Frobenius Exact Categories

• Frobenius exact categories are defined as exact categories with enough projectives and injectives that coincide, enabling the formation of stable triangulated categories.
• n-Frobenius categories generalize the classical notion by incorporating higher vanishing conditions on extensions, which leads to rich higher-dimensional stable and phantom category structures.
• These frameworks have practical applications in module theory, tensor categories, and representation theory, linking algebraic structures with geometric insights.

A Frobenius exact category, and its higher-dimensional generalizations such as the $n$-Frobenius category, form a foundational structure in modern homological algebra. These categories rigorously interpolate between abelian and triangulated settings, supporting the construction of stable or derived categories with explicit, highly-structured properties. The contemporary theory encompasses exact categories in the sense of Quillen, tensor categories, $n$-angulated generalizations, and stable phenomena, with characteristic triangulations and universal properties.

1. Quillen Exact Categories and the Frobenius Condition

An exact category in the sense of Quillen is an additive category $\mathcal{C}$ equipped with a distinguished class of kernel–cokernel pairs $(i,p): A \to B \to C$, called conflations, that satisfy axioms mimicking the short exact sequences of abelian categories:

• The class of conflations is closed under isomorphism, direct sums, pull-backs along arbitrary maps for deflations, and push-outs along arbitrary maps for inflations.
• Each conflation is a pair with $i$ a kernel of $p$, and $p$ a cokernel of $i$.
• For any such structure, one defines $\Ext^1_{\mathcal{C}}(C,A)$ as the set of equivalence classes of conflations $0 \to A \to B \to C \to 0$, and, dually, higher $\Ext^n_{\mathcal{C}}(C,A)$ by splicing $n$ conflations to form a sequence of length $n$.

A Frobenius exact category is an exact category in which

• There are enough projectives (every object $M$ admits a deflation $P \to M$ with $P$ projective) and enough injectives (dually, each $M$ admits an inflation $M \to I$ into an injective).
• The classes of projective and injective objects coincide.

In such categories, the stable category $\mathcal{C}/\mathrm{Proj}$ (objects as in $\mathcal{C}$, morphisms modulo maps factoring through projective-injectives) acquires a canonical triangulated structure (Arentz-Hansen, 2017, Liu et al., 2019).

2. $n$-Frobenius Categories: Higher-Dimensional Generalization

For a fixed non-negative integer $n$, the notion of $n$-Frobenius category extends the classical theory:

• An object $P$ in $\mathcal{C}$ is $n$-projective if $\Ext^i_{\mathcal{C}}(P,X) = 0$ for all $i > n$ and all $X$.
• $I \in \mathcal{C}$ is $n$-injective if $\Ext^i_{\mathcal{C}}(X,I) = 0$ for $i > n$ and all $X$.
• The category has enough $n$-projectives (resp. $n$-injectives) if every object admits a deflation from an $n$-projective (resp. an inflation into an $n$-injective).
• $n$-Frobenius category: $\mathcal{C}$ has enough $n$-projectives and $n$-injectives, and the subcategories $n$-proj $\mathcal{C} = n$-inj $\mathcal{C}$ (Bahlekeh et al., 2023).

When $n=0$, this is the classical Frobenius category. For $n=1$, projectivity coincides with vanishing of $\Ext^2(P,-)$ (classical projectivity), and the standard stable triangulated category is recovered.

3. Phantom Stable Categories and the Universal Stable Quotient

A distinctive higher-dimensional refinement is the phantom stable category of an $n$-Frobenius category. Let $\mathcal{C}$ be $n$-Frobenius, and define a subfunctor $\mathcal{P}(A,B)\subseteq \Ext^n_{\mathcal{C}}(A,B)$ comprising length $n$ conflations factoring through $n$-projective objects. A morphism $f: M\to N$ is $n$-Ext-phantom if it annihilates $\Ext^n_{\mathcal{C}}/\mathcal{P}$, i.e., $(\Ext^n_{\mathcal{C}}/\mathcal{P})(f,-)=0$ and $(\Ext^n_{\mathcal{C}}/\mathcal{P})(-,f)=0$.

The phantom stable category $(\mathcal{C}_{\mathcal{P}},T)$ is characterized by:

• $T(f)=0$ for all $n$-Ext-phantom $f$;
• $T(s)$ is an isomorphism for every quasi-invertible $s$ (i.e., induces an isomorphism on $\Ext^n_{\mathcal{C}}/\mathcal{P}$);
• $T$ is universal: any other additive functor with these properties factors uniquely through $T$ (Bahlekeh et al., 2023).

The construction relies on a localization (calculus of fractions) with respect to quasi-invertibles and annihilation of phantom maps. When $n=0$, $\mathcal{P}$ is the ideal of maps factoring through projectives, and the phantom stable category is the usual stable category.

4. Triangulated Structure and Stable Categories

For classical ($n=0$) Frobenius exact categories, the stable category $\mathcal{C}/\mathrm{Proj}$ admits an explicit triangulated structure (Arentz-Hansen, 2017, Liu et al., 2019):

• The suspension (shift) functor $\Omega$ is given by $\Omega M = \ker(P\to M)$ where $P\to M$ is a projective cover.
• Distinguished triangles arise from conflations (short exact sequences): a conflation $A \rightarrowtail B \twoheadrightarrow C$ yields a triangle $A \to B \to C \to \Omega A$ in $\mathcal{C}/\mathrm{Proj}$.
• In the higher $n$-Frobenius context, this triangulated construction is replaced by a suitable higher analog: for $n>1$, the phantom stable category is the natural generalization, organizing higher extensions (length-$n$ conflations) rather than only the classical short exact sequences (Bahlekeh et al., 2023).

5. Key Examples and Applications

Frobenius and $n$-Frobenius exact categories are pervasive in algebra and geometry.

Example Class	Structure Type	Comments
mod–$A$, $A$ self-injective	Frobenius (n=0)	Stable module category: triangulated (Bahlekeh et al., 2023)
GProj $R$, $R$ Gorenstein	Frobenius (n=0)	Gorenstein-projectives: triangulated stable cat.
Coh $X$, $X$ proj. dim $d$	$d$-Frobenius	$d$-Frobenius via locally free sheaves
Ch(Flat $X$), $X$ noetherian	$n$-Frobenius	Flat complexes, higher extensions

Cluster categories (e.g., of Dynkin type) arise as stable categories of 2-Calabi–Yau Frobenius exact 2-cluster tilting subcategories. More generally, any abelian category with non-zero $n$-projective objects admits a non-trivial $n$-Frobenius subcategory (Bahlekeh et al., 2023).

6. Relation to Tensor and Exangulated Categories

In symmetric tensor categories over a field of positive characteristic $p>0$, the notion of Frobenius-exact is formulated in terms of the exactness of the Frobenius functor $$F: \mathcal{C} \to \mathcal{C} \boxtimes \mathrm{Ver}_p$$ (Verlinde category):

• $\mathcal{C}$ is Frobenius-exact if $F$ is exact;
• Equivalently, if $\mathcal{C}$ admits a symmetric tensor functor to a semisimple category (e.g., fusion categories);
• The pre-Tannakian category $\mathcal{C}$ admits a fiber functor to $\mathrm{Ver}_p$ if and only if it has moderate growth and is Frobenius-exact (Etingof et al., 2019, Coulembier et al., 2021).

The concept of $n$-Frobenius categories is further linked to the framework of $n$-exangulated categories: for $n=1$, the exact category coincides with the extriangulated setting, and a Frobenius exact category yields a stable triangulated category (Liu et al., 2019).

7. Classification, Factor Categories, and Structural Properties

Structural theorems describe when factor categories of Frobenius exact categories (by suitable subcategories of projective-injectives) remain Frobenius, and when extension-closed exact subcategories are equivalent to subcategories of Cohen–Macaulay modules over additive categories (Chen, 2010). Classification results characterize thick/triangulated subcategories in stable categories of Frobenius type (Arentz-Hansen, 2017). Furthermore, orbit and completed orbit categories (under auto-equivalences) can be constructed to be Frobenius under explicit, checkable conditions (Chávez, 2015).

The construction of new $n$-Frobenius categories is flexible: for instance, the exact category of complexes in an additive category admits additional Frobenius structures parametrized by suitable endofunctors and natural transformations (Ben et al., 2014, Frank et al., 8 Apr 2025).

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This synthesis situates Frobenius and $n$-Frobenius exact categories as central objects that mediate between projective/injective theory, extension groups, triangulated (and $n$-angulated) categories, and tensor category representation theory, providing a categorical foundation for many contemporary themes in higher homological algebra and representation theory (Bahlekeh et al., 2023).