Categorical Entropy in Triangulated Categories

• Categorical entropy is a measure quantifying the asymptotic growth of invariants in triangulated categories via exact endofunctors and split-generators.
• It connects growth under Bridgeland stability conditions with spectral radii, topological, and combinatorial entropies across diverse categorical frameworks.
• Computational tools involving t-structures, dg invariants, and functorial constructions enable explicit entropy calculations in geometry and algebra.

Categorical entropy is a family of invariants that quantify complexity in categorical settings. In the current literature on triangulated and derived categories, the term most commonly denotes the Dimitrov–Haiden–Katzarkov–Kontsevich growth invariant of an exact endofunctor on a triangulated category with a split-generator, measuring the asymptotic growth of the complexity of iterated images of generators. In this form it has been related to mass growth under Bridgeland stability conditions, spectral radii on Grothendieck or cohomological invariants, topological entropy, and symbolic or graph-theoretic entropy. The same phrase also appears in other categorical frameworks, including entropy functions on abelian categories, functorial relative entropy on standard Borel spaces, and Shannon-type entropy on finite categories (Kikuta et al., 2016, Ikeda, 2016, Dikranjan et al., 2010, Gagne et al., 2017, Chen et al., 2023).

1. DHKK entropy on triangulated categories

Let $\mathcal T$ be a triangulated category with a split-generator $G$, and let $\Phi:\mathcal T\to\mathcal T$ be an exact endofunctor. The DHKK complexity $\delta_t(E,F)$ is defined by taking an infimum over towers of distinguished triangles expressing $F$ from shifts $E[n_i]$, with weighted cost $\sum_i e^{n_i t}$. The categorical entropy is then

$$h_t(\Phi)=\lim_{n\to\infty}\frac{1}{n}\log \delta_t(G,\Phi^n(G)).$$

The limit exists and is independent of the chosen split-generator. The value at $t=0$,

$h_0(\Phi),$

is the usual categorical entropy (Lu et al., 2021).

In $G$0-finite triangulated categories, this invariant admits a more concrete expression. If

$G$1

then

$G$2

for split-generators $G$3. This reformulation turns entropy into an asymptotic growth rate of total weighted $G$4-dimensions and is the basis of many explicit computations (Yoshida, 20 Jan 2026).

Several formal properties recur across the literature. The complexity function is submultiplicative, exact functors do not increase complexity, and entropy is invariant under the choice of split-generator. In saturated settings there are also opposite-category and inverse-functor symmetries, such as

$G$5

under the stated hypotheses (Kikuta et al., 2020, Kim, 2022).

2. Computational formalisms and structural avatars

One important computational avatar is mass growth. For a Bridgeland stability condition $G$6, the mass of an object $G$7 with parameter $G$8 is

$G$9

where $\Phi:\mathcal T\to\mathcal T$0 are the Harder–Narasimhan factors of $\Phi:\mathcal T\to\mathcal T$1. The associated mass growth rate

$\Phi:\mathcal T\to\mathcal T$2

satisfies

$\Phi:\mathcal T\to\mathcal T$3

If the relevant connected component of $\Phi:\mathcal T\to\mathcal T$4 contains an algebraic stability condition, then

$\Phi:\mathcal T\to\mathcal T$5

so categorical entropy can be computed directly from mass growth (Ikeda, 2016).

A second formalism uses bounded $\Phi:\mathcal T\to\mathcal T$6-structures and bounded co-$\Phi:\mathcal T\to\mathcal T$7-structures with finite hearts or finite co-hearts. For a bounded $\Phi:\mathcal T\to\mathcal T$8-structure $\Phi:\mathcal T\to\mathcal T$9, the invariant $\delta_t(E,F)$0 defined from the unique $\delta_t(E,F)$1-filtration of objects satisfies

$\delta_t(E,F)$2

Dually, for a bounded co-$\delta_t(E,F)$3-structure $\delta_t(E,F)$4, one has

$\delta_t(E,F)$5

In ST-triples, this produces a duality phenomenon: if $\delta_t(E,F)$6 is an autoequivalence preserving the two sides of the triple, then the entropies on the $\delta_t(E,F)$7-side and the co-$\delta_t(E,F)$8-side coincide (Kim, 2022).

Further calculational tools come from functorial constructions. For a spherical functor $\delta_t(E,F)$9 with twist $F$0 and cotwist $F$1, if the essential image of the right adjoint contains a split-generator of $F$2, then

$F$3

This mechanism generalizes computations for spherical twists and $F$4-twists (Kim, 2021).

Entropy also behaves well under categorical localization. If $F$5 is a full pretriangulated subcategory preserved by $F$6, and $F$7 and $F$8 are the induced functors on the subcategory and the quotient, then

$F$9

For admissible subcategories this becomes a max-formula. This suggests a close formal analogy with the behavior of topological entropy under factor maps and invariant pieces (Bae et al., 2022).

3. Spectral radius, Gromov–Yomdin type formulas, and algebraic growth

A recurrent problem is whether categorical entropy is governed by the spectral radius of the induced action on Hochschild homology, cohomology, or the numerical Grothendieck group. A categorical Gromov–Yomdin type conjecture asks for formulas of the form

$E[n_i]$0

for Fourier–Mukai endofunctors with invertible Hochschild action, and

$E[n_i]$1

for projective autoequivalences, where $E[n_i]$2 denotes the induced endomorphism on the numerical Grothendieck group (Kikuta et al., 2016).

In geometric dynamics this principle is realized for pullback functors. If $E[n_i]$3 is a surjective endomorphism of a smooth projective variety, then the categorical entropy of the derived pullback satisfies

$E[n_i]$4

where $E[n_i]$5 are the dynamical degrees. Thus the derived pullback reproduces classical topological entropy exactly (Kikuta et al., 2016).

Several noncommutative and representation-theoretic settings exhibit analogous formulas. For a finitely presented monomial algebra $E[n_i]$6, the bounded derived category $E[n_i]$7 carries the Serre twist $E[n_i]$8, and

$E[n_i]$9

This entropy matches the topological entropy of the associated subshift, the entropy of the language of legal words, the entropy of the Ufnarovski graph, and the growth rate of the graph path algebra. In the quiver case $\sum_i e^{n_i t}$0,

$\sum_i e^{n_i t}$1

with $\sum_i e^{n_i t}$2 the spectral radius of the adjacency matrix (Lu et al., 2021).

For generic abelian surfaces, the entropy of Fourier–Mukai autoequivalences is explicitly computable from the induced matrix on the algebraic Mukai lattice. In the class treated there, one has

$\sum_i e^{n_i t}$3

and more generally an upper bound $\sum_i e^{n_i t}$4 for composites of Fourier–Mukai equivalences, pullbacks $\sum_i e^{n_i t}$5, and pushforwards $\sum_i e^{n_i t}$6 for finite morphisms. Combined with known lower bounds, this yields equality at $\sum_i e^{n_i t}$7 (Yoshioka, 2017).

The lower-bound side is quite general. In the presence of stability conditions one has

$\sum_i e^{n_i t}$8

so the numerical spectral radius gives a categorical Yomdin-type lower bound even when equality fails (Ikeda, 2016).

4. Explicit computations and counterexamples

A large body of work consists of exact entropy computations for specific functors. In commutative algebra, for a $\sum_i e^{n_i t}$9-dimensional $$h_t(\Phi)=\lim_{n\to\infty}\frac{1}{n}\log \delta_t(G,\Phi^n(G)).$$0-finite noetherian local ring of characteristic $$h_t(\Phi)=\lim_{n\to\infty}\frac{1}{n}\log \delta_t(G,\Phi^n(G)).$$1, the Frobenius pushforward functor

$$h_t(\Phi)=\lim_{n\to\infty}\frac{1}{n}\log \delta_t(G,\Phi^n(G)).$$2

has entropy

$$h_t(\Phi)=\lim_{n\to\infty}\frac{1}{n}\log \delta_t(G,\Phi^n(G)).$$3

The formula splits into a dimension term and a residue-field term, and it completely determines the categorical entropy of Frobenius pushforward in this setting (2207.13774).

For spherical functors one recovers familiar formulas. If $$h_t(\Phi)=\lim_{n\to\infty}\frac{1}{n}\log \delta_t(G,\Phi^n(G)).$$4 is a $$h_t(\Phi)=\lim_{n\to\infty}\frac{1}{n}\log \delta_t(G,\Phi^n(G)).$$5-spherical object, then the entropy of the associated spherical twist satisfies

$$h_t(\Phi)=\lim_{n\to\infty}\frac{1}{n}\log \delta_t(G,\Phi^n(G)).$$6

The same framework also recovers the entropy behavior of $$h_t(\Phi)=\lim_{n\to\infty}\frac{1}{n}\log \delta_t(G,\Phi^n(G)).$$7-twists (Kim, 2021).

Counterexamples to naive Gromov–Yomdin expectations are equally prominent. On any smooth projective surface containing a $$h_t(\Phi)=\lim_{n\to\infty}\frac{1}{n}\log \delta_t(G,\Phi^n(G)).$$8-curve $$h_t(\Phi)=\lim_{n\to\infty}\frac{1}{n}\log \delta_t(G,\Phi^n(G)).$$9, the autoequivalence

$t=0$0

has

$t=0$1

so positive categorical entropy can coexist with trivial cohomological spectral radius (Mattei, 2019).

More recent work shows that such failures are widespread. If a smooth projective surface $t=0$2 fails the Gromov–Yomdin property, then $t=0$3 also fails it for every $t=0$4, and no hyperkähler or Enriques manifold satisfies the Gromov–Yomdin property because one can construct an explicit autoequivalence with positive categorical entropy and unipotent action on cohomology. In particular, for a projective hyperkähler manifold the autoequivalence

$t=0$5

has

$t=0$6

since the $t=0$7-twist acts trivially on cohomology and tensoring by $t=0$8 acts unipotently (Yoshida, 20 Jan 2026).

Surface geometry also exhibits both rigidity and failure. For every autoequivalence of a bielliptic surface $t=0$9,

$h_0(\Phi),$0

and for types $h_0(\Phi),$1 there are explicit autoequivalences with positive categorical entropy even though bielliptic surfaces have no automorphism of positive topological entropy and no spherical objects in $h_0(\Phi),$2. By contrast, the same paper constructs an Enriques-surface example with

$h_0(\Phi),$3

showing that the equality does not persist across all Kodaira-dimension-zero surfaces (Yoshida, 5 Jul 2025).

5. Refinements, dg invariants, and symplectic/topological comparisons

Categorical entropy admits secondary refinements. Categorical polynomial entropy is defined by

$h_0(\Phi),$4

and measures polynomial-order growth after the exponential rate has been removed. It satisfies a Yomdin-type lower bound in terms of the induced endomorphism on the numerical Grothendieck group, and for a surjective endomorphism $h_0(\Phi),$5 one has

$h_0(\Phi),$6

or $h_0(\Phi),$7 in the complex case. This produces a categorical trichotomy into elliptic, parabolic, and loxodromic behavior (Fan et al., 2020).

A related dg-theoretic invariant is Hochschild entropy. For a quasi-endofunctor $h_0(\Phi),$8 of a smooth proper dg category, the Hochschild cohomological and homological entropies are defined from the asymptotic growth of $h_0(\Phi),$9 and $G$00, and both satisfy

$G$01

This places categorical entropy above a purely Hochschild-theoretic growth invariant and supplies another obstruction to Gromov–Yomdin type formulas (Kikuta et al., 2020).

In symplectic topology, compactly supported exact symplectic automorphisms induce autoequivalences of Fukaya categories. For a Weinstein manifold $G$02 and a compactly supported exact symplectic automorphism $G$03, the induced autoequivalence $G$04 of the wrapped Fukaya category satisfies

$G$05

The proof uses a fully stopped partially wrapped category, a Lefschetz-fibration generator, intersection-count estimates via a Lagrangian tomograph and Crofton inequality, and Yomdin’s volume-growth bound. The same work introduces barcode entropy with

$G$06

for suitable Lagrangian pairs (Bae et al., 2022).

A parallel symplectic viewpoint proves localization formulas and Floer-theoretic entropy formulas for compact and wrapped Fukaya categories, especially under a duality condition relating compact generators $G$07 and wrapped generators $G$08. In plumbing spaces $G$09, Penner-type symplectic automorphisms satisfy

$G$10

reducing entropy to the spectral radius of an explicit nonnegative matrix (Bae et al., 2022).

A more conjectural extension connects categorical entropy to statistical mechanics. For a saturated $G$11-category, the expression for

$G$12

is reinterpreted through a one-dimensional lattice model built from a bimodule resolution. Under Condition B, the exponential of the entropy density is an algebraic integer, and the paper states that categorical entropy corresponds to the von Neumann entropy of a gauged lattice model (Wu et al., 24 May 2025).

6. Other categorical meanings of entropy

Outside the DHKK framework, categorical entropy also refers to structurally different notions. In a well-powered cocomplete abelian category $G$13, an entropy function is a map

$G$14

satisfying isomorphism invariance, an exactness criterion for vanishing entropy, and a coproduct criterion for vanishing entropy. The associated Pinsker radical

$G$15

is a hereditary radical, and binary entropy functions are in bijective order-preserving correspondence with hereditary torsion theories (Dikranjan et al., 2010).

In categorical probability theory, relative entropy is realized as a functor

$G$16

on a category of standard Borel probability spaces and coherent measurement-and-hypothesis pairs $G$17. For a coherent morphism,

$G$18

and functoriality gives

$G$19

The same work proves convex linearity, lower semicontinuity, and uniqueness up to a multiplicative constant (Gagne et al., 2017).

For finite categories equipped with probabilistic data, one can define

$G$20

This recovers Shannon entropy for discrete categories and recovers $G$21 when $G$22 is the normalized weighting associated to magnitude. The construction is additive under products and satisfies a chain rule for weighted coproducts (Chen et al., 2023).

In a distinct statistical usage, entropy over finite categorical state spaces is the objective in maximum-entropy reconstruction of categorical distributions under marginal constraints. For a joint distribution $G$23 on the finite state space $G$24,

$G$25

and among all distributions consistent with the prescribed marginals there exists a unique maximizer

$G$26

Iterative proportional fitting converges to this unique maximum-entropy distribution directly in probability space (Loukas et al., 2022).

Across these usages, the common theme is that entropy encodes complexity, uncertainty, or information loss through categorical structure. In the DHKK-derived setting, that structure is dynamical and triangulated; in the other settings, it is abelian, probabilistic, combinatorial, or statistical.