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Journal articles on the topic 'Morita categories'

Author: Grafiati

Published: 14 September 2024

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1

Caviglia, Giovanni, and Javier J. Gutiérrez. "Morita homotopy theory for (∞,1)-categories and ∞-operads." Forum Mathematicum 31, no. 3 (May 1, 2019): 661–84. http://dx.doi.org/10.1515/forum-2018-0033.

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Abstract We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of {(\infty,1)} -categories and {\infty} -operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structure on simplicial categories and the Cisinski–Moerdijk model structure on simplicial operads, respectively.

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2

Gómez Pardo, J. L., and P. A. Guil Asensio. "Morita duality for Grothendieck categories." Publicacions Matemàtiques 36 (July 1, 1992): 625–35. http://dx.doi.org/10.5565/publmat_362a92_22.

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3

Rickard, Jeremy. "Morita Theory for Derived Categories." Journal of the London Mathematical Society s2-39, no. 3 (June 1989): 436–56. http://dx.doi.org/10.1112/jlms/s2-39.3.436.

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4

Greenlees, J. P. C., and Greg Stevenson. "Morita theory and singularity categories." Advances in Mathematics 365 (May 2020): 107055. http://dx.doi.org/10.1016/j.aim.2020.107055.

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5

Cline, E., B. Parshall, and L. Scott. "Derived categories and Morita theory." Journal of Algebra 104, no. 2 (December 1986): 397–409. http://dx.doi.org/10.1016/0021-8693(86)90224-3.

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6

DellʼAmbrogio, Ivo, and Gonçalo Tabuada. "Morita homotopy theory ofC⁎-categories." Journal of Algebra 398 (January 2014): 162–99. http://dx.doi.org/10.1016/j.jalgebra.2013.09.022.

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7

Anh, P. N., and R. Wiegandt. "Morita Duality for Grothendieck Categories." Journal of Algebra 168, no. 1 (August 1994): 273–93. http://dx.doi.org/10.1006/jabr.1994.1229.

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8

HOLSTEIN, JULIAN V. S. "Morita cohomology." Mathematical Proceedings of the Cambridge Philosophical Society 158, no. 1 (December 5, 2014): 1–26. http://dx.doi.org/10.1017/s0305004114000516.

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AbstractWe consider two categorifications of the cohomology of a topological spaceXby taking coefficients in the category of differential graded categories. We consider both derived global sections of a constant presheaf and singular cohomology and find the resulting dg-categories are quasi-equivalent and moreover quasi-equivalent to representations in perfect complexes of chains on the loop space ofX.

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9

Mazorchuk, Volodymyr, and Vanessa Miemietz. "Morita theory for finitary 2-categories." Quantum Topology 7, no. 1 (2016): 1–28. http://dx.doi.org/10.4171/qt/72.

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10

Wang, Pei. "Morita context functors on cellular categories." Communications in Algebra 47, no. 4 (January 31, 2019): 1773–84. http://dx.doi.org/10.1080/00927872.2018.1517360.

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11

Miyachi, Jun-ichi. "Derived categories and Morita duality theory." Journal of Pure and Applied Algebra 128, no. 2 (June 1998): 153–70. http://dx.doi.org/10.1016/s0022-4049(97)00046-7.

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12

Neshveyev, Sergey, and Makoto Yamashita. "A Few Remarks on the Tube Algebra of a Monoidal Category." Proceedings of the Edinburgh Mathematical Society 61, no. 3 (May 8, 2018): 735–58. http://dx.doi.org/10.1017/s0013091517000426.

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AbstractWe prove two results on the tube algebras of rigid C*-tensor categories. The first is that the tube algebra of the representation category of a compact quantum groupGis a full corner of the Drinfeld double ofG. As an application, we obtain some information on the structure of the tube algebras of the Temperley–Lieb categories 𝒯ℒ(d) ford> 2. The second result is that the tube algebras of weakly Morita equivalent C*-tensor categories are strongly Morita equivalent. The corresponding linking algebra is described as the tube algebra of the 2-category defining the Morita context.

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13

Dey, Krishanu, Sugato Gupta, and Sujit Kumar Sardar. "Morita invariants of semirings related to a Morita context." Asian-European Journal of Mathematics 12, no. 02 (April 2019): 1950023. http://dx.doi.org/10.1142/s1793557119500232.

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The main purpose of the paper is to consider two Morita equivalent semirings [Formula: see text] and [Formula: see text] via Morita context [Formula: see text] instead of considering them via the equivalence of the resulting semimodule categories and then to investigate various Morita invariants related to each of the pairs [Formula: see text]; [Formula: see text]; [Formula: see text]; [Formula: see text], etc.

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14

Hanihara, Norihiro. "Morita theorem for hereditary Calabi-Yau categories." Advances in Mathematics 395 (February 2022): 108092. http://dx.doi.org/10.1016/j.aim.2021.108092.

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15

Berbec, Ioan. "The Morita-Takeuchi Theory for Quotient Categories." Communications in Algebra 31, no. 2 (January 4, 2003): 843–58. http://dx.doi.org/10.1081/agb-120017346.

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16

Balaba, I. N. "Morita equivalences of categories of graded modules." Russian Mathematical Surveys 42, no. 3 (June 30, 1987): 209–10. http://dx.doi.org/10.1070/rm1987v042n03abeh001422.

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17

Naidu, Deepak. "Categorical Morita Equivalence for Group-Theoretical Categories." Communications in Algebra 35, no. 11 (October 23, 2007): 3544–65. http://dx.doi.org/10.1080/00927870701511996.

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18

Berger, Clemens, and Kruna Ratkovic. "Gabriel-Morita Theory for Excisive Model Categories." Applied Categorical Structures 27, no. 1 (August 25, 2018): 23–54. http://dx.doi.org/10.1007/s10485-018-9539-x.

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19

Décoppet, Thibault D. "The Morita Theory of Fusion 2-Categories." Higher Structures 7, no. 1 (May 21, 2023): 234–92. http://dx.doi.org/10.21136/hs.2023.07.

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20

费, 卿. "Generators of Module Categories over Morita Ring." Pure Mathematics 13, no. 07 (2023): 2136–41. http://dx.doi.org/10.12677/pm.2023.137221.

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21

Laan, Valdis, and Ülo Reimaa. "Morita equivalence of factorizable semigroups." International Journal of Algebra and Computation 29, no. 04 (June 2019): 723–41. http://dx.doi.org/10.1142/s0218196719500243.

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A semigroup is called factorizable if each of its elements can be written as a product. We study equivalences and adjunctions between various categories of acts over a fixed factorizable semigroup. We prove that two factorizable semigroups are Morita equivalent if and only if they are strongly Morita equivalent. We also show that Morita equivalence of finite factorizable semigroups is algorithmically decidable in finite time.

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22

Hu, Wei, and Changchang Xi. "Derived equivalences and stable equivalences of Morita type, I." Nagoya Mathematical Journal 200 (December 2010): 107–52. http://dx.doi.org/10.1215/00277630-2010-014.

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AbstractFor self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this article, we first show that each derived equivalence F between the derived categories of Artin algebras A and B arises naturally as a functor between their stable module categories, which can be used to compare certain homological dimensions of A with that of B. We then give a sufficient condition for the functor to be an equivalence. Moreover, if we work with finite-dimensional algebras over a field, then the sufficient condition guarantees the existence of a stable equivalence of Morita type. In this way, we extend the classical result of Rickard. Furthermore, we provide several inductive methods for constructing those derived equivalences that induce stable equivalences of Morita type. It turns out that we may produce a lot of (usually not self-injective) finite-dimensional algebras that are both derived-equivalent and stably equivalent of Morita type; thus, they share many common invariants.

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23

Hu, Wei, and Changchang Xi. "Derived equivalences and stable equivalences of Morita type, I." Nagoya Mathematical Journal 200 (December 2010): 107–52. http://dx.doi.org/10.1017/s0027763000010199.

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AbstractFor self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this article, we first show that each derived equivalenceFbetween the derived categories of Artin algebrasAandBarises naturally as a functorbetween their stable module categories, which can be used to compare certain homological dimensions ofAwith that ofB. We then give a sufficient condition for the functorto be an equivalence. Moreover, if we work with finite-dimensional algebras over a field, then the sufficient condition guarantees the existence of a stable equivalence of Morita type. In this way, we extend the classical result of Rickard. Furthermore, we provide several inductive methods for constructing those derived equivalences that induce stable equivalences of Morita type. It turns out that we may produce a lot of (usually not self-injective) finite-dimensional algebras that are both derived-equivalent and stably equivalent of Morita type; thus, they share many common invariants.

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24

MARSH, ROBERT J., and YANN PALU. "NEARLY MORITA EQUIVALENCES AND RIGID OBJECTS." Nagoya Mathematical Journal 225 (August 19, 2016): 64–99. http://dx.doi.org/10.1017/nmj.2016.27.

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If $T$ and $T^{\prime }$ are two cluster-tilting objects of an acyclic cluster category related by a mutation, their endomorphism algebras are nearly Morita equivalent (Buan et al., Cluster-tilted algebras, Trans. Amer. Math. Soc. 359(1) (2007), 323–332 (electronic)); that is, their module categories are equivalent “up to a simple module”. This result has been generalized by Yang, using a result of Plamondon, to any simple mutation of maximal rigid objects in a 2-Calabi–Yau triangulated category. In this paper, we investigate the more general case of any mutation of a (non-necessarily maximal) rigid object in a triangulated category with a Serre functor. In that setup, the endomorphism algebras might not be nearly Morita equivalent, and we obtain a weaker property that we call pseudo-Morita equivalence. Inspired by Buan and Marsh (From triangulated categories to module categories via localization II: calculus of fractions, J. Lond. Math. Soc. (2) 86(1) (2012), 152–170; From triangulated categories to module categories via localisation, Trans. Amer. Math. Soc. 365(6) (2013), 2845–2861), we also describe our result in terms of localizations.

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25

Grossman, Pinhas, Masaki Izumi, and Noah Snyder. "The Asaeda–Haagerup fusion categories." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 743 (October 1, 2018): 261–305. http://dx.doi.org/10.1515/crelle-2015-0078.

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Abstract The classification of subfactors of small index revealed several new subfactors. The first subfactor above index 4, the Haagerup subfactor, is increasingly well understood and appears to lie in a (discrete) infinite family of subfactors where the \mathbb{Z} /3 \mathbb{Z} symmetry is replaced by other finite abelian groups. The goal of this paper is to give a similarly good description of the Asaeda–Haagerup subfactor which emerged from our study of its Brauer–Picard groupoid. More specifically, we construct a new subfactor {\mathcal{S}} which is a \mathbb{Z} /4 \mathbb{Z} \times \mathbb{Z} /2 \mathbb{Z} analogue of the Haagerup subfactor and we show that the even parts of the Asaeda–Haagerup subfactor are higher Morita equivalent to an orbifold quotient of {\mathcal{S}} . This gives a new construction of the Asaeda–Haagerup subfactor which is much more symmetric and easier to work with than the original construction. As a consequence, we can settle many open questions about the Asaeda–Haagerup subfactor: calculating its Drinfeld center, classifying all extensions of the Asaeda–Haagerup fusion categories, finding the full higher Morita equivalence class of the Asaeda–Haagerup fusion categories, and finding intermediate subfactor lattices for subfactors coming from the Asaeda–Haagerup categories. The details of the applications will be given in subsequent papers.

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26

Iglesias, F. Castaño, and J. Gómez Torrecillas. "Wide Morita contexts and equivalences of comodule categories." Journal of Pure and Applied Algebra 131, no. 3 (October 1998): 213–25. http://dx.doi.org/10.1016/s0022-4049(97)00100-x.

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27

López-Permouth, Sergio R. "Lifting Morita equivalence to categories of fuzzy modules." Information Sciences 64, no. 3 (October 1992): 191–201. http://dx.doi.org/10.1016/0020-0255(92)90100-m.

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28

Kong, Liang, and Ingo Runkel. "Morita classes of algebras in modular tensor categories." Advances in Mathematics 219, no. 5 (December 2008): 1548–76. http://dx.doi.org/10.1016/j.aim.2008.07.004.

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29

Pardo, J. L. Gómez, and P. A. Guil Asensio. "Linear compactness and Morita duality for Grothendieck categories." Journal of Algebra 148, no. 1 (May 1992): 53–67. http://dx.doi.org/10.1016/0021-8693(92)90236-f.

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30

Coconeţ, Tiberiu, Andrei Marcus, and Constantin-Cosmin Todea. "Block Extensions, Local Categories and Basic Morita Equivalences." Quarterly Journal of Mathematics 71, no. 2 (April 28, 2020): 703–28. http://dx.doi.org/10.1093/qmathj/haaa008.

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Abstract Let $(\mathcal{K},\mathcal{O},k)$ be a $p$-modular system where $p$ is a prime and $k$ algebraically closed, let $b$ be a $G$-invariant block of the normal subgroup $H$ of a finite group $G$, having defect pointed group $Q_\delta$ in $H$ and $P_\gamma$ in $G$ and consider the block extension $b\mathcal{O}G$. One may attach to $b$ an extended local category $\mathcal{E}_{(b,H,G)}$, a group extension $L$ of $Z(Q)$ by $N_G(Q_\delta )/C_H(Q)$ having $P$ as a Sylow $p$-subgroup, and a cohomology class $[\alpha ]\in H^2(N_G(Q_\delta )/QC_H(Q),k^\times )$. We prove that these objects are invariant under the $G/H$-graded basic Morita equivalences. Along the way, we give alternative proofs of the results of Külshammer and Puig (1990), and Puig and Zhou (2012) on extensions of nilpotent blocks. We also deduce by our methods a result of Zhou (2016) on $p^{\prime}$-extensions of inertial blocks.

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31

Galindo, César, and Julia Yael Plavnik. "Tensor functors between Morita duals of fusion categories." Letters in Mathematical Physics 107, no. 3 (November 24, 2016): 553–90. http://dx.doi.org/10.1007/s11005-016-0914-y.

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32

Ohtake, K. "Morita Duality for Grothendieck Categories and Its Application." Journal of Algebra 174, no. 3 (June 1995): 801–22. http://dx.doi.org/10.1006/jabr.1995.1154.

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33

Ben-Zvi, David, Sam Gunningham, and Hendrik Orem. "Highest Weights for Categorical Representations." International Mathematics Research Notices 2020, no. 24 (December 5, 2018): 9988–10004. http://dx.doi.org/10.1093/imrn/rny258.

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Abstract We present a criterion for establishing Morita equivalence of monoidal categories and apply it to the categorical representation theory of reductive groups $G$. We show that the “de Rham group algebra” $\mathcal D(G)$ (the monoidal category of $\mathcal D$-modules on $G$) is Morita equivalent to the universal Hecke category $\mathcal D({N}\backslash{G}/{N})$ and to its monodromic variant $\widetilde{\mathcal D}({B}\backslash{G}/{B})$. In other words, de Rham $G$-categories, that is, module categories for $\mathcal D(G)$, satisfy a “highest weight theorem”—they all appear in the decomposition of the universal principal series representation $\mathcal D(G/N)$ or in twisted $\mathcal D$-modules on the flag variety $\widetilde{\mathcal D}(G/B)$.

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34

Blecher, David P., Paul S. Muhly, and Vern I. Paulsen. "Categories of operator modules (Morita equivalence and projective modules)." Memoirs of the American Mathematical Society 143, no. 681 (2000): 0. http://dx.doi.org/10.1090/memo/0681.

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35

Năstăsescu, C., and B. Torrecillas. "Morita Duality for Grothendieck Categories with Applications to Coalgebras." Communications in Algebra 33, no. 11 (October 2005): 4083–96. http://dx.doi.org/10.1080/00927870500261397.

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36

Gao, Nan, and Chrysostomos Psaroudakis. "Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings." Algebras and Representation Theory 20, no. 2 (November 3, 2016): 487–529. http://dx.doi.org/10.1007/s10468-016-9652-1.

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37

Liu, Miantao, Ruixin Li, and Nan Gao. "Morphism Categories of Gorenstein-projective Modules." Algebra Colloquium 25, no. 03 (August 14, 2018): 377–86. http://dx.doi.org/10.1142/s1005386718000275.

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Let Λ be an algebra of finite Cohen-Macaulay type and Γ its Cohen-Macaulay Auslander algebra. We are going to characterize the morphism category Mor(Λ-Gproj) of Gorenstein-projective Λ-modules in terms of the module category Γ-mod by a categorical equivalence. Based on this, we obtain that some factor category of the epimorphism category Epi(Λ-Gproj) is a Frobenius category, and also, we clarify the relations among Mor(Λ-Gproj), Mor(T2Λ-Gproj) and Mor(Δ-Gproj), where T2(Λ) and Δ are respectively the lower triangular matrix algebra and the Morita ring closely related to Λ.

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38

KOIKE, KAZUTOSHI. "MORITA DUALITY AND RING EXTENSIONS." Journal of Algebra and Its Applications 12, no. 02 (December 16, 2012): 1250160. http://dx.doi.org/10.1142/s0219498812501605.

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In this paper, we show that there exists a category equivalence between certain categories of A-rings (respectively, ring extensions of A) and B-rings (respectively, ring extensions of B), where A and B are Morita dual rings. In this category equivalence, corresponding two A-ring and B-ring are Morita dual. This is an improvement of a result of Müller, which state that if a ring A has a Morita duality induced by a bimodule BQA and R is a ring extension of A such that RA and Hom A(R, Q)A are linearly compact, then R has a Morita duality induced by the bimodule S End R( Hom A(R, Q))R with S = End R( Hom A(R, Q)). We also investigate relationships between Morita duality and finite ring extensions. Particularly, we show that if A and B are Morita dual rings with B basic, then every finite triangular (respectively, normalizing) extension R of A is Morita dual to a finite triangular (respectively, normalizing) extension S of B, and we give a result about finite centralizing free extensions, which unify a result of Mano about self-duality and a result of Fuller–Haack about semigroup rings.

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39

Zhang, Bo-Ye, and Ji-Wei He. "Graded Derived Equivalences." Mathematics 10, no. 1 (December 29, 2021): 103. http://dx.doi.org/10.3390/math10010103.

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We consider the equivalences of derived categories of graded rings over different groups. A Morita type equivalence is established between two graded algebras with different group gradings. The results obtained here give a better understanding of the equivalences of derived categories of graded rings.

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40

Brochier, Adrien, David Jordan, and Noah Snyder. "On dualizability of braided tensor categories." Compositio Mathematica 157, no. 3 (March 2021): 435–83. http://dx.doi.org/10.1112/s0010437x20007630.

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We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $q$.

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41

Toën, Bertrand. "The homotopy theory of dg-categories and derived Morita theory." Inventiones mathematicae 167, no. 3 (December 20, 2006): 615–67. http://dx.doi.org/10.1007/s00222-006-0025-y.

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42

Słomińska, Jolanta. "Dold–Kan type theorems and Morita equivalences of functor categories." Journal of Algebra 274, no. 1 (April 2004): 118–37. http://dx.doi.org/10.1016/j.jalgebra.2003.10.025.

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43

Breaz, Simion. "A Morita type theorem for a sort of quotient categories." Czechoslovak Mathematical Journal 55, no. 1 (March 2005): 133–44. http://dx.doi.org/10.1007/s10587-005-0009-x.

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44

Rogers, Morgan. "Toposes of Topological Monoid Actions." Compositionality 5 (January 10, 2023): 1. http://dx.doi.org/10.32408/compositionality-5-1.

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We demonstrate that categories of continuous actions of topological monoids on discrete spaces are Grothendieck toposes. We exhibit properties of these toposes, giving a solution to the corresponding Morita-equivalence problem. We characterize these toposes in terms of their canonical points. We identify natural classes of representatives with good topological properties, 'powder monoids' and then 'complete monoids', for the Morita-equivalence classes of topological monoids. Finally, we show that the construction of these toposes can be made (2-)functorial by considering geometric morphisms induced by continuous semigroup homomorphisms.

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45

TSEMENTZIS, DIMITRIS. "A SYNTACTIC CHARACTERIZATION OF MORITA EQUIVALENCE." Journal of Symbolic Logic 82, no. 4 (December 2017): 1181–98. http://dx.doi.org/10.1017/jsl.2017.59.

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AbstractWe characterize Morita equivalence of theories in the sense of Johnstone in terms of a new syntactic notion of a common definitional extension developed by Barrett and Halvorson for cartesian, regular, coherent, geometric and first-order theories. This provides a purely syntactic characterization of the relation between two theories that have equivalent categories of models naturally in any Grothendieck topos.

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46

Tart, Lauri. "On Morita equivalence of partially ordered semigroups with local units." Acta et Commentationes Universitatis Tartuensis de Mathematica 15, no. 2 (December 11, 2020): 15–33. http://dx.doi.org/10.12697/acutm.2011.15.07.

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We show that for two partially ordered semigroups S and T with common local units, there exists a unitary Morita context with surjective maps if and only if the categories of closed right S- and T-posets are equivalent.

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47

Müger, Michael. "From subfactors to categories and topology I: Frobenius algebras in and Morita equivalence of tensor categories." Journal of Pure and Applied Algebra 180, no. 1-2 (May 2003): 81–157. http://dx.doi.org/10.1016/s0022-4049(02)00247-5.

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48

Kashu, Alexei. "Euclidean Combinatorial Configurations: Typology, Continuous Extensions and Representations." Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1(98) (July 2022): 83–98. http://dx.doi.org/10.56415/basm.y2022.i1.p83.

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The preradicals and closure operators in module categories are studied. The concordance is shown between the mappings connecting the classes of preradicals and of closure operators of two module categories $R$-Mod and $S$-Mod in the case of a Morita context $(R,\, _{\ind R}\,U_{\ind S},\, _{\ind S}V_{\ind R},S)$, using the functors $Hom_{\ind R}(U,\mbox{-})$ and $Hom_{\ind S}(V,\mbox{-})$.

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49

Blecher, David P. "On Morita's fundamental theorem for $C^*$-algebras." MATHEMATICA SCANDINAVICA 88, no. 1 (March 1, 2001): 137. http://dx.doi.org/10.7146/math.scand.a-14319.

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We give a solution, via operator spaces, of an old problem in the Morita equivalence of $C^*$-algebras. Namely, we show that $C^*$-algebras are strongly Morita equivalent in the sense of Rieffel if and only if their categories of left operator modules are isomorphic via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product (= interior tensor product) with a strong Morita equivalence bimodule. An operator module over a $C^*$-algebra $\mathcal A$ is a closed subspace of some B(H) which is left invariant under multiplication by $\pi(\mathcal\ A)$, where $\pi$ is a*-representation of $\mathcal A$ on $H$. The category $_{\mathcal{AHMOD}}$ of *-representations of $\mathcal A$ on Hilbert space is a full subcategory of the category $_{\mathcal{AOMOD}}$ of operator modules. Our main result remains true with respect to subcategories of $OMOD$ which contain $HMOD$ and the $C^*$-algebra itself. It does not seem possible to remove the operator space framework; in the very simplest cases there may exist no bounded equivalence functors on categories with bounded module maps as morphisms (as opposed to completely bounded ones). Our proof involves operator space techniques, together with a $C^*$-algebra argument using compactness of the quasistate space of a $C^*$-algebra, and lowersemicontinuity in the enveloping von Neumann algebra.

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50

Tabuada, Gonçalo. "The fundamental theorem via derived Morita invariance, localization, and 1-homotopy invariance." Journal of K-theory 9, no. 3 (May 24, 2011): 407–20. http://dx.doi.org/10.1017/is011004009jkt155.

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AbstractWe prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and 1-homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and Kassel's fundamental theorems in homotopy algebraic K-theory, and periodic cyclic homology, respectively.

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