Deligne Products and Categorical Eilenberg–Watts

Fix the finite basis $(v_1,\dots ,v_n)$ of $V$ and let $A=Y^n$ be the supplied $n$-fold biproduct of $Y$ with injections $i_1,\dots ,i_n$; for $n=0$ this is the empty biproduct, the zero object (Biproduct, Additive category). For each object $Z$ the evaluations $f\mapsto (f(v_1),\dots ,f(v_n))$ and $g\mapsto (g\circ i_1,\dots ,g\circ i_n)$ are bijections ${\mathrm{Hom}}_k(V,\mathcal{C}(Y,Z))\to \mathcal{C}(Y,Z{)}^n$ and $\mathcal{C}(A,Z)\to \mathcal{C}(Y,Z{)}^n$ by [F1] and [F2], so their composite is a bijection ${\eta }_Z:\mathcal{C}(A,Z)⟶{\mathrm{Hom}}_k(V,\mathcal{C}(Y,Z))\mathrm{.}$ For $u:Z\to Z^{\mathrm{\prime }}$ both ${\eta }_{Z^{\mathrm{\prime }}}(u\circ g)$ and the componentwise composite $u\circ {\eta }_Z(g)$ have $j$-th entry $u\circ g\circ i_j$, so $\eta$ is natural in $Z$ (Natural transformation and its components, The hom-bifunctor of a preadditive category takes values in abelian groups); the universal element $u_A={\eta }_A(1_A)$ is the linear map with $u_A(v_j)=i_j$, so $(A,u_A)$ represents the functor $Z\mapsto {\mathrm{Hom}}_k(V,\mathcal{C}(Y,Z))$ (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding, Natural isomorphism).

Let $(v_1^{\mathrm{\prime }},\dots ,v_n^{\mathrm{\prime }})$ be a second finite basis and write $v_k^{\mathrm{\prime }}={\sum }_j{s}_{jk}{v}_j$ for the invertible scalar matrix $S=(s_{jk})$ (Linear map between vector spaces over the same field). The associated representation is $(A,u_A^{\mathrm{\prime }})$ with $u_A^{\mathrm{\prime }}(v_k^{\mathrm{\prime }})=i_k$; by linearity and [F2] there is a unique endomorphism $\varphi :A\to A$ with $\varphi \circ u_A=u_A^{\mathrm{\prime }}$, its matrix being determined by $S$, and the same construction with the two bases interchanged gives a two-sided inverse, so $\varphi$ is invertible. By [F4] applied to the two universal elements of the functor of step 1.1, $\varphi$ is the unique compatible isomorphism between the two representing objects; hence $V\odot Y$ is determined up to a unique compatible isomorphism and is independent of the chosen basis of $V$.

Let $h:Y\to Y^{\mathrm{\prime }}$ and let $(A^{\mathrm{\prime }},u^{\mathrm{\prime }})$ be the representation of $Z\mapsto {\mathrm{Hom}}_k(V,\mathcal{C}(Y^{\mathrm{\prime }},Z))$ produced by step 1.1, with natural bijections ${\eta }_Z^{\mathrm{\prime }}$. Precomposition with $h$ gives maps $h_{\ast }:\mathcal{C}(Y^{\mathrm{\prime }},Z)\to \mathcal{C}(Y,Z)$, $g\mapsto g\circ h$, componentwise linear, and the composite ${\theta }_Z:=({\eta }_Z{)}^{-1}\circ h_{\ast }\circ {\eta }_Z^{\mathrm{\prime }}$ is a natural transformation $\mathcal{C}(A^{\mathrm{\prime }},-)\Rightarrow \mathcal{C}(A,-)$ (Natural transformation and its components, Covariant functor, identity functor, composite functor, and contravariant functor). By [F3] applied to the objects $A^{\mathrm{\prime }}$ and $A$ there is a unique morphism ${\varphi }_h:A\to A^{\mathrm{\prime }}$ with ${\theta }_Z(f)=f\circ {\varphi }_h$ for every $f:A^{\mathrm{\prime }}\to Z$, and this is the structural morphism $V\odot Y\to V\odot Y^{\mathrm{\prime }}$ induced by the representing property.

Let $\lambda :V\to V^{\mathrm{\prime }}$ be a $k$-linear map. Precomposition gives ${\lambda }^{\ast }:{\mathrm{Hom}}_k(V^{\mathrm{\prime }},\mathcal{C}(Y,Z))\to {\mathrm{Hom}}_k(V,\mathcal{C}(Y,Z))$, $g\mapsto g\circ \lambda$, and the composite $({\eta }_Z^V{)}^{-1}\circ {\lambda }^{\ast }\circ {\eta }_Z^{V^{\mathrm{\prime }}}$ is a natural transformation $\mathcal{C}(A^{V^{\mathrm{\prime }}},-)\Rightarrow \mathcal{C}(A^V,-)$; by [F3] it is induced by a unique morphism $V\odot Y\to V^{\mathrm{\prime }}\odot Y$, the structural morphism in the coefficient variable (k-linear categories and k-linear functors, Linear map between vector spaces over the same field).

For natural transformations $\alpha :\mathcal{C}(A,-)\Rightarrow \mathcal{C}(B,-)$ and $\beta :\mathcal{C}(B,-)\Rightarrow \mathcal{C}(C,-)$, [F3] gives ${\alpha }_c(f)=f\circ E(\alpha )$ and ${\beta }_c(g)=g\circ E(\beta )$, hence $E(\beta \circ \alpha )=E(\alpha )\circ E(\beta )$; identities correspond to identities. Now take the family of representing objects and universal elements in the statement as supplied data. The transformations in steps 2.2 and 2.3 go opposite to the corresponding maps of pairs, and their composites act by $g\mapsto g\circ h\circ h^{\mathrm{\prime }}$ in the object variable and by $f\mapsto f\circ {\lambda }^{\mathrm{\prime }}\circ \lambda$ in the coefficient variable. These operations commute with one another, so their representing morphisms preserve identities and composition and give the asserted functor $(V,Y)\mapsto V\odot Y$. This proves functoriality of the supplied family, without selecting one globally from objectwise existence.

The isomorphism $\mathcal{C}(V\odot Y,Z)\cong {\mathrm{Hom}}_k(V,\mathcal{C}(Y,Z))$ is $k$-linear, since basis evaluation and composition with the biproduct injections are $k$-linear. It is therefore the enriched tensor isomorphism of [F5] over all $k$-vector spaces. A finite-dimensional enriching base is available only when all hom-spaces of $\mathcal{C}$ are finite-dimensional. The objectwise existence and basis comparison require only finite data; functoriality uses the supplied family as in step 3.1.

Put $\rho (a)=H(r_a,1_S)$ and $\sigma (c)=H(1_R,r_c)$, where $r_a(x)=xa$ and $r_c(y)=yc$. These are $k$-linear in $a,c$, unital, and satisfy $\rho (ab)=\rho (b)\rho (a)$ and $\sigma (c{c}^{\mathrm{\prime }})=\sigma (c^{\mathrm{\prime }})\sigma (c)$. The two families commute by functoriality on the product category, so the bilinear map $(a,c)\mapsto \sigma (c)\rho (a)$ descends through $R{\otimes }_{k}S$ to a unital anti-homomorphism $\tau :T\to \mathrm{End}(W)$. This is the right $T$-action on $W$. Naturality shows that ${\eta }_{R,S}$ commutes with $\tau (t)$ for every $t$.

On finite free left $T$-modules put $K(T^n)=W^n$. A left-linear map $\phi :T^m\to T^n$ is determined by $\phi (e_j)={\sum }_i{p}_{ij}{e}_i$, hence $\phi (x{)}_i={\sum }_j{x}_j{p}_{ij}$. Define $K(\phi )$ to have $(i,j)$-entry $\tau (p_{ij})$. If $\psi$ has entries $q_{\mathrm{\ell }i}$, then $\psi \phi$ has entries ${\sum }_i{p}_{ij}{q}_{\mathrm{\ell }i}$, and $\tau (p_{ij}{q}_{\mathrm{\ell }i})=\tau (q_{\mathrm{\ell }i})\tau (p_{ij})$, proving $K(\psi \phi )=K(\psi )K(\phi )$. Identities are preserved, so this is a $k$-linear functor on free modules.

For a presentation $P=(T^m\stackrel{\delta }{\to }{T}^n\stackrel{\pi }{\to }Z\to 0)$ put $K(P)=\mathrm{coker}K(\delta )$, with projection $q_P$. A map $u:Z\to Z^{\mathrm{\prime }}$ lifts to $g:T^n\to T^{n^{\mathrm{\prime }}}$ by lifting its finitely many generator images through ${\pi }^{\mathrm{\prime }}$. The map $g\delta$ lands in $\mathrm{im}{\delta }^{\mathrm{\prime }}$, so lifting the finitely many generator images again gives $g\delta ={\delta }^{\mathrm{\prime }}v$. Thus $q_{P^{\mathrm{\prime }}}K(g)K(\delta )=0$, and $K(g)$ descends to a map $K(P)\to K(P^{\mathrm{\prime }})$. Two lifts differ by ${\delta }^{\mathrm{\prime }}{v}^{\mathrm{\prime }}$ for the same reason, so their descended maps agree. Identity lifts and composites of lifts give identities and composition.

Applying step 3.1 to the identity of $Z$ gives canonical mutually inverse comparisons between $K(P)$ and $K(P^{\mathrm{\prime }})$ for any two presentations. On the small module source, choose one presentation per object and one cokernel per resulting map, and put $\bar{H}(Z)=K(P_Z)$. These simultaneous choices use The Axiom of Choice, not merely the finite choice used for each lift. For a class-sized definable target, collection first bounds a set of witnesses for this set-indexed family and AC selects them. The lift-independent maps of step 3.1 define a $k$-linear functor; other choices give a canonical natural isomorphism. Fix these presentation and cokernel data for the construction.

Choose presentations $R^s\stackrel{d}{\to }{R}^r\to X\to 0$ and $S^{s^{\mathrm{\prime }}}\stackrel{d^{\mathrm{\prime }}}{\to }{S}^{r^{\mathrm{\prime }}}\to Y\to 0$. Then $X{\otimes }_{k}Y$ has the presentation $T^{s{r}^{\mathrm{\prime }}}\oplus T^{r{s}^{\mathrm{\prime }}}\stackrel{(d\otimes 1,1\otimes d^{\mathrm{\prime }})}{\to }{T}^{r{r}^{\mathrm{\prime }}}⟶X{\otimes }_{k}Y⟶0.$ Indeed its last term is the quotient of $R^r{\otimes }_k{S}^{r^{\mathrm{\prime }}}$ by the two images: sending $(\bar{x},\bar{y})$ to the class of $x\otimes y$ is well defined and bilinear, and the tensor universal property gives an inverse to the induced quotient map. The identifications $R^a{\otimes }_k{S}^b\cong T^{ab}$ preserve the left $T$-actions.

Right exactness and additivity in each variable identify $H(R^a,S^b)$ with $W^{ab}$ and compute $H(X,Y)$ as the successive cokernel of the maps induced by $d^{\mathrm{\prime }}$ and $d$. Their entries are precisely the action endomorphisms in step 1.1. These successive cokernels are the cokernel of the pair of maps in step 5.1 after applying $K$: a map out of $W^{r{r}^{\mathrm{\prime }}}$ factors through either description exactly when it kills both maps. The universal property [F5] therefore gives $\bar{H}(X{\otimes }_{k}Y)\cong H(X,Y)$. Lifting maps between the presentations shows that both sides use the same matrices; step 3.1 removes dependence on the lifts. The comparison is consequently natural in both variables.

Naturality against biproduct injections and projections determines ${\eta }_{R^a,S^b}$ from ${\eta }_{R,S}$. Naturality against the presentation surjections, which $H,H^{\mathrm{\prime }}$ send to epimorphisms, then determines ${\eta }_{X,Y}$, proving injectivity. Conversely a morphism $f:W\to W^{\mathrm{\prime }}$ commuting with the right $T$-actions gives componentwise maps $W^{ab}\to W^{\mathrm{\prime }ab}$ commuting with all free-pair matrices. They descend through the successive cokernels of step 6.1. Lifts as in step 3.1 show these components are independent of presentations and natural in $X,Y$, and the component at $(R,S)$ is $f$. Thus evaluation is a bijection onto compatible morphisms of right $T$-module objects.

Steps 4.1 and 6.1 prove (i), and step 7.1 proves (ii). The construction uses AC for its set-indexed object data; all lift choices are finite and induce unique maps on cokernels. No commutativity of $R$ or $S$ is used.

By [F1] the tensor product $X{\otimes }_{k}Y$ carries commuting left actions of $R$ and $S$, hence the left $T$-action $(r\otimes s)(x\otimes y)=rx\otimes sy$ of the $k$-algebra $T=R{\otimes }_{k}S$ (Algebras over a commutative ring, central structure maps, and algebra homomorphisms), and it is functorial in both variables with the identity and composition laws [F2]; choosing a finite basis of $X$ identifies $X{\otimes }_{k}Y$ with a finite direct sum of copies of $Y$ by the universal property of [F2], so ${\dim }_k(X{\otimes }_{k}Y)={\dim }_{k}X\cdot {\dim }_{k}Y$ [F6] and $⊠$ lands in the finite-dimensional category $T\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis). Each partial functor is $k$-linear because additivity and scalar multiplication are read off on generators, where $(f+f^{\mathrm{\prime }})\otimes 1=f\otimes 1+f^{\mathrm{\prime }}\otimes 1$ and $(\lambda f)\otimes 1=\lambda (f\otimes 1)$, and the maps are determined by their values on generators [F2]; hence $(X,Y)\mapsto X{\otimes }_{k}Y$ is $k$-linear in each variable (k-linear categories and k-linear functors, Product category and its projection functors).

For fixed $Y$ the functor $X\mapsto X{\otimes }_{k}Y$ preserves cokernels: if $q:X\to X^{\mathrm{\prime }}$ is the cokernel of $f:X^{\mathrm{\prime }\mathrm{\prime }}\to X$, so that $\ker q=\mathrm{im}f$ (Exact sequences and short exact sequences of modules, Module homomorphism and isomorphism, kernel, image and cokernel), then $q\otimes 1$ is surjective with $\mathrm{im}(f\otimes 1)\subseteq \ker (q\otimes 1)$, and the balanced map sending $(x^{\mathrm{\prime }},y)$ with $q(x)=x^{\mathrm{\prime }}$ to the class of $x\otimes y$ is well defined because representatives differ by an element of $\ker q=\mathrm{im}f$, and it induces a two-sided inverse of the map on generators [F2]; hence $\ker (q\otimes 1)=\mathrm{im}(f\otimes 1)$. The functor is additive by the same generator computation, so it preserves finite coproducts and cokernels and therefore every finite colimit, that is, it is right exact (Left exact and right exact functors, An additive functor preserves finite biproducts); the argument in the first variable is identical, so $⊠$ is right exact in each variable.

Let $G,G^{\mathrm{\prime }}:T\text{-}\mathrm{m}\mathrm{o}\mathrm{d}\to \mathcal{E}$ be $k$-linear and right exact. Both $G\circ ⊠$ and $G^{\mathrm{\prime }}\circ ⊠$ are $k$-linear and right exact in each variable by step 2.1, so [F3] applies to the pair $(G\circ ⊠,G^{\mathrm{\prime }}\circ ⊠)$ and gives a bijection $\mathrm{Nat}(G\circ ⊠,G^{\mathrm{\prime }}\circ ⊠)\cong {\mathrm{Hom}}_T(G(T),G^{\mathrm{\prime }}(T))$ onto right $T$-module maps, by evaluation at $(R,S)$. Applying [F3] with the second algebra $k$ to the bifunctors $(Z,V)\mapsto G(Z{\otimes }_{k}V)$ and $(Z,V)\mapsto G^{\mathrm{\prime }}(Z{\otimes }_{k}V)$, whose value at $(T,k)$ is $G(T{\otimes }_{k}k)\cong G(T)$, gives in the same way a bijection $\mathrm{Nat}(G,G^{\mathrm{\prime }})\cong {\mathrm{Hom}}_T(G(T),G^{\mathrm{\prime }}(T))$: injectivity because the component at $(Z,k)$ recovers ${\eta }_Z$ under the natural isomorphism $Z{\otimes }_{k}k\cong Z$, and surjectivity because the extension of a right $T$-module map constructed by [F3] is natural and has the prescribed component. Since $(\eta \circ ⊠{)}_{R,S}={\eta }_T$, composition with $⊠$ identifies these two bijections, so restriction along $⊠$ is fully faithful.

Let $H$ be a $k$-linear bifunctor right exact in each variable and let $\bar{H}$ be the functor constructed from $W=H(R,S)$ in [F3]. Write $K$ for the functor on finite free $T$-modules used in that construction. Then $\bar{H}$ is $k$-linear: if $g_u,g_v$ are lifts of $u,v:Z\to Z^{\mathrm{\prime }}$ between the chosen presentations, then $g_u+g_v$ lifts $u+v$ and the induced map on cokernels is additive in the lift by the uniqueness in [F7], so $\bar{H}(u+v)=\bar{H}(u)+\bar{H}(v)$, and $g_{\lambda u}=\lambda g_u$ gives $\bar{H}(\lambda u)=\lambda \bar{H}(u)$ similarly. It is right exact: for a presentation $T^m\stackrel{\delta }{\to }{T}^n\to Z\to 0$ the universal property of the cokernel [F7] gives a bijection $\mathcal{E}(\bar{H}(Z),E^{\mathrm{\prime }})\cong \{h:W^n\to E^{\mathrm{\prime }}:hK(\delta )=0\}$ natural in $E^{\mathrm{\prime }}$, and the right-hand side is the set of $T$-linear maps $Z\to \mathcal{E}(W,E^{\mathrm{\prime }})$ under the left $T$-action $(t\cdot \phi )(w)=\phi (w\cdot t)$ on the hom-object, a map $h$ corresponding to the family of its components $h{i}_j:W\to E^{\mathrm{\prime }}$ and the condition $hK(\delta )=0$ saying exactly that this family annihilates $\delta (T^m)$; hence a cokernel sequence $Z^{\mathrm{\prime }\mathrm{\prime }}\to Z\to Z^{\mathrm{\prime }}\to 0$ in $T\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ gives exact sequences $0\to \mathcal{E}(\bar{H}(Z^{\mathrm{\prime }}),E^{\mathrm{\prime }})\to \mathcal{E}(\bar{H}(Z),E^{\mathrm{\prime }})\to \mathcal{E}(\bar{H}(Z^{\mathrm{\prime }\mathrm{\prime }}),E^{\mathrm{\prime }})$ for all $E^{\mathrm{\prime }}$, which by the characterization in [F7] says that $\bar{H}(Z^{\mathrm{\prime }\mathrm{\prime }})\to \bar{H}(Z)\to \bar{H}(Z^{\mathrm{\prime }})$ is again a cokernel sequence. So $\bar{H}$ preserves cokernels and, being additive, all finite colimits, that is, it is right exact (Left exact and right exact functors, An additive functor preserves finite biproducts, Abelian category). By [F3] there is a natural isomorphism $\bar{H}\circ ⊠\cong H$. Fix presentation data once on the small source. The construction on compatible action maps in the determination lemma then makes $H\mapsto \bar{H}$ a functor, with $\bar{H}\circ ⊠\cong H$ natural in $H$. Full faithfulness from step 3.1 lifts this comparison uniquely to $\overline{G\circ ⊠}\cong G$, natural in $G$, so restriction and this functor are quasi-inverse, giving an equivalence of categories (Equivalence, quasi-inverse, and adjoint equivalence of categories, Functor category $[\mathcal{C},\mathcal{D}]$, Natural transformation and its components, Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{C}\mathbf{A}\mathbf{T}$ is not formed).

By [F4] $T\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ is an abelian $k$-linear category, and because ${\dim }_{k}T={\dim }_{k}R\cdot {\dim }_{k}S$ is finite [F6] while the module category of a finite-dimensional $k$-algebra is intrinsically finite [F4], $T\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ is a finite $k$-linear abelian category in the sense of Finite k-linear abelian categories; the equivalence of step 4.1, required for every $k$-linear abelian $\mathcal{E}$, is exactly the universal property of The Deligne product of finite linear categories, so $T\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ with $⊠$ is a Deligne product of $R\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ and $S\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$. For abstract finite $k$-linear abelian categories $\mathcal{A},\mathcal{B}$, [F5] provides equivalences $\mathcal{A}\simeq R\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$, $\mathcal{B}\simeq S\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ with $R,S$ finite-dimensional, and transporting $⊠$, the finite direct sums, the algebra $T$ and the universal property along them yields a Deligne product of $\mathcal{A}$ and $\mathcal{B}$, well defined up to an equivalence respecting the universal bifunctors as follows directly from the universal property: for two products $P,Q$, extend their universal bifunctors to right exact functors $P\to Q$ and $Q\to P$. The restrictions of both composites are isomorphic to the corresponding universal bifunctors, so full faithfulness of restriction lifts these isomorphisms to composites isomorphic to the identities. No commutativity of $R$ or $S$ was used. The construction inherits the set-indexed choices of [F3], made under the stated AC assumption.

Use the supplied finite-dimensional unital $k$-algebras $R,S$ and $k$-linear equivalences of [F1] $\mathcal{A}\simeq R\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$, $\mathcal{B}\simeq S\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ (Equivalence, quasi-inverse, and adjoint equivalence of categories, Algebras over a commutative ring, central structure maps, and algebra homomorphisms, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis). The $k$-dual $X^{\ast }={\mathrm{Hom}}_k(X,k)$ with the right $R$-action $(\lambda \cdot a)(x)=\lambda (ax)$, equivalently the left $R^{\mathrm{o}\mathrm{p}}$-action $a\cdot \lambda :=\lambda \cdot a$, is a contravariant exact equivalence to $R^{\mathrm{o}\mathrm{p}}\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ [F2] (Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$), so it identifies ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}$ with $R^{\mathrm{o}\mathrm{p}}\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ (Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$, The opposite ring $R^{\mathrm{o}\mathrm{p}}$); $R^{\mathrm{o}\mathrm{p}}\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ and $S\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ are finite $k$-linear abelian categories [F6] (Abelian category, k-linear categories and k-linear functors).