Monoidal Triangulated Functors: Theory & Applications

• Monoidal triangulated functors are exact functors between tensor-triangulated categories that preserve distinguished triangles and monoidal structures.
• They induce continuous, contravariant maps on prime spectra using support formulas to relate tensor ideals between categories.
• They have applications in tensor-triangular geometry, noncommutative geometry, and stable homotopy theory via exotic equivalences.

A monoidal triangulated functor is a structured functor between monoidal triangulated categories (tensor-triangulated categories) that is simultaneously exact with respect to distinguished triangles and (lax or strong) symmetric monoidal with respect to the tensor structure. Such functors play a central role in tensor-triangular geometry, the theory of spectral supports, and categorical representation theory. The interplay between triangulated and monoidal structures leads to refined notions of support, universality, and functoriality, particularly in the context of noncommutative geometry and stable homotopy theory.

1. Definition and Fundamental Properties

Let $(K,\Sigma,\otimes,\mathbf{1}_K)$ and $(L,\Sigma',\otimes',\mathbf{1}_L)$ be essentially small monoidal triangulated categories (abbreviated as MΔCs), i.e., $\otimes$ is exact in each variable and compatible with the triangulated structure. An exact monoidal triangulated functor

$F: K \longrightarrow L$

consists of:

• A triangulated functor $F$, meaning $F\circ \Sigma \cong \Sigma'\circ F$ and $F$ preserves distinguished triangles;
• A strong monoidal structure: a natural isomorphism

$$F(x) \otimes' F(y) \cong F(x \otimes y) \quad \text{for all } x, y \in K,$$

compatible with associators, units, and satisfying $F(\mathbf{1}_K) \cong \mathbf{1}_L$.

Exactness of $\otimes$ in $K$ and the requirement that $F$ commutes with $\otimes$ ensures that $F$ is exact in each variable and preserves the full tensor-triangulated structure (Miller, 3 May 2025, Nikandros et al., 2023).

A “lax” or “oplax” monoidal structure allows the monoidal structural maps to be non-invertible, but most constructions of interest in tensor-triangular geometry work with strong monoidal functors.

2. Induced Maps on (Complete) Prime Spectra

Given an exact monoidal functor $F: K \to L$, it is natural to ask how it acts on the spectra of thick tensor ideals. The completely prime spectrum $\Spc^{cp}(K)$ consists of proper thick two-sided tensor-ideals $P \subset K$ with the property: $$x \otimes y \in P \implies (x \in P) \text{ or } (y \in P).$$ Endowed with the Zariski topology specified by $U^{cp}(x) = \{P \mid x \in P\}$ being open for each $x \in K$, this construction refines Balmer's prime spectrum, particularly in the noncommutative setting.

For any exact monoidal functor $F: K \to L$, the inverse image $F^{-1}(Q)$ of a completely prime ideal $Q \subset L$ is again a completely prime ideal of $K$, yielding a continuous, contravariant map

$\Spc^{cp}(F) : \Spc^{cp}(L) \to \Spc^{cp}(K), \qquad Q \mapsto F^{-1}(Q).$

This map satisfies the support formula: $\Spc^{cp}(F)^{-1}(\supp^{cp}_K(x)) = \supp^{cp}_L(F(x)) \quad \text{for each } x \in K.$ However, the analogous construction for the noncommutative Balmer spectrum $\Spc(K)$ (which only requires the lattice-theoretic primality of thick tensor ideals) is not always functorial: it requires $Q \cap F(K) \subset L$ to remain prime in the essential image $F(K) \subset L$ (Miller, 3 May 2025).

3. Universality and Multiplicative Support Data

The pair $(\Spc^{cp}(K), \supp^{cp})$ forms the universal, or final, object in the category $Scp(K)$ of multiplicative support data: for any topological space $Y$ and assignment $\sigma : K \to \{$closed subsets of$~Y\}$ satisfying the usual triangulated axioms and the tensor product property $\sigma(x\otimes y) = \sigma(x) \cap \sigma(y)$, there is a unique continuous map $f: Y \to \Spc^{cp}(K)$ with

$\sigma(x) = f^{-1}(\supp^{cp}(x)).$

This universality ensures that $\Spc^{cp}(-)$ defines a functor

$\Spc^{cp}: \mathbf{mon}_\Delta^{op} \longrightarrow \mathbf{Top}$

on the 2-category of (essentially small) monoidal triangulated categories and exact monoidal functors. This is the cornerstone for constructing tensor-triangular support theories with robust functorial behavior (Miller, 3 May 2025).

4. Injectivity and Surjectivity of Induced Maps

The action of exact monoidal functors on spectra admits precise injectivity and surjectivity criteria:

• Injectivity of $\Spc^{cp}(F)$: If $F$ is essentially surjective on objects, then the induced map on prime spectra is injective.
• Surjectivity of $\Spc^{cp}(F)$: If $K$ is rigid (every object has a dual), $L$ is duo (all thick one-sided tensor-ideals are two-sided), and $F$ detects tensor-nilpotence of morphisms (i.e., $F(f) = 0 \implies f^{\otimes n} = 0$ for some $n$), then $\Spc^{cp}(F)$ is surjective.

Furthermore, when $F$ admits a right adjoint $U: L \to K$, the image of $\Spc^{cp}(F)$ is precisely the support of the object $U(\mathbf{1}_L)$; surjectivity coincides with $F$ detecting tensor-nilpotence (Miller, 3 May 2025).

5. Examples: Crossed Products and Weight Complex Functors

Crossed Product Categories

Let $K$ be an MΔC and $G$ a finite group acting by monoidal auto-equivalences. The crossed product category $K \rtimes G$ inherits a monoidal triangulated structure, and there is an equivariant homeomorphism

$\Spc^{cp}(K \rtimes G) \cong G\text{-fixed points of } \Spc^{cp}(K) = G\!-\!\Spc^{cp}(K).$

Thus, the structure of the spectrum under group actions is tractable. Notably, if $K$ has no completely prime ideals, neither does $K \rtimes G$ (Miller, 3 May 2025).

Weight Complex Functor

Bondarko's weight complex functor $w$ is an archetype of a monoidal triangulated functor between categories of motives and bounded chain complexes in the heart of a weight structure. Given a stable symmetric monoidal $\infty$-category with a bounded, compatible weight structure, $w$ admits a symmetric monoidal refinement

$$w^{\otimes}: C^{\otimes} \to K(hC_{w}^\heartsuit)^{\otimes}$$

respecting tensor products and units up to canonical isomorphism. Applications include effective Chow motives and Voevodsky motives, as well as various contexts such as mixed Tate motives and mixed Hodge modules (Aoki, 2019).

6. Universal and Functorial Support Theories Beyond the Braided Case

While $\Spc^{cp}(K)$ is universal among multiplicative support data, it can be empty when $K$ lacks sufficiently many completely prime ideals. Miller introduced the partial-prime spectrum $p\Spc(K)$, consisting of "partial" prime tensor-ideals, which is always nonempty for $K \neq 0$ and is contravariantly functorial on all monoidal triangulated categories. For braided MΔCs, $p\Spc(K) \cong \Spc(K)$ recovers Balmer's spectrum, and $p\Spc$ is characterized as the right Kan extension of $\Spc$ from the braided to the general monoidal triangulated context (Miller, 3 May 2025).

7. Exotic Equivalences and Monoidal Compatibility

A notable phenomenon in stable homotopy theory involves exotic triangulated equivalences that are compatible with monoidal structures up to isomorphism but are not genuinely tensor-triangulated at the level of model categories. Franke's reconstruction functor,

$R: D([1],1)(A) \to Ho(M),$

is a triangulated equivalence between homotopy categories which admits a natural isomorphism

$R(X) \wedge R(Y) \cong R(X \otimes Y),$

fulfilling associativity and coherence up to isomorphism. However, it may fail to be strongly monoidal (lack of isomorphism on the unit object or commutativity in certain cases), so it is classified as lax or oplax monoidal, rather than a genuine tensor-triangulated equivalence (Nikandros et al., 2023).

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References:

(Miller, 3 May 2025): On functoriality and the tensor product property in noncommutative tensor-triangular geometry (Aoki, 2019): The weight complex functor is symmetric monoidal (Nikandros et al., 2023): Monoidal Properties of Franke's Exotic Equivalence