Pseudo-Tensor Categories and Duality

• Pseudo-tensor categories are algebraic structures that generalize tensor calculus from vector spaces to flexible linear categories.
• They utilize operations like direct sums, tensor products, and symmetric as well as exterior powers within a rigid, pseudo-abelian, Q-linear ACU framework.
• The abstract Poincaré duality theorem establishes isomorphisms between alternating powers and their duals, extending classical vector space duality.

A pseudo-tensor category is an algebraic structure in which the familiar operations of tensor calculus, such as direct sums, tensor products, dualization, and formation of exterior and symmetric powers, are abstracted and generalized from the classical setting of vector spaces to more flexible linear categories. In the context of rigid pseudo-abelian $\mathbb{Q}$-linear ACU (associative, commutative, with unit) tensor categories, this abstraction enables the definition and study of analogs of important algebraic constructs and dualities, such as the Poincaré duality isomorphisms for exterior powers. The foundational definitions, key constructions, and Poincaré duality theorem in this abstract setting are examined comprehensively in (Masdeu et al., 2014).

1. Structural Foundations: ACU, Rigidity, and Pseudo-Abelianity

In a broad categorical framework, let $\mathcal{C}$ denote an additive, $\mathbb{Q}$-linear category equipped with:

• A biadditive tensor product $$\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$$
• Strict associativity constraint: $$(a \otimes b) \otimes c \cong a \otimes (b \otimes c)$$
• Commutativity (symmetry) constraint: $\tau_{X,Y} \colon X \otimes Y \to Y \otimes X$, with $$\tau_{Y,X} \circ \tau_{X,Y} = \mathrm{id}_{X \otimes Y}$$
• A unit object $\mathbb{I}$, with left and right unit isomorphisms $\ell_X \colon \mathbb{I} \otimes X \cong X$ and $r_X \colon X \otimes \mathbb{I} \cong X$

The ACU structure is supplemented by additional properties:

• Internal Homs: For every $X, Y \in \mathcal{C}$, the object $\mathrm{Hom}(X, Y)$ exists, together with the evaluation map $$\mathrm{ev}_{X,Y} \colon \mathrm{Hom}(X, Y) \otimes X \to Y$$ satisfying $$\mathrm{Hom}_{\mathcal{C}}(S, \mathrm{Hom}(X, Y)) \cong \mathrm{Hom}_{\mathcal{C}}(S \otimes X, Y)$$.
• Rigidity: Each $X$ admits a dual $X^\vee = \mathrm{Hom}(X, \mathbb{I})$, with evaluation $$\mathrm{ev}_X \colon X^\vee \otimes X \to \mathbb{I}$$ and coevaluation $$\mathrm{coev}_X \colon \mathbb{I} \to X \otimes X^\vee$$, satisfying the zig-zag (triangular) identities.
• Pseudo-Abelianity: Every idempotent $e \colon Z \to Z$ splits; that is, $Z \cong Z^+ \oplus Z^-$, where these summands are the kernel and cokernel.
• $\mathbb{Q}$-Linearity: Each Hom-space is a $\mathbb{Q}$-vector space, with bilinear composition and tensoring.

In this environment—termed a "rigid pseudo-abelian $\mathbb{Q}$-linear ACU tensor category"—abstract versions of familiar algebraic constructions can be implemented and studied.

2. Alternating and Symmetric Algebras in Pseudo-Tensor Categories

Given an object $X \in \mathcal{C}$, its tensor algebra is defined by $T^\bullet X := \bigoplus_{i \geq 0} T^i X$, where $T^i X = X^{\otimes i}$, with multiplication given by the canonical associator. The symmetric group $S_i$ acts on $T^i X$ by permuting tensor factors via the categorical symmetry $\tau$. Idempotents in the group algebra $\mathbb{Q}[S_i]$ corresponding to the sign and trivial characters are constructed as: $$e_{a,X}^i := \frac{1}{i!} \sum_{\sigma \in S_i} \mathrm{sign}(\sigma)\cdot\sigma\ \qquad e_{s,X}^i := \frac{1}{i!} \sum_{\sigma \in S_i} 1\cdot\sigma$$ By pseudo-abelianity, these idempotents split, yielding the decompositions: $$T^i X \cong \bigwedge^i X \oplus \cdots \qquad \text{(alternating)}$$

$$T^i X \cong \vee^i X \oplus \cdots \qquad \text{(symmetric)}$$

Here, $\bigwedge^i X$ (resp. $\vee^i X$) denotes the alternating (resp. symmetric) power, which inherits a graded (super-)commutative algebra structure with the natural product: $$\bigwedge^i X \otimes \bigwedge^j X \longrightarrow \bigwedge^{i+j} X$$

$$\vee^i X \otimes \vee^j X \longrightarrow \vee^{i+j} X$$

This algebraic structure allows the transfer of many classical constructions for vector spaces to this more general categorical context.

3. Internal Multiplication, Duality, and Casimir Elements

Fix a graded algebra $$A^\bullet = (A_i, \cdot_{i,j} \colon A_i \otimes A_j \to A_{i+j})$$ in $\mathcal{C}$. For $j \geq i$, there is a multiplication map $f_{i,j-i} \colon A_i \to \mathrm{Hom}(A_{j-i},A_j)$, with

$$\mathrm{ev}_{j-i,j} \circ (f_{i,j-i} \otimes \mathrm{id}_{A_{j-i}}) = \cdot_{i,j-i}$$

Making use of the duality morphism $$d_{X,Y}: \mathrm{Hom}(X,Y) \to \mathrm{Hom}(Y^\vee, X^\vee)$$, one defines the internal multiplication

$$m_{i,j} \colon A_i \otimes A_j^\vee \longrightarrow A_{j-i}^\vee$$

as the composite

$$A_i \otimes A_j^\vee \xrightarrow{f_{i,j-i} \otimes \mathrm{id}} \mathrm{Hom}(A_{j-i}, A_j) \otimes A_j^\vee \xrightarrow{\mathrm{ev}} A_{j-i}^\vee$$

The Casimir element associated to $X$ in a rigid category, $C_X : \mathbb{I} \to X \otimes X^\vee$, satisfies $$(\mathrm{ev}_X \otimes \mathrm{id}_X)\circ(\mathrm{id}_{X^\vee} \otimes C_X) = \mathrm{id}_X$$. This central object encodes duality symmetries essential for the definition of abstract Poincaré isomorphisms.

4. Abstract Poincaré Duality in Pseudo-Tensor Categories

Consider the alternating algebra $A^\bullet = \bigwedge^\bullet X$ and suppose the top exterior power $\bigwedge^g X$ is invertible (rank $g$). For $0 \leq i \leq g$, define Poincaré morphisms: $$D^{i,g} : \bigwedge^i X \to \bigwedge^{g-i} X^\vee \otimes (\bigwedge^g X^\vee)^\vee$$ as the composite

$$\bigwedge^i X \xrightarrow{m_{i,g-i}} \mathrm{Hom}(\bigwedge^g X^\vee,\bigwedge^{g-i} X^\vee) \cong \bigwedge^{g-i} X^\vee \otimes (\bigwedge^g X^\vee)^\vee$$

Dually, define $$D_{i,g} : \bigwedge^i X^\vee \to \bigwedge^{g-i} X \otimes \bigwedge^g X^\vee$$.

Abstract Poincaré duality theorem: In a rigid, $\mathbb{Q}$-linear, pseudo-abelian tensor category with $\bigwedge^g X$ invertible of rank $r \in \mathrm{End}(\mathbb{I})$, the following hold: $$D_{g-i,g} \circ D^{i,g} = (-1)^{i(g-i)} \binom{g}{g-i}^{-1} \mathrm{Id}_{\bigwedge^i X}$$

$$D^{i,g} \circ D_{g-i,g} = (-1)^{i(g-i)} \binom{g}{i}^{-1} \mathrm{Id}_{\bigwedge^{g-i} X^\vee}$$

with the usual meaning of binomial coefficients and invertibility assumptions on the relevant scalars. Consequently, if all these binomials are invertible, $D^{i,g}$ is an isomorphism, yielding: $$\bigwedge^i X^\vee \otimes (\bigwedge^g X) \cong \bigwedge^{g-i} X$$ This establishes a categorical analog of classical Poincaré duality, fundamental for cohomological theories and representation theory in abstract tensor contexts (Masdeu et al., 2014).

5. Recovery of the Classical Vector Space Case

In the canonical concrete context where $\mathcal{C} = \mathrm{Vect}_K$ and $X = V$ is a $K$-vector space of dimension $g$, $\bigwedge^i V$ is the classical exterior power and $V^\vee$ the linear dual. The above Poincaré morphism specializes to

$$v_1 \wedge \cdots \wedge v_i \mapsto \left( \omega \mapsto \omega(v_1 \wedge \ldots \wedge v_i) \right) \in \bigwedge^{g-i} V^\vee$$

up to normalization by $\frac{g!}{i!(g-i)!}$. The theorem then recovers the isomorphism

$\bigwedge^i V^\vee \cong \bigwedge^{g-i} V$

via identification of $\bigwedge^g V$ with $K$. This alignment with the classical case underscores the conceptual completeness of the categorical abstraction (Masdeu et al., 2014).

6. Mathematical Significance and Applications

Rigid pseudo-abelian $\mathbb{Q}$-linear ACU tensor categories ("pseudo-tensor categories," Editor's term) provide a robust framework for generalizing algebraic and geometric duality phenomena. Their structure supports the development of exterior and symmetric power objects, idempotent splittings, and associated duality isomorphisms necessary for advanced representation theory, category-theoretic algebraic geometry, and abstract cohomology theories.

The existence of isomorphisms such as those in the abstract Poincaré duality theorem enables the extension of central results from linear algebra into broader categorical contexts, facilitating later advances in Tannakian duality and the study of motives, as well as applications in quantum algebra and categorical representation theory (Masdeu et al., 2014).