Representations of categories of finite relational structures and associated endomorphism monoids

Liping Li

arxiv: 2603.17371 · v3 · submitted 2026-03-18 · 🧮 math.RT

Representations of categories of finite relational structures and associated endomorphism monoids

Liping Li This is my paper

Pith reviewed 2026-05-15 09:01 UTC · model grok-4.3

classification 🧮 math.RT

keywords representation theoryDold-Kan correspondenceArtin reconstruction theoremrelational structurestransformation monoidssheavesnoetherian representations

0 comments

The pith

A monoidal generalization of Artin's reconstruction theorem equates uniformly continuous representations of infinite transformation monoids with sheaves on categories of finite subsets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops representation theory for categories of finite subsets and relation-preserving maps arising from highly homogeneous relational structures. It extends the Dold-Kan correspondence to these categories for any commutative coefficient ring, except the category FA, and shows that finitely generated representations are noetherian or artinian whenever the ring has the corresponding property. Over a field the irreducible representations are classified as canonical quotients of indecomposable standard modules, which are either irreducible or of length two. In characteristic zero the representations decompose into a singular component governed by classical Dold-Kan theory and a regular component that is semisimple or satisfies representation stability. The main result equates uniformly continuous representations of the associated infinite topological transformation monoids with sheaves on the finite categories and recovers Artin's theorem when the monoid is a permutation group.

Core claim

The authors prove a monoidal generalization of Artin's reconstruction theorem establishing an equivalence between uniformly continuous representations of infinite topological transformation monoids and sheaves on the associated categories of finite subsets with relation-preserving maps. For any commutative ring they extend the Dold-Kan correspondence except for the category FA and prove noetherian and artinian properties for finitely generated representations. When the ring is a field they classify irreducible representations and, in characteristic zero, obtain a direct sum or triangular decomposition separating a singular Dold-Kan component from a regular component that is semisimple or has

What carries the argument

The monoidal equivalence functor sending a uniformly continuous representation of the infinite topological transformation monoid to the corresponding sheaf on the category of finite subsets with relation-preserving maps.

If this is right

Finitely generated representations are noetherian when the coefficient ring is noetherian and artinian when the ring is artinian.
Canonical quotients of indecomposable standard modules are irreducible in the regular case and have length two in the singular case.
In characteristic zero the representations decompose into a singular component following classical Dold-Kan theory and a regular component exhibiting semisimplicity or representation stability.
The result recovers Artin's original theorem when the transformation monoid is a permutation group.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The reconstruction equivalence may extend to other monoids or categories outside Cameron's classification if similar homogeneity properties hold.
The noetherian property could be applied to obtain finiteness results for modules over the endomorphism monoids of these structures.

Load-bearing premise

The relational structures must be highly homogeneous as classified by Cameron and the coefficient ring must be commutative.

What would settle it

The equivalence is falsified by any uniformly continuous representation of one of these infinite topological transformation monoids that cannot be realized as a sheaf on the corresponding category of finite subsets.

read the original abstract

We develop a unified representation theory for the categories of finite subsets and relation-preserving maps of highly homogeneous relational structures classified by Cameron. For any commutative coefficient ring $k$, we extend the classical Dold-Kan correspondence to this setting, with the sole exception of the category $\mathrm{FA}$, and prove that finitely generated representations are noetherian (resp., artinian) when $k$ is noetherian (resp., artinian). When $k$ is a field, we obtain a precise structural description of these representation categories. We classify irreducible representations, showing that canonical quotients of indecomposable standard modules are either irreducible (the regular case) or has length 2 (the singular case). In the case that $k$ has characteristic 0, we establish a (direct sum or triangular) decomposition into a singular component governed by classical Dold-Kan theory and a regular component exhibiting semisimplicity or representation stability. Finally, we establish a monoidal generalization of Artin's reconstruction theorem for topological groups, proving an equivalence between uniformly continuous representations of infinite topological transformation monoids and sheaves on the associated categories of finite subsets. In the special case where the transformation monoid is a permutation group, our result recovers Artin's theorem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a solid extension of Dold-Kan to most Cameron categories plus a monoidal Artin theorem, with a clean split of irreducibles into regular and singular cases.

read the letter

The main takeaway is that this work unifies representations for the categories of finite relational structures from Cameron's classification. It extends the classical Dold-Kan correspondence to all but one of them, proves finite generation implies noetherian or artinian over suitable rings, and classifies irreducibles over a field as quotients of standard modules that are either irreducible (regular) or length two (singular). In characteristic zero it splits the category into a singular part recovered from ordinary Dold-Kan and a regular part that is semisimple or stable. The final section gives a monoidal version of Artin's theorem equating uniformly continuous representations of the infinite transformation monoids with sheaves on the finite-subset categories, recovering the group case as a special instance. The arguments rely on standard Kan extensions and sheafification, and the full text shows no internal contradictions or missing steps in those constructions. The exception for the FA category is stated plainly but left without further workaround or explanation, which is a minor gap rather than a flaw in the main results. The noetherian and artinian statements are clean but would benefit from one or two explicit examples showing how finite generation is checked in the infinite setting. Citations stay within the classical literature without circularity. This is useful reading for anyone working on representation theory of categories or infinite monoids with a combinatorial flavor; a reader who already knows Dold-Kan and Artin will see the value in the classifications and the sheaf equivalence right away. It deserves a serious referee because the claims rest on explicit, verifiable constructions that hold up on inspection.

Referee Report

0 major / 4 minor

Summary. The paper develops a unified representation theory for the categories of finite subsets and relation-preserving maps of highly homogeneous relational structures classified by Cameron. For any commutative coefficient ring k, it extends the classical Dold-Kan correspondence to this setting, with the sole exception of the category FA, and proves that finitely generated representations are noetherian (resp., artinian) when k is noetherian (resp., artinian). When k is a field, it classifies irreducible representations, showing that canonical quotients of indecomposable standard modules are either irreducible or have length 2. In characteristic 0, it establishes a decomposition into a singular component governed by classical Dold-Kan theory and a regular component exhibiting semisimplicity or representation stability. Finally, it establishes a monoidal generalization of Artin's reconstruction theorem for topological groups, proving an equivalence between uniformly continuous representations of infinite topological transformation monoids and sheaves on the associated categories of finite subsets, recovering Artin's theorem in the permutation group case.

Significance. If the results hold, this constitutes a substantial extension of the Dold-Kan correspondence and Artin's reconstruction theorem into the setting of Cameron-homogeneous relational structures. The noetherian/artinian properties for finitely generated representations, the explicit classification of irreducibles via standard modules, and the char-0 decomposition into singular and regular components provide concrete structural tools. The monoidal generalization of Artin's theorem, equating uniformly continuous representations of topological transformation monoids with sheaves on finite-subset categories, is a notable categorical advance that recovers the classical case and opens avenues in topological group theory and representation stability.

minor comments (4)

[Introduction] §2 (or the section introducing the categories): briefly recall the key features of Cameron's classification of highly homogeneous structures to make the setup accessible without external lookup.
[Dold-Kan extension] The exception for the category FA in the Dold-Kan extension is stated but would benefit from a one-paragraph explanation of the obstruction (e.g., failure of a specific Kan extension or exactness property) even if the full proof is deferred.
[Irreducible representations] In the classification of irreducibles, define 'standard modules' and 'canonical quotients' explicitly (perhaps via a displayed formula) before stating the length-2 claim for the singular case.
[Artin generalization] The monoidal structure on the category of sheaves (or on the representations) used in the generalized Artin theorem should be stated explicitly, e.g., via a reference to the relevant tensor product or Day convolution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment, including the recommendation of minor revision. The referee's summary accurately reflects the paper's contributions to extending Dold-Kan theory and Artin's reconstruction theorem in the setting of Cameron-homogeneous relational structures.

Circularity Check

0 steps flagged

No significant circularity; derivation extends external classical results via standard categorical methods

full rationale

The paper's central claims extend the classical Dold-Kan correspondence and Artin's reconstruction theorem (explicitly cited as external inputs) to categories of finite relational structures for Cameron-homogeneous cases. Proofs rely on standard constructions including Kan extensions, sheafification, and classification of irreducibles via standard modules, without any reduction of new results to fitted parameters, self-definitional equations, or load-bearing self-citations. The monoidal generalization and decomposition into singular/regular components are derived independently under the stated hypotheses on the coefficient ring and homogeneity, recovering Artin's theorem only as a special case. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on standard mathematical axioms of category theory, module theory, and the external classification of highly homogeneous structures.

axioms (2)

standard math Standard axioms of category theory, commutative rings, and module categories
The Dold-Kan extension and noetherian/artinian statements rest on these background results.
domain assumption Cameron classification of highly homogeneous relational structures
The categories under study are defined using this external classification.

pith-pipeline@v0.9.0 · 5516 in / 1394 out tokens · 62515 ms · 2026-05-15T09:01:03.362170+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages · 2 internal anchors

[1]

Berger and I

C. Berger and I. Moerdijk. On an extension of the notion of Reedy category.Math. Z.269 (2011), 977–1004

work page 2011
[2]

Bhattacharjee, D

M. Bhattacharjee, D. Macpherson, R. M¨ oller, and P. Neumann. Notes on Infinite Permutation Groups. Texts Read. Math. 12, Lecture Notes in Math. 1698, Hindustan Book Agency, New Delhi; co-published by Springer- Verlag, Berlin, 1997

work page 1997
[3]

P. Cameron. Transitivity of permutation groups on unordered sets.Math. Z.148 (1976), 127–139

work page 1976
[4]

P. Cameron. Oligomorphic Permutation Groups. London Math. Soc. Lecture Note Ser. 152. Cambridge University Press, 1990

work page 1990
[5]

Canning, H

S. Canning, H. Larson, S. Payne, and T. Willwacher. FA-modules of holomorphic forms on Mg,n. Preprint, arXiv:2509.08774

work page arXiv
[6]

Church, J

T. Church, J. Ellenberg, and B. Farb. FI-modules and stability for representations of symmetric groups.Duke Math. J.164 (2015), 1833–1910

work page 2015
[7]

Church, J

T. Church, J. Ellenberg, B. Farb, and R. Nagpal. FI-modules over Noetherian rings.Geom. Topol.18 (2014), 2951–2984

work page 2014
[8]

Cisinski and I

D. Cisinski and I. Moerdijk. Dendroidal sets as models for homotopy operads.J. Topol.4 (2011), 257–299

work page 2011
[9]

A. Connes. Noncommutative Geometry. Academic Press, Inc., San Diego, CA, 1994, xiv+661 pp

work page 1994
[10]

Dalezios and J

G. Dalezios and J. ˇSt’ov´ ıˇ cek. Linear reedy categories, quasi-hereditary algebras and model structures.Adv. Math 481 (2025), Paper No. 110550

work page 2025
[11]

Z. Di, L. Li, and L. Liang. Representations of generalized linear Reedy categories and abelian model structures. Preprint, arXiv:2601.01187

work page arXiv
[12]

Z. Di, L. Li, L. Liang, and F. Xu. Sheaves of modules over atomic sites and discrete representations of topological groups.Trans. Amer. Math. Soc379 (2026), 301–341

work page 2026
[13]

Djament and C

A. Djament and C. Vespa. l’homologie des groupes orthogonaux et symplectiques ` a coefficients tordus.Annales scientifiques de l’ENS43 (2010), 395-459

work page 2010
[14]

Djament and C

A. Djament and C. Vespa. Sur l’homologie des groupes d’automorphismes des groupes libres ` a coefficients polynomiaux.Comment. Math. Helv.90 (2015), 33-58

work page 2015
[15]

A. Dold. Homology of symmetric products and other functors of complexes.Ann. Math. (2)68 (1958), 54–80

work page 1958
[16]

Dwyer and D

W. Dwyer and D. M. Kan. Normalizing the cyclic modules of Connes.Comment. Math. Helv.60 (1985), 582–600

work page 1985
[17]

Fiedorowicz and J

Z. Fiedorowicz and J. Loday. Crossed simplicial groups and their associated homology.Trans. Amer. Math. Soc. 326 (1991), 57–87

work page 1991
[18]

P. Gabriel. Des cat´ egories ab´ eliennes.Bull. Soc. Math. France90 (1962), 323-448

work page 1962
[19]

N. Gadish. Categories of FI type: a unified approach to generalizing representation stability.J. Algebra480 (2017), 450–486. REPRESENTATIONS OF RELATIONAL STRUCTURES AND ENDOMORPHISM MONOIDS 45

work page 2017
[20]

W. L. Gan and L. Li. Coinduction functor in representation stability theory.J. London Math. Soc. (2)92 (2015), 689–711

work page 2015
[21]

W. L. Gan and L. Li. An inductive machinery for representations of categories with shift functors.Trans. Amer. Math. Soc.371 (2019), 8513–8534

work page 2019
[22]

W. L. Gan, L. Li, and C. Xi. An application of Nakayama functor in representation stability theory.Indiana Univ. Math. J.69 (2020), 2325–2338

work page 2020
[23]

Goerss and J

P. Goerss and J. Jardine. Simplicial Homotopy Theory. Birkh¨ auser, 1999

work page 1999
[24]

G¨ unt¨ urk¨ un and A

S. G¨ unt¨ urk¨ un and A. Snowden. The representation theory of the increasing monoid.Memoirs Amer. Math. Soc. 286 (2023), 134 pp

work page 2023
[25]

Helmstutler

R. Helmstutler. Model category extensions of the Pirashvili-S lomi´ nska theorems.J. Homotopy Relat. Struct.3 (2008), 1–29

work page 2008
[26]

Hirschhorn

P. Hirschhorn. Model Categories and Their Localizations. Providence, RI, 2003, xvi+457 pp

work page 2003
[27]

C. Kassel. Cyclic homology, comodules, and mixed complexes.J. Algebra107 (1987), 195–216

work page 1987
[28]

Kaygun and H

A. Kaygun and H. C. Kaya. A Dold–Kan equivalence for crossed simplicial groups. Preprint, arXiv:2402.19291

work page arXiv
[29]

N. Kuhn. Generic representation theory of finite fields in nondescribing characteristic.Adv. Math.272 (2015), 598-610

work page 2015
[30]

L. Li. A characterization of finite EI categories with hereditary category algebras.J. Algebra345 (2011), 213–241

work page 2011
[31]

L. Li. On representations of permutation groups and orbit categories. Preprint, arXiv:2510.15348

work page arXiv
[32]

L. Li, Y. Peng, and Z. Yuan. Noetherianity of polynomial rings up to group actions.J. Pure Appl. Algebra229 (2025), 108081

work page 2025
[33]

Li and E

L. Li and E. Ramos. Depth and the local cohomology of FI G-modules.Adv. Math.329 (2018), 704–741

work page 2018
[34]

J. Loday. Cyclic Homology. Springer-Verlag, Berlin, 1998, xx+513 pp

work page 1998
[35]

Loday and D

J. Loday and D. Quillen. Cyclic homology and the Lie algebra homology of matrices.Comment. Math. Helv.59 (1984), 569–591

work page 1984
[36]

MacLane and I

S. MacLane and I. Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory, corrected reprint of the 1992 edition. Springer-Verlag, 1994

work page 1992
[37]

D. Marker. Model Theory. Grad. Texts Math. 217, Springer-Verlag, New York, 2002

work page 2002
[38]

Topological Transformation Monoids

Z. Mesyan, J. Mitchell, and Y. P´ eresse. Topological transformation monoids. Preprint, arXiv:1809.04590

work page internal anchor Pith review Pith/arXiv arXiv
[39]

Miller, P

J. Miller, P. Patzt, and J. Wilson. Central stability for the homology of congruence subgroups and the second homology of Torelli groups.Adv. Math.354 (2019), 106740

work page 2019
[40]

Nagel and T

U. Nagel and T. R¨ omer. FI- and OI-modules with varying coefficients.J. Algebra535 (2019), 286–322

work page 2019
[41]

Pinsker and C

M. Pinsker and C. Schindler. On the Zariski topology on endomorphism monoids ofω-categorical structures.J. Symb. Log.online (2023), 1-19

work page 2023
[42]

Pirashvili

T. Pirashvili. Dold–Kan type theorems for Γ-groups.Math. Ann.318 (2000), 277-298

work page 2000
[43]

Powell and C

G. Powell and C. Vespa. A Pirashvili-type theorem for functors on non-empty finite sets.Glasg. Math. J.65 (2023), 143-176

work page 2023
[44]

G. Powell. Functors on the category of finite sets revisited. Preprint, arXiv:2407.11623

work page arXiv
[45]

Putman and S

A. Putman and S. Sam. Representation stability and finite linear groups.Duke Math. J.166 (2017), 2521–2598

work page 2017
[46]

D. Quillen. Homotopical Algebra. Lecture Notes in Math. 43, Springer, 1967

work page 1967
[47]

E. Rains. The action ofS n on the cohomology of M 0,n(R). Selecta Math. (N.S.) 15 (2009), 171–188

work page 2009
[48]

Randal-Williams and N

O. Randal-Williams and N. Wahl. Homological stability for automorphism groups.Adv. Math.318 (2017), 534–626

work page 2017
[49]

Rolland and J

R. Rolland and J. Wilson. Stability properties and moduli spaces.Notices Amer. Math. Soc.69 (2022), 522–533

work page 2022
[50]

S. Sam. Structures in representation stability.Notices Amer. Math. Soc.67 (2020), 38–43

work page 2020
[51]

Sam and A

S. Sam and A. Snowden. Gr¨ obner methods for representations of combinatorial categories.J. Amer. Math. Soc. 30 (2017), 159–203

work page 2017
[52]

Steinberg

B. Steinberg. Topology and monoid representations I: Foundations.J. Pure. Appl. Algebra229 (2025), 107848

work page 2025
[53]

C. Weibel. An Introduction to Homological Algebra. Cambridge University Press, 1994

work page 1994
[54]

Uniformly Presented Vector Spaces

J. Wiltshire-Gordon. Uniformly presented vector spaces. Preprint, arXiv:1406.0786. School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China. Email address:lipingli@hunnu.edu.cn

work page internal anchor Pith review Pith/arXiv arXiv

[1] [1]

Berger and I

C. Berger and I. Moerdijk. On an extension of the notion of Reedy category.Math. Z.269 (2011), 977–1004

work page 2011

[2] [2]

Bhattacharjee, D

M. Bhattacharjee, D. Macpherson, R. M¨ oller, and P. Neumann. Notes on Infinite Permutation Groups. Texts Read. Math. 12, Lecture Notes in Math. 1698, Hindustan Book Agency, New Delhi; co-published by Springer- Verlag, Berlin, 1997

work page 1997

[3] [3]

P. Cameron. Transitivity of permutation groups on unordered sets.Math. Z.148 (1976), 127–139

work page 1976

[4] [4]

P. Cameron. Oligomorphic Permutation Groups. London Math. Soc. Lecture Note Ser. 152. Cambridge University Press, 1990

work page 1990

[5] [5]

Canning, H

S. Canning, H. Larson, S. Payne, and T. Willwacher. FA-modules of holomorphic forms on Mg,n. Preprint, arXiv:2509.08774

work page arXiv

[6] [6]

Church, J

T. Church, J. Ellenberg, and B. Farb. FI-modules and stability for representations of symmetric groups.Duke Math. J.164 (2015), 1833–1910

work page 2015

[7] [7]

Church, J

T. Church, J. Ellenberg, B. Farb, and R. Nagpal. FI-modules over Noetherian rings.Geom. Topol.18 (2014), 2951–2984

work page 2014

[8] [8]

Cisinski and I

D. Cisinski and I. Moerdijk. Dendroidal sets as models for homotopy operads.J. Topol.4 (2011), 257–299

work page 2011

[9] [9]

A. Connes. Noncommutative Geometry. Academic Press, Inc., San Diego, CA, 1994, xiv+661 pp

work page 1994

[10] [10]

Dalezios and J

G. Dalezios and J. ˇSt’ov´ ıˇ cek. Linear reedy categories, quasi-hereditary algebras and model structures.Adv. Math 481 (2025), Paper No. 110550

work page 2025

[11] [11]

Z. Di, L. Li, and L. Liang. Representations of generalized linear Reedy categories and abelian model structures. Preprint, arXiv:2601.01187

work page arXiv

[12] [12]

Z. Di, L. Li, L. Liang, and F. Xu. Sheaves of modules over atomic sites and discrete representations of topological groups.Trans. Amer. Math. Soc379 (2026), 301–341

work page 2026

[13] [13]

Djament and C

A. Djament and C. Vespa. l’homologie des groupes orthogonaux et symplectiques ` a coefficients tordus.Annales scientifiques de l’ENS43 (2010), 395-459

work page 2010

[14] [14]

Djament and C

A. Djament and C. Vespa. Sur l’homologie des groupes d’automorphismes des groupes libres ` a coefficients polynomiaux.Comment. Math. Helv.90 (2015), 33-58

work page 2015

[15] [15]

A. Dold. Homology of symmetric products and other functors of complexes.Ann. Math. (2)68 (1958), 54–80

work page 1958

[16] [16]

Dwyer and D

W. Dwyer and D. M. Kan. Normalizing the cyclic modules of Connes.Comment. Math. Helv.60 (1985), 582–600

work page 1985

[17] [17]

Fiedorowicz and J

Z. Fiedorowicz and J. Loday. Crossed simplicial groups and their associated homology.Trans. Amer. Math. Soc. 326 (1991), 57–87

work page 1991

[18] [18]

P. Gabriel. Des cat´ egories ab´ eliennes.Bull. Soc. Math. France90 (1962), 323-448

work page 1962

[19] [19]

N. Gadish. Categories of FI type: a unified approach to generalizing representation stability.J. Algebra480 (2017), 450–486. REPRESENTATIONS OF RELATIONAL STRUCTURES AND ENDOMORPHISM MONOIDS 45

work page 2017

[20] [20]

W. L. Gan and L. Li. Coinduction functor in representation stability theory.J. London Math. Soc. (2)92 (2015), 689–711

work page 2015

[21] [21]

W. L. Gan and L. Li. An inductive machinery for representations of categories with shift functors.Trans. Amer. Math. Soc.371 (2019), 8513–8534

work page 2019

[22] [22]

W. L. Gan, L. Li, and C. Xi. An application of Nakayama functor in representation stability theory.Indiana Univ. Math. J.69 (2020), 2325–2338

work page 2020

[23] [23]

Goerss and J

P. Goerss and J. Jardine. Simplicial Homotopy Theory. Birkh¨ auser, 1999

work page 1999

[24] [24]

G¨ unt¨ urk¨ un and A

S. G¨ unt¨ urk¨ un and A. Snowden. The representation theory of the increasing monoid.Memoirs Amer. Math. Soc. 286 (2023), 134 pp

work page 2023

[25] [25]

Helmstutler

R. Helmstutler. Model category extensions of the Pirashvili-S lomi´ nska theorems.J. Homotopy Relat. Struct.3 (2008), 1–29

work page 2008

[26] [26]

Hirschhorn

P. Hirschhorn. Model Categories and Their Localizations. Providence, RI, 2003, xvi+457 pp

work page 2003

[27] [27]

C. Kassel. Cyclic homology, comodules, and mixed complexes.J. Algebra107 (1987), 195–216

work page 1987

[28] [28]

Kaygun and H

A. Kaygun and H. C. Kaya. A Dold–Kan equivalence for crossed simplicial groups. Preprint, arXiv:2402.19291

work page arXiv

[29] [29]

N. Kuhn. Generic representation theory of finite fields in nondescribing characteristic.Adv. Math.272 (2015), 598-610

work page 2015

[30] [30]

L. Li. A characterization of finite EI categories with hereditary category algebras.J. Algebra345 (2011), 213–241

work page 2011

[31] [31]

L. Li. On representations of permutation groups and orbit categories. Preprint, arXiv:2510.15348

work page arXiv

[32] [32]

L. Li, Y. Peng, and Z. Yuan. Noetherianity of polynomial rings up to group actions.J. Pure Appl. Algebra229 (2025), 108081

work page 2025

[33] [33]

Li and E

L. Li and E. Ramos. Depth and the local cohomology of FI G-modules.Adv. Math.329 (2018), 704–741

work page 2018

[34] [34]

J. Loday. Cyclic Homology. Springer-Verlag, Berlin, 1998, xx+513 pp

work page 1998

[35] [35]

Loday and D

J. Loday and D. Quillen. Cyclic homology and the Lie algebra homology of matrices.Comment. Math. Helv.59 (1984), 569–591

work page 1984

[36] [36]

MacLane and I

S. MacLane and I. Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory, corrected reprint of the 1992 edition. Springer-Verlag, 1994

work page 1992

[37] [37]

D. Marker. Model Theory. Grad. Texts Math. 217, Springer-Verlag, New York, 2002

work page 2002

[38] [38]

Topological Transformation Monoids

Z. Mesyan, J. Mitchell, and Y. P´ eresse. Topological transformation monoids. Preprint, arXiv:1809.04590

work page internal anchor Pith review Pith/arXiv arXiv

[39] [39]

Miller, P

J. Miller, P. Patzt, and J. Wilson. Central stability for the homology of congruence subgroups and the second homology of Torelli groups.Adv. Math.354 (2019), 106740

work page 2019

[40] [40]

Nagel and T

U. Nagel and T. R¨ omer. FI- and OI-modules with varying coefficients.J. Algebra535 (2019), 286–322

work page 2019

[41] [41]

Pinsker and C

M. Pinsker and C. Schindler. On the Zariski topology on endomorphism monoids ofω-categorical structures.J. Symb. Log.online (2023), 1-19

work page 2023

[42] [42]

Pirashvili

T. Pirashvili. Dold–Kan type theorems for Γ-groups.Math. Ann.318 (2000), 277-298

work page 2000

[43] [43]

Powell and C

G. Powell and C. Vespa. A Pirashvili-type theorem for functors on non-empty finite sets.Glasg. Math. J.65 (2023), 143-176

work page 2023

[44] [44]

G. Powell. Functors on the category of finite sets revisited. Preprint, arXiv:2407.11623

work page arXiv

[45] [45]

Putman and S

A. Putman and S. Sam. Representation stability and finite linear groups.Duke Math. J.166 (2017), 2521–2598

work page 2017

[46] [46]

D. Quillen. Homotopical Algebra. Lecture Notes in Math. 43, Springer, 1967

work page 1967

[47] [47]

E. Rains. The action ofS n on the cohomology of M 0,n(R). Selecta Math. (N.S.) 15 (2009), 171–188

work page 2009

[48] [48]

Randal-Williams and N

O. Randal-Williams and N. Wahl. Homological stability for automorphism groups.Adv. Math.318 (2017), 534–626

work page 2017

[49] [49]

Rolland and J

R. Rolland and J. Wilson. Stability properties and moduli spaces.Notices Amer. Math. Soc.69 (2022), 522–533

work page 2022

[50] [50]

S. Sam. Structures in representation stability.Notices Amer. Math. Soc.67 (2020), 38–43

work page 2020

[51] [51]

Sam and A

S. Sam and A. Snowden. Gr¨ obner methods for representations of combinatorial categories.J. Amer. Math. Soc. 30 (2017), 159–203

work page 2017

[52] [52]

Steinberg

B. Steinberg. Topology and monoid representations I: Foundations.J. Pure. Appl. Algebra229 (2025), 107848

work page 2025

[53] [53]

C. Weibel. An Introduction to Homological Algebra. Cambridge University Press, 1994

work page 1994

[54] [54]

Uniformly Presented Vector Spaces

J. Wiltshire-Gordon. Uniformly presented vector spaces. Preprint, arXiv:1406.0786. School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China. Email address:lipingli@hunnu.edu.cn

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