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Journal articles on the topic 'Homotopy coherence'

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Published: 2 June 2025

Last updated: 31 July 2025

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1

Porter, Tim, and Jim Stasheff. "Homotopy Coherent Representations." Symmetry 14, no. 3 (2022): 553. http://dx.doi.org/10.3390/sym14030553.

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Homotopy coherence has a considerable history, albeit also by other names. For this volume highlighting symmetries, the appropriate use is homotopy coherence of representations, at one time known as representations up to homotopy/homotopy coherent representations. We present a brief semi-historical survey, providing some links that may not be common knowledge.

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2

Geoghegan, Ross, and Andrew Nicas. "Homotopy periodicity and coherence." Proceedings of the American Mathematical Society 124, no. 9 (1996): 2889–95. http://dx.doi.org/10.1090/s0002-9939-96-03543-5.

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3

Hardie, K. A., K. H. Kamps, and P. J. Witbooi. "A Note On Homotopy Pushout And Homotopy Coherence." Quaestiones Mathematicae 26, no. 4 (2003): 399–403. http://dx.doi.org/10.2989/16073600309486070.

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4

Yanofsky, Noson S. "Coherence, Homotopy and 2-Theories." K-Theory 23, no. 3 (2001): 203–35. http://dx.doi.org/10.1023/a:1011893700822.

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5

YALIN, SINAN. "Simplicial localisation of homotopy algebras over a prop." Mathematical Proceedings of the Cambridge Philosophical Society 157, no. 3 (2014): 457–68. http://dx.doi.org/10.1017/s0305004114000437.

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AbstractWe prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer–Kan equivalence between the simplicial localisations of the associated categories of algebras. This homotopy invariance under base change implies that the homotopy category of homotopy algebras over a propPdoes not depend on the choice of a cofibrant resolution ofP, and gives thus a coherence to the notion of algebra up to homotopy in this setting. The result is established more generally for algebras in combinatorial monoidal dg categories.

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6

Dugger, Daniel. "Coherence for invertible objects and multigraded homotopy rings." Algebraic & Geometric Topology 14, no. 2 (2014): 1055–106. http://dx.doi.org/10.2140/agt.2014.14.1055.

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7

Kashiwabara, Takuji. "Mod p K-theory of Ω∞Σ∞X revisited". Mathematical Proceedings of the Cambridge Philosophical Society 114, № 2 (1993): 219–21. http://dx.doi.org/10.1017/s0305004100071553.

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In this note we present a new proof of a theorem of McClure on K*(Ω∞Σ∞X, Z/p) [11], in the special case when X is a finite complex with K1(X; Z/p) = 0. Although our method does not work in the full generality covered by his work, our argument requires neither a geometric interpretation of complex k-theory nor all the delicate coherence properties of its multiplication. Since BP-theory is not likely to possess such coherence properties [9], the possibility of generalizing his approach to the case of higher Morava K-theory does not seem feasible. On the contrary, the main ingredient of our appro

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8

FIEDLER, THOMAS, and ARNAUD MORTIER. "ON HOMOTOPIES WITH TRIPLE POINTS OF CLASSICAL KNOTS." Journal of Knot Theory and Its Ramifications 21, no. 04 (2012): 1250038. http://dx.doi.org/10.1142/s0218216511009911.

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We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point p of the cylinder is called coherent if all three branches intersect at p pairwise with the same intersection index. A triple unknotting of a classical knot K is a homotopy which connects K with the trivial knot and which has as singularities only coherent triple points. We give a new formula for the first Vassiliev invariant v2(K) by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we show that

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9

Karar, Faten Ragab, Fatma Elzhraa Ahmed Mohammed, and A. A. El Fattah. "The excision theory for homology theory through A_∞-algebras." Edelweiss Applied Science and Technology 8, no. 6 (2024): 9472–86. https://doi.org/10.55214/25768484.v8i6.4026.

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This paper investigated A_∞-algebras, which are generalizations of associative algebras that incorporate higher homotopy structures. We began by revisiting the fundamental definitions and properties of A_∞-algebras and their associated homological theories, providing a solid foundation for understanding these complex structures. The study included an in-depth analysis of simplicial homology as it relates to A_∞-algebras, focusing on significant results, particularly those concerning excision theory. In this context, we introduced new insights into the relationship between bar homology and simp

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10

Fisette, Robert, and Alexander Polishchuk. "-algebras associated with curves and rational functions on . I." Compositio Mathematica 150, no. 4 (2014): 621–67. http://dx.doi.org/10.1112/s0010437x13007574.

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AbstractWe consider the natural$A_{\infty }$-structure on the$\mathrm{Ext}$-algebra$\mathrm{Ext}^*(G,G)$associated with the coherent sheaf$G=\mathcal{O}_C\oplus \mathcal{O}_{p_1}\oplus \cdots \oplus \mathcal{O}_{p_n}$on a smooth projective curve$C$, where$p_1,\ldots,p_n\in C$are distinct points. We study the homotopy class of the product$m_3$. Assuming that$h^0(p_1+\cdots +p_n)=1$, we prove that$m_3$is homotopic to zero if and only if$C$is hyperelliptic and the points$p_i$are Weierstrass points. In the latter case we show that$m_4$is not homotopic to zero, provided the genus of$C$is greater th

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11

Cordier, Jean-Marc, and Timothy Porter. "Homotopy coherent category theory." Transactions of the American Mathematical Society 349, no. 1 (1997): 1–54. http://dx.doi.org/10.1090/s0002-9947-97-01752-2.

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12

Mardešić, Sibe. "Coherent homotopy and localization." Topology and its Applications 94, no. 1-3 (1999): 253–74. http://dx.doi.org/10.1016/s0166-8641(98)00034-0.

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13

Cordier, Jean-Marc, and Timothy Porter. "Vogt's theorem on categories of homotopy coherent diagrams." Mathematical Proceedings of the Cambridge Philosophical Society 100, no. 1 (1986): 65–90. http://dx.doi.org/10.1017/s0305004100065877.

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Let Top be the category of compactly generated topological spaces and continuous maps. The category, Top, can be given the structure of a simplicially enriched category (or S-category, S being the category of simplicial sets). For A a small category, Vogt (in [22]) constructed a category, Coh (A, Top), of homotopy coherent A-indexed diagrams in Top and homotopy classes of homotopy coherent maps, and proved a theorem identifying this as being equivalent to Ho (TopA), the category obtained from the category of commutative A-indexed diagrams by localizing with respect to the level homotopy equiva

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14

Cordier, Jean Marc, and Timothy Porter. "Maps between homotopy coherent diagrams." Topology and its Applications 28, no. 3 (1988): 255–75. http://dx.doi.org/10.1016/0166-8641(88)90046-6.

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15

Hosseini, Esmaeil. "Flat quasi-coherent sheaves of finite cotorsion dimension." Journal of Algebra and Its Applications 16, no. 01 (2017): 1750015. http://dx.doi.org/10.1142/s0219498817500153.

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Let [Formula: see text] be a quasi-compact and semi-separated scheme. If every flat quasi-coherent sheaf has finite cotorsion dimension, we prove that [Formula: see text] is [Formula: see text]-perfect for some [Formula: see text]. If [Formula: see text] is coherent and [Formula: see text]-perfect (not necessarily of finite Krull dimension), we prove that every flat quasi-coherent sheaf has finite pure injective dimension. Also, we show that there is an equivalence [Formula: see text] of homotopy categories, whenever [Formula: see text] is the homotopy category of pure injective flat quasi-coh

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16

Bašić, Matija, and Thomas Nikolaus. "Dendroidal sets as models for connective spectra." Journal of K-theory 14, no. 3 (2014): 387–421. http://dx.doi.org/10.1017/is014005003jkt265.

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AbstractDendroidal sets have been introduced as a combinatorial model for homotopy coherent operads. We introduce the notion of fully Kan dendroidal sets and show that there is a model structure on the category of dendroidal sets with fibrant objects given by fully Kan dendroidal sets. Moreover we show that the resulting homotopy theory is equivalent to the homotopy theory of connective spectra.

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17

Chu, Hongyi, and Rune Haugseng. "Homotopy-coherent algebra via Segal conditions." Advances in Mathematics 385 (July 2021): 107733. http://dx.doi.org/10.1016/j.aim.2021.107733.

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18

Oh, Yong-Geun, and Hiro Lee Tanaka. "Smooth constructions of homotopy-coherent actions." Algebraic & Geometric Topology 22, no. 3 (2022): 1177–216. http://dx.doi.org/10.2140/agt.2022.22.1177.

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19

Krause, Henning. "Coherent functors in stable homotopy theory." Fundamenta Mathematicae 173, no. 1 (2002): 33–56. http://dx.doi.org/10.4064/fm173-1-3.

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20

Gille, Stefan. "Homotopy invariance of coherent Witt groups." Mathematische Zeitschrift 244, no. 2 (2003): 211–33. http://dx.doi.org/10.1007/s00209-003-0489-z.

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21

Arakawa, Kensuke. "Classifying space via homotopy coherent nerve." Homology, Homotopy and Applications 25, no. 2 (2023): 373–81. http://dx.doi.org/10.4310/hha.2023.v25.n2.a16.

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22

Estrada, Sergio, and James Gillespie. "The projective stable category of a coherent scheme." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 149, no. 1 (2018): 15–43. http://dx.doi.org/10.1017/s0308210517000385.

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We define the projective stable category of a coherent scheme. It is the homotopy category of an abelian model structure on the category of unbounded chain complexes of quasi-coherent sheaves. We study the cofibrant objects of this model structure, which are certain complexes of flat quasi-coherent sheaves satisfying a special acyclicity condition.

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23

Walde, Tashi. "Homotopy coherent theorems of Dold–Kan type." Advances in Mathematics 398 (March 2022): 108175. http://dx.doi.org/10.1016/j.aim.2021.108175.

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24

Campbell, Alexander. "A homotopy coherent cellular nerve for bicategories." Advances in Mathematics 368 (July 2020): 107138. http://dx.doi.org/10.1016/j.aim.2020.107138.

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25

Nagata, Masato. "Homotopy between exact coherent structures in shear flows." Meccanica 51, no. 12 (2016): 3015–23. http://dx.doi.org/10.1007/s11012-016-0518-8.

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26

WALEFFE, FABIAN. "Exact coherent structures in channel flow." Journal of Fluid Mechanics 435 (May 25, 2001): 93–102. http://dx.doi.org/10.1017/s0022112001004189.

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Exact coherent states in no-slip plane Poiseuille flow are calculated by homotopy from free-slip to no-slip boundary conditions. These coherent states are unstable travelling waves. They consist of wavy low-speed streaks flanked by staggered streamwise vortices closely resembling the asymmetric coherent structures observed in the near-wall region of turbulent flows. The travelling waves arise from a saddle-node bifurcation at a sub-turbulent Reynolds number with wall-normal, spanwise and streamwise dimensions smaller than but comparable to 50+, 100+ and 250+, respectively. These coherent solut

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27

Nikolaus, Thomas. "Algebraic K-Theory of ∞-Operads." Journal of K-theory 14, no. 3 (2014): 614–41. http://dx.doi.org/10.1017/is014008019jkt277.

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AbstractThe theory of dendroidal sets has been developed to serve as a combinatorial model for homotopy coherent operads, see [MW07, CM13b]. An ∞-operad is a dendroidal setDsatisfying certain lifting conditions.In this paper we give a definition of K-groupsKn(D) for a dendroidal setD. These groups generalize the K-theory of symmetric monoidal (resp. permutative) categories and algebraic K-theory of rings. We establish some useful properties like invariance under the appropriate equivalences and long exact sequences which allow us to compute these groups in some examples. Using results from [He

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28

Baues, H. J., K. A. Hardie, and K. H. Kamps. "The self-equivalence groups in certain coherent homotopy categories." Tsukuba Journal of Mathematics 21, no. 1 (1997): 213–28. http://dx.doi.org/10.21099/tkbjm/1496163173.

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29

Schwänzl, R., and R. M. Vogt. "E∞-monoids with coherent homotopy inverses are Abelian groups." Topology 28, no. 4 (1989): 481–84. http://dx.doi.org/10.1016/0040-9383(89)90006-2.

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30

Riehl, Emily, and Dominic Verity. "Homotopy coherent adjunctions and the formal theory of monads." Advances in Mathematics 286 (January 2016): 802–88. http://dx.doi.org/10.1016/j.aim.2015.09.011.

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31

Waleffe, Fabian. "Homotopy of exact coherent structures in plane shear flows." Physics of Fluids 15, no. 6 (2003): 1517. http://dx.doi.org/10.1063/1.1566753.

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32

Hardie, K. A., K. H. Kamps, and P. J. Witbooi. "A Coherent Homotopy Category of 2-track Commutative Cubes." Applied Categorical Structures 19, no. 1 (2008): 39–60. http://dx.doi.org/10.1007/s10485-008-9174-z.

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33

Groth, Moritz, and Jan Šťovíček. "Tilting theory via stable homotopy theory." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 743 (2018): 29–90. http://dx.doi.org/10.1515/crelle-2015-0092.

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Abstract We show that certain tilting results for quivers are formal consequences of stability, and as such are part of a formal calculus available in any abstract stable homotopy theory. Thus these results are for example valid over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory and also in the equivariant, motivic or parametrized variant thereof. In further work, we will continue developing this calculus and obtain additional abstract tilting results. Here, we also deduce an additional characterization of stability

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34

Jiang, Yi-Jing, Ping-Hong Lai, and Xin Huang. "Interhemispheric functional in age-related macular degeneration patient: a resting-state functional MRI study." NeuroReport 35, no. 10 (2024): 621–26. http://dx.doi.org/10.1097/wnr.0000000000002045.

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Age-related macular degeneration (AMD) is a prevalent disease leading to severe visual impairment in the elderly population. Despite this, the pathogenesis of AMD remains largely unexplored. The application of resting-state functional MRI (rs-fMRI) allows for the detection of coherent intrinsic brain activities along with the interactions taking place between the two hemispheres. In the frame of our study, we utilize voxel-mirrored homotopic connectivity (VMHC) as an rs-fMRI method to carry out a comparative analysis of functional homotopy between the two hemispheres with the aim of further un

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35

Batanin, Mikhail A. "Homotopy coherent category theory and A∞-structures in monoidal categories." Journal of Pure and Applied Algebra 123, no. 1-3 (1998): 67–103. http://dx.doi.org/10.1016/s0022-4049(96)00084-9.

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36

Riehl, Emily, and Dominic Verity. "Recognizing Quasi-Categorical Limits and Colimits in Homotopy Coherent Nerves." Applied Categorical Structures 28, no. 4 (2020): 669–716. http://dx.doi.org/10.1007/s10485-020-09594-x.

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37

Ševera, Pavol, and Michal Širaň. "Integration of Differential Graded Manifolds." International Mathematics Research Notices 2020, no. 20 (2019): 6769–814. http://dx.doi.org/10.1093/imrn/rnz004.

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Abstract We consider the problem of integration of $L_\infty $-algebroids (differential non-negatively graded manifolds) to $L_\infty $-groupoids. We first construct a “big” Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer–Cartan equation. The main analytic trick in our work is an integral transformation sending the solutions of the Maurer–Cartan equation to closed differential forms. Following the ideas of Ezra Getzler, we then impose a gauge condition that cuts out a finite-dimensional simplicial submanifold. This “smaller” simplicial manifold

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38

Liu, Chenyi, Alexander Wong, Kostadinka Bizheva, Paul Fieguth, and Hongxia Bie. "Homotopic, non-local sparse reconstruction of optical coherence tomography imagery." Optics Express 20, no. 9 (2012): 10200. http://dx.doi.org/10.1364/oe.20.010200.

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39

Alonso Tarrío, Leovigildo, Ana Jeremías López, Marta Pérez Rodríguez, and María J. Vale Gonsalves. "The derived category of quasi-coherent sheaves and axiomatic stable homotopy." Advances in Mathematics 218, no. 4 (2008): 1224–52. http://dx.doi.org/10.1016/j.aim.2008.03.011.

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40

GILLESPIE, JAMES. "AC-GORENSTEIN RINGS AND THEIR STABLE MODULE CATEGORIES." Journal of the Australian Mathematical Society 107, no. 02 (2018): 181–98. http://dx.doi.org/10.1017/s1446788718000290.

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We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring that is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo–Gillespie–Hovey. It is also compatible with the notion of $n$ -coherent rings introduced by Bravo–Perez. So a $0$ -coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$ -coherent AC-Gorenstein ring is precisely a Ding–Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category o

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41

Hosseini, E., and Sh Salarian. "A cotorsion theory in the homotopy category of flat quasi-coherent sheaves." Proceedings of the American Mathematical Society 141, no. 3 (2012): 753–62. http://dx.doi.org/10.1090/s0002-9939-2012-11364-4.

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42

BRADLOW, S. B., O. GARCÍA-PRADA, V. MERCAT, V. MUÑOZ, and P. E. NEWSTEAD. "ON THE GEOMETRY OF MODULI SPACES OF COHERENT SYSTEMS ON ALGEBRAIC CURVES." International Journal of Mathematics 18, no. 04 (2007): 411–53. http://dx.doi.org/10.1142/s0129167x07004151.

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Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E,V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent systems for different values of α when k ≤ n and the variation of the moduli spaces when we vary α. As a consequence, for sufficiently large α, we compute the Picard groups and the first and second homotopy groups of the moduli spaces of coherent systems i

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43

Wang, Junpeng, and Zhenxing Di. "Relative Gorenstein rings and duality pairs." Journal of Algebra and Its Applications 19, no. 08 (2019): 2050147. http://dx.doi.org/10.1142/s0219498820501479.

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Let [Formula: see text] be a ring (not necessarily commutative) and [Formula: see text] a bi-complete duality pair. We investigate the notions of (flat-typed) [Formula: see text]-Gorenstein rings, which unify Iwanaga–Gorenstein rings, Ding–Chen rings, AC-Gorenstein rings and Gorenstein [Formula: see text]-coherent rings. Using an abelian model category approach, we show that [Formula: see text] and [Formula: see text], the homotopy categories of all exact complexes of projective and injective [Formula: see text]-modules, are triangulated equivalent whenever [Formula: see text] is a flat-typed

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44

Braunling, Oliver. "K-Theory of Locally Compact Modules over Rings of Integers." International Mathematics Research Notices 2020, no. 6 (2018): 1748–93. http://dx.doi.org/10.1093/imrn/rny083.

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Abstract We generalize a recent result of Clausen; for a number field with integers $\mathcal{O}$, we compute the K-theory of locally compact $\mathcal{O}$-modules. For the rational integers this recovers Clausen’s result as a special case. Our method of proof is quite different; instead of a homotopy coherent cone construction in $\infty$-categories, we rely on calculus of fraction type results in the style of Schlichting. This produces concrete exact category models for certain quotients, a fact that might be of independent interest. As in Clausen’s work, our computation works for all locali

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45

RIEHL, EMILY. "On the structure of simplicial categories associated to quasi-categories." Mathematical Proceedings of the Cambridge Philosophical Society 150, no. 3 (2011): 489–504. http://dx.doi.org/10.1017/s0305004111000053.

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AbstractThe homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint are important to the study of (∞, 1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor also gives a cofibrant replacement for ordinary categories, regarded as trivial simplicial categories. However, the hom-spaces of the simplicial category X arising from a quasi-category X are not well understood. We show that when X is a

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46

Vladimirov, Igor G., and Ian R. Petersen. "A Homotopy Approach to Coherent Quantum LQG Control Synthesis Using Discounted Performance Criteria." IFAC-PapersOnLine 54, no. 9 (2021): 166–71. http://dx.doi.org/10.1016/j.ifacol.2021.06.072.

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Hardie, K. A., K. H. Kamps, and T. Porter. "The coherent homotopy category over a fixed space is a category of fractions." Topology and its Applications 40, no. 3 (1991): 265–74. http://dx.doi.org/10.1016/0166-8641(91)90109-y.

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48

Schweigert, Christoph, and Lukas Woike. "Homotopy coherent mapping class group actions and excision for Hochschild complexes of modular categories." Advances in Mathematics 386 (August 2021): 107814. http://dx.doi.org/10.1016/j.aim.2021.107814.

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49

Wall, D. P., and M. Nagata. "Exact coherent states in channel flow." Journal of Fluid Mechanics 788 (January 8, 2016): 444–68. http://dx.doi.org/10.1017/jfm.2015.685.

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Three spatially extended travelling wave exact coherent states, together with one spanwise-localised state, are presented for channel flow. Two of the extended flows are derived by homotopy from solutions to the problem of channel flow subject to a spanwise rotation investigated by Wall & Nagata (J. Fluid Mech., vol. 727, 2013, pp. 523–581). Both these flows are asymmetric with respect to the channel centreplane, and feature streaky structures in streamwise velocity flanked by staggered vortical structures. One of these flows features two streak/vortex systems per spanwise wavelength, whil

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50

Deguchi, Kengo, and Philip Hall. "Free-stream coherent structures in parallel boundary-layer flows." Journal of Fluid Mechanics 752 (July 9, 2014): 602–25. http://dx.doi.org/10.1017/jfm.2014.282.

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AbstractOur concern in this paper is with high-Reynolds-number nonlinear equilibrium solutions of the Navier–Stokes equations for boundary-layer flows. Here we consider the asymptotic suction boundary layer (ASBL) which we take as a prototype parallel boundary layer. Solutions of the equations of motion are obtained using a homotopy continuation from two known types of solutions for plane Couette flow. At high Reynolds numbers, it is shown that the first type of solution takes the form of a vortex–wave interaction (VWI) state, see Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666),

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