizing and using $Q\cong Q^{\ast \ast }$ gives $Q\cong {\mathrm{Hom}}_R(X,a{)}^{\ast }$. The tensor universal property gives $(b{\otimes }_k{a}^{\ast }){\otimes }_{R}X\cong b{\otimes }_{k}Q$. Likewise currying gives ${\mathrm{Hom}}_R(b^{\ast }{\otimes }_{k}a,X)\cong {\mathrm{Hom}}_k(b^{\ast },{\mathrm{Hom}}_R(a,X))\cong {\mathrm{Hom}}_R(a,X){\otimes }_{k}b$; the last map and its inverse are finite-dual-basis evaluation maps. Thus the two external formulas hold naturally in $a,b,X$, with the $S$-action on the factor $b$ corresponding on Hom to precomposition by the right $S$-action of $b^{\ast }$.

Transporting these universal families through [F3] and the supplied module equivalences gives ${\mathrm{\Psi }}^l(F)={\int }^{a\in \mathcal{A}}\bar{a}⊠F(a)$ for left exact $F$ and ${\mathrm{\Psi }}^r(G)={\int }_{a\in \mathcal{A}}\bar{a}⊠G(a)$ for right exact $G$. Their model kernels are respectively $F(R^{\ast })$ and $G(R)$ by steps 2.2–2.3. For a natural transformation $\eta$, naturality at ${\mathrm{\ell }}_x$ and $t_{\lambda }$ identifies the induced maps of these universal objects with ${\eta }_R$ and ${\eta }_{R^{\ast }}$, respectively. Hence these are precisely the quasi-inverse functors of [F1] and [F2], including their natural comparison isomorphisms in both composites. Together with steps 1.1 and 3.1 this proves the claimed triangle and external formulas. No commutativity or algebraic closure is needed. The Deligne-product data carry the stated Axiom of Choice assumption; the displayed maps are canonical and use no additional choice or infinite-dimensional (co)limits.