Fernando Muro

We prove that the (homotopy) hypercommutative algebra structure on the de Rham cohomology of a Poisson or Jacobi manifold defined by several authors is (homotopically) trivial, i.e. it reduces to the underlying (homotopy) commutative algebra structure. We do so by showing that the DG operads which codify the algebraic structure on the de Rham complex of Poisson and Jacobi manifolds, generated by the exterior product and the interior products with the structure polyvector fields, are quasi-isomorphic to the commutative suboperad. Hence, there is no hope to endow the de Rham cohomology of such manifolds with any (higher) structure beyond the well-known (homotopy) commutative algebra structure, which exists for any smooth manifold. We proceed similarly with the commutative Batalin-Vilkovisky-infinity algebra structure on the de Rham complex of a generalized Poisson supermanifold.

@article{zbMATH08105648,
     author = {Guan, Ai and Muro, Fernando},
     title = {{Operations on the de {Rham} cohomology of {Poisson} and {Jacobi} manifolds}},
     fjournal = {Selecta Mathematica. New Series},
     journal = {Sel. Math., New Ser.},
     issn = {1022-1824},
     volume = {31},
     number = {5},
     pages = {36},
     note = {Id/No 94},
     year = {2025},
     language = {English},
     doi = {10.1007/s00029-025-01087-w},
     keywords = {18M60,53D17},
     }

Let $d$ be a positive integer. In a previous article we established a bijective correspondence between the following classes of objects, considered up to the appropriate notion of equivalence: differential graded algebras with finite-dimensional $0$-th cohomology such that the canonical generator of their perfect derived category is a basic $d\mathbb{Z}$-cluster tilting object, and basic Frobenius algebras that are twisted $(d+2)$-periodic as bimodules. For $d=1$ this correspondence specialises to previous work of the second-named author on algebraic triangulated categories of finite type. In this article, we prove a variant of our general correspondence for bimodule right Calabi--Yau dg algebras. A novel ingredient is a new cohomology theory which contains obstructions to the existence and uniqueness of minimal $A_\infty$-bimodule structures on a graded bimodule.

@misc{arXiv:2509.22625,
     author = {Jasso, Gustavo and Muro, Fernando},
     title = {{The {Derived} {Auslander}--{Iyama} {Correspondence} {II}: {Bimodule} {Calabi}--{Yau} {Structures}}},
     year = {2025},
     howpublished = {Preprint, {arXiv}:2509.22625 [math.{RT}] (2025)},
     keywords = {18G80,18N40},
     url = {https://arxiv.org/abs/2509.22625},
     }

The Derived Auslander--Iyama Corresponence, a recent result of the authors, provides a classification up to quasi-isomorphism of the derived endomorphism algebras of basic $d\mathbb{Z}$-cluster tilting objects in $\operatorname{Hom}$-finite algebraic triangulated categories in terms of a small amount of algebraic data. In this note we highlight the role of minimal $A_\infty$-algebra structures in the proof of this result, as well as the crucial role of the enhanced $A_\infty$-obstruction theory developed by the second-named author.

@misc{arXiv:2508.18852,
     author = {Jasso, Gustavo and Muro, Fernando},
     title = {{Minimal ${A}_{\infty}$-algebras of endomorphisms: {The} case of $d\mathbb{Z}$-cluster tilting objects}},
     year = {2025},
     howpublished = {Preprint, {arXiv}:2508.18852 [math.{RT}] (2025)},
     keywords = {18G80,18N40},
     url = {https://arxiv.org/abs/2508.18852},
     }

We develop an obstruction theory for the extension of truncated minimal $A$-infinity bimodule structures over truncated minimal $A$-infinity algebras. Obstructions live in far-away pages of a (truncated) fringed spectral sequence of Bousfield--Kan type. The second page of this spectral sequence is mostly given by a new cohomology theory associated to a pair consisting of a graded algebra and a graded bimodule over it. This new cohomology theory fits in a long exact sequence involving the Hochschild cohomology of the algebra and the self-extensions of the bimodule. We show that the second differential of this spectral sequence is given by the Gerstenhaber bracket with a bimodule analogue of the universal Massey product of a minimal $A$-infinity algebra. We also develop a closely-related obstruction theory for truncated minimal $A$-infinity bimodule structures over (the truncation of) a fixed minimal $A$-infinity algebra; the second page of the corresponding spectral sequence is now mostly given by the vector spaces of self-extensions of the underlying graded bimodule and the second differential is described analogously to the previous one. We also establish variants of the above for graded algebras and graded bimodules that are $d$-sparse, that is they are concentrated in degrees that are multiples of a fixed integer $d\geq1$. These obstruction theories are used to establish intrinsic formality and almost formality theorems for differential graded bimodules over differential graded algebras. Our results hold, more generally, in the context of graded operads with multiplication equipped with an associative operadic ideal, examples of which are the endomorphism operad of a graded algebra and the linear endomorphism operad of a pair consisting of a graded algebra and a graded bimodule over it.

@misc{arXiv:2507.17568,
     author = {Jasso, Gustavo and Muro, Fernando},
     title = {{Obstruction theory for ${A}$-infinity bimodules}},
     year = {2025},
     howpublished = {Preprint, {arXiv}:2507.17568 [math.{AT}] (2025)},
     keywords = {18M65,18N40,55S35,18G40},
     url = {https://arxiv.org/abs/2507.17568},
     }

We provide an outline of the proof of the Donovan--Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds. The proof relies on results of August, of Hua and the second-named author, Wemyss, and on the Derived Auslander--Iyama Correspondence -- a recent result by the first- and third-named authors.

@incollection{zbMATH07950525,
     author = {Jasso, Gustavo and Keller, Bernhard and Muro, Fernando},
     title = {{The {Donovan}-{Wemyss} conjecture via the derived {Auslander}-{Iyama} correspondence}},
     booktitle = {{Triangulated categories in representation theory and beyond. The Abel symposium 2022, Ålesund, Norway, June 6--10, 2022}},
     isbn = {978-3-031-57788-8; 978-3-031-57791-8; 978-3-031-57789-5},
     pages = {105--140},
     year = {2024},
     publisher = {Cham: Springer},
     language = {English},
     doi = {10.1007/978-3-031-57789-5_4},
     keywords = {14E30,13D03,14F08},
     }

We develop a theory of minimal models for algebras over an operad defined over a commutative ring, not necessarily a field, extending and supplementing the work of Sagave in the associative case.

@article{zbMATH07719245,
     author = {Maes, Jeroen and Muro, Fernando},
     title = {{Derived homotopy algebras}},
     fjournal = {Proceedings of the Royal Society of Edinburgh. Section A. Mathematics},
     journal = {Proc. R. Soc. Edinb., Sect. A, Math.},
     issn = {0308-2105},
     volume = {153},
     number = {4},
     pages = {1198--1243},
     year = {2023},
     language = {English},
     doi = {10.1017/prm.2022.42},
     keywords = {18M70,14A30,16E05,18G10,55U35},
     }

We define a generalization of Massey products for algebras over a Koszul operad in characteristic zero, extending Massey's and Allday's and Retah's in the associative and Lie cases, respectively. We establish connections with minimal models and with Dimitrova's universal operadic cohomology class. We compute a Gerstenhaber algebra example and a hypercommutative algebra example related to the Chevalley-Eilenberg complex of the Heisenberg Lie algebra.

@article{zbMATH07700052,
     author = {Muro, Fernando},
     title = {{Massey products for algebras over operads}},
     fjournal = {Communications in Algebra},
     journal = {Commun. Algebra},
     issn = {0092-7872},
     volume = {51},
     number = {8},
     pages = {3298--3313},
     year = {2023},
     language = {English},
     doi = {10.1080/00927872.2023.2181780},
     keywords = {18M70,55S20,13D03,16E40,17B56,55P48},
     }

We define an obstruction to the formality of a differential graded algebra over a graded operad defined over a commutative ground ring. This obstruction lives in the derived operadic cohomology of the algebra. Moreover, it determines all operadic Massey products induced on the homology algebra, hence the name of derived universal Massey product.

@article{zbMATH07681901,
     author = {Muro, Fernando},
     title = {{Derived universal {Massey} products}},
     fjournal = {Homology, Homotopy and Applications},
     journal = {Homology Homotopy Appl.},
     issn = {1532-0073},
     volume = {25},
     number = {1},
     pages = {189--218},
     year = {2023},
     language = {English},
     doi = {10.4310/HHA.2023.v25.n1.a10},
     keywords = {18M70,13D03,16E40,17B56,55S20},
     }

We give a necessary and sufficient condition for the existence of an enhancement of a finite triangulated category. Moreover, we show that enhancements are unique when they exist, up to Morita equivalence.

@article{zbMATH07506910,
     author = {Muro, Fernando},
     title = {{Enhanced finite triangulated categories}},
     fjournal = {Journal of the Institute of Mathematics of Jussieu},
     journal = {J. Inst. Math. Jussieu},
     issn = {1474-7480},
     volume = {21},
     number = {3},
     pages = {741--783},
     year = {2022},
     language = {English},
     doi = {10.1017/S1474748020000250},
     keywords = {16E40,18G40,55S35},
     }

We work over a perfect field. Recent work of the third-named author established a Derived Auslander Correspondence that relates finite-dimensional self-injective algebras that are twisted $3$-periodic to algebraic triangulated categories of finite type. Moreover, the aforementioned work also shows that the latter triangulated categories admit a unique differential graded enhancement. In this article we prove a higher-dimensional version of this result that, given an integer $d\geq1$, relates twisted $(d+2)$-periodic algebras to algebraic triangulated categories with a $d\mathbb{Z}$-cluster tilting object. We also show that the latter triangulated categories admit a unique differential graded enhancement. Our result yields recognition theorems for interesting algebraic triangulated categories, such as the Amiot cluster category of a self-injective quiver with potential in the sense of Herschend and Iyama and, more generally, the Amiot-Guo-Keller cluster category associated with a $d$-representation finite algebra in the sense of Iyama and Oppermann. As an application of our result, we obtain infinitely many triangulated categories with a unique differential graded enhancement that is not strongly unique. In the appendix, B. Keller explains how -- combined with crucial results of August and Hua-Keller -- our main result yields the last key ingredient to prove the Donovan-Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds.

@misc{arXiv:2208.14413,
     author = {Jasso, Gustavo and Keller, Bernhard and Muro, Fernando},
     title = {{The {Derived} {Auslander}-{Iyama} {Correspondence}}},
     year = {2022},
     howpublished = {Preprint, {arXiv}:2208.14413 [math.{RT}] (2022)},
     keywords = {18G80,18N40},
     url = {https://arxiv.org/abs/2208.14413},
     }

Several model structures related to the homotopy theory of locally constant factorization algebras are constructed. This answers a question raised by D. Calaque in his habilitation thesis. Our methods also solve a problem related to cosheafification and factorization algebras identified by O. Gwilliam - K. Rejzner in the locally constant case.

@misc{arXiv:2107.14174,
     author = {Carmona, Victor and Flores, Ramon and Muro, Fernando},
     title = {{A model structure for locally constant factorization algebras}},
     year = {2021},
     howpublished = {Preprint, {arXiv}:2107.14174 [math.{AT}] (2021)},
     url = {https://arxiv.org/abs/2107.14174},
     }

In this paper we relate triangulated category structures to the cohomology of small categories and define initial obstructions to the existence of an algebraic or topological enhancement. We show that these obstructions do not vanish in an example of triangulated category without models. We also obtain cohomological characterizations of pre-triangulated DG, A-infinity, and spectral categories.

@article{zbMATH07242463,
     author = {Muro, Fernando},
     title = {{The first obstructions to enhancing a triangulated category}},
     fjournal = {Mathematische Zeitschrift},
     journal = {Math. Z.},
     issn = {0025-5874},
     volume = {296},
     number = {1-2},
     pages = {719--759},
     year = {2020},
     language = {English},
     doi = {10.1007/s00209-019-02438-y},
     keywords = {18E30,16E40,18G50,18G60,18E30,16G70},
     }

We extend the Bousfield-Kan spectral sequence for the computation of the homotopy groups of the space of minimal A-infinity algebra structures on a graded projective module. We use the new part to define obstructions to the extension of truncated minimal A-infinity algebra structures. We also consider the Bousfield-Kan spectral sequence for the moduli space of A-infinity algebras. We compute up to the second page, terms and differentials, of these spectral sequences in terms of Hochschild cohomology.

@article{zbMATH07190145,
     author = {Muro, Fernando},
     title = {{Enhanced {{\(A_{\infty}\)}}-obstruction theory}},
     fjournal = {Journal of Homotopy and Related Structures},
     journal = {J. Homotopy Relat. Struct.},
     issn = {2193-8407},
     volume = {15},
     number = {1},
     pages = {61--112},
     year = {2020},
     language = {English},
     doi = {10.1007/s40062-019-00245-0},
     keywords = {18G40,55S35,16E40},
     }

In this note we describe the role of the Schur multiplier in the structure of the $p$-torsion of discrete groups. More concretely, we show how the knowledge of $H_2G$ allows to approximate many groups by colimits of copies of finite $p$-groups. Our examples include interesting families of non-commutative infinite groups, including Burnside groups, certain solvable examples and the first Grigorchuk group. We also provide a counterexample for a conjecture of E. Farjoun.

@article{zbMATH07053579,
     author = {Flores, Ram{\'o}n and Muro, Fernando},
     title = {{Torsion homology and cellular approximation}},
     fjournal = {Algebraic \& Geometric Topology},
     journal = {Algebr. Geom. Topol.},
     issn = {1472-2747},
     volume = {19},
     number = {1},
     pages = {457--476},
     year = {2019},
     language = {English},
     doi = {10.2140/agt.2019.19.457},
     keywords = {20F99,55P60},
     }

We define two model structures on the category of bicomplexes concentrated in the right half plane. The first model structure has weak equivalences detected by the totalisation functor. The second model structure's weak equivalences are detected by the $E^2$-term of the spectral sequence associated to the filtration of the total complex by the horizontal degree. We then extend this result to twisted complexes.

@article{zbMATH07006249,
     author = {Muro, Fernando and Roitzheim, Constanze},
     title = {{Homotopy theory of bicomplexes}},
     fjournal = {Journal of Pure and Applied Algebra},
     journal = {J. Pure Appl. Algebra},
     issn = {0022-4049},
     volume = {223},
     number = {5},
     pages = {1913--1939},
     year = {2019},
     language = {English},
     doi = {10.1016/j.jpaa.2018.08.007},
     keywords = {18G55,55U35},
     url = {kar.kent.ac.uk/67611/1/bi_and_twisted_complexes_revision.pdf},
     }

We correct a mistake in the construction of push-outs along free morphisms of algebras over a nonsymmetric operad in arXiv:1101.1634 [math.AT], and we fix the affected results in arXiv:1101.1634 [math.AT] and arXiv:1304.6641 [math.AT].

@article{zbMATH06791664,
     author = {Muro, Fernando},
     title = {{Correction to: ``{Homotopy} theory of nonsymmetric operads. {I}--{II}''}},
     fjournal = {Algebraic \& Geometric Topology},
     journal = {Algebr. Geom. Topol.},
     issn = {1472-2747},
     volume = {17},
     number = {6},
     pages = {3837--3852},
     year = {2017},
     language = {English},
     doi = {10.2140/agt.2017.17.3837},
     keywords = {18M60,55U35,18M05,18C40,18D20},
     }

We define a $K$-theory for pointed right derivators and show that it agrees with Waldhausen $K$-theory in the case where the derivator arises from a good Waldhausen category. This $K$-theory is not invariant under general equivalences of derivators, but only under a stronger notion of equivalence that is defined by considering a simplicial enrichment of the category of derivators. We show that derivator $K$-theory, as originally defined, is the best approximation to Waldhausen $K$-theory by a functor that is invariant under equivalences of derivators.

@article{zbMATH06673997,
     author = {Muro, Fernando and Raptis, Georgios},
     title = {{{{\(K\)}}-theory of derivators revisited}},
     fjournal = {Annals of \(K\)-Theory},
     journal = {Ann. \(K\)-Theory},
     issn = {2379-1683},
     volume = {2},
     number = {2},
     pages = {303--340},
     year = {2017},
     language = {English},
     doi = {10.2140/akt.2017.2.303},
     keywords = {19D99,55U35},
     }

We construct small cylinders for cellular non-symmetric DG-operads over an arbitrary commutative ring by using the basic perturbation lemma from homological algebra. We show that our construction, applied to the A-infinity operad, yields the operad parametrizing A-infinity maps whose linear part is the identity. We also compute some other examples with non-trivial operations in arities 1 and 0.

@article{zbMATH06571599,
     author = {Muro, Fernando},
     title = {{Cylinders for non-symmetric {DG}-operads via homological perturbation theory}},
     fjournal = {Journal of Pure and Applied Algebra},
     journal = {J. Pure Appl. Algebra},
     issn = {0022-4049},
     volume = {220},
     number = {9},
     pages = {3248--3281},
     year = {2016},
     language = {English},
     doi = {10.1016/j.jpaa.2016.02.013},
     keywords = {18D50,18G55,03E10},
     url = {idus.us.es/xmlui/handle/11441/43063},
     }

We show that the canonical map from the associative operad to the unital associative operad is a homotopy epimorphism for a wide class of symmetric monoidal model categories. As a consequence, the space of unital associative algebra structures on a given object is up to homotopy a subset of connected components of the space of non-unital associative algebra structures.

@article{zbMATH06550647,
     author = {Muro, Fernando},
     title = {{Homotopy units in {{\(A\)}}-infinity algebras}},
     fjournal = {Transactions of the American Mathematical Society},
     journal = {Trans. Am. Math. Soc.},
     issn = {0002-9947},
     volume = {368},
     number = {3},
     pages = {2145--2184},
     year = {2016},
     language = {English},
     doi = {10.1090/tran/6545},
     keywords = {18D50,55P48,18G55},
     }

In a well generated triangulated category T, given a regular cardinal a, we consider the following problems: given a functor from the category of a-compact objects to abelian groups that preserves products of <a objects and takes exact triangles to exact sequences, is it the restriction of a representable functor in T? Is every natural transformation between two such restricted representable functors induced by a map between the representatives? If the answer to both questions is positive we say that T satisfies a-Adams representability. A classical result going back to Brown and Adams shows that the stable homotopy category satisfies Adams representability for the first infinite cardinal. For that cardinal, Adams representability is well understood thanks to the work of Christensen, Keller and Neeman. In this paper, we develop an obstruction theory to decide when T satisfies a-Adams representability. We derive necessary and sufficient conditions of homological nature, and we compute several examples. In particular, we show that there are rings satisfying a-Adams representability for all non-countable cardinals a and rings which do not satisfy a-Adams representability for any infinite cardinal a. Moreover, we exhibit rings for which the answer to both questions is no for infinite many cardinals. As a side result, we give an example of an infinite phantom map.

@article{zbMATH06548165,
     author = {Muro, Fernando and Ravent{\'o}s, Oriol},
     title = {{Transfinite {Adams} representability}},
     fjournal = {Advances in Mathematics},
     journal = {Adv. Math.},
     issn = {0001-8708},
     volume = {292},
     pages = {111--180},
     year = {2016},
     language = {English},
     doi = {10.1016/j.aim.2016.01.009},
     keywords = {18E30,55S35,55S45,18G20},
     }

In this paper we show how to modify cofibrations in a monoidal model category so that the tensor unit becomes cofibrant while keeping the same weak equivalences. We obtain aplications to enriched categories and coloured operads in stable homotopy theory.

@article{zbMATH06454207,
     author = {Muro, Fernando},
     title = {{On the unit of a monoidal model category}},
     fjournal = {Topology and its Applications},
     journal = {Topology Appl.},
     issn = {0166-8641},
     volume = {191},
     pages = {37--47},
     year = {2015},
     language = {English},
     doi = {10.1016/j.topol.2015.05.006},
     keywords = {55P42,55U35},
     }

We construct a model structure on the category of small categories enriched over a combinatorial closed symmetric monoidal model category satisfying the monoid axiom. Weak equivalences are Dwyer-Kan equivalences, i.e. enriched functors which induce weak equivalences on morphism objects and equivalences of ordinary categories when we take sets of connected components on morphism objects.

@article{zbMATH06451815,
     author = {Muro, Fernando},
     title = {{Dwyer-Kan homotopy theory of enriched categories}},
     fjournal = {Journal of Topology},
     journal = {J. Topol.},
     issn = {1753-8416},
     volume = {8},
     number = {2},
     pages = {377--413},
     year = {2015},
     language = {English},
     doi = {10.1112/jtopol/jtu029},
     keywords = {18D20,18G55,55U35},
     url = {idus.us.es/xmlui/handle/11441/43027},
     }

In this paper we introduce a new approach to determinant functors which allows us to extend Deligne's determinant functors for exact categories to Waldhausen categories, (strongly) triangulated categories, and derivators. We construct universal determinant functors in all cases by original methods which are interesting even for the known cases. Moreover, we show that the target of each universal determinant functor computes the corresponding $K$-theory in dimensions 0 and 1. As applications, we answer open questions by Maltsiniotis and Neeman on the $K$-theory of (strongly) triangulated categories and a question of Grothendieck to Knudsen on determinant functors. We also prove additivity theorems for low-dimensional $K$-theory and obtain generators and (some) relations for various $K_{1}$-groups.

@article{zbMATH06417698,
     author = {Muro, Fernando and Tonks, Andrew and Witte, Malte},
     title = {{On determinant functors and {{\(K\)}}-theory}},
     fjournal = {Publicacions Matem{\`a}tiques},
     journal = {Publ. Mat., Barc.},
     issn = {0214-1493},
     volume = {59},
     number = {1},
     pages = {137--233},
     year = {2015},
     language = {English},
     doi = {10.5565/PUBLMAT_59115_07},
     keywords = {19A99,18E10,18E30,18G55,19B99},
     }

We construct a topological cellular operad such that the algebras over its cellular chains are the homotopy unital A-infinity algebras of Fukaya-Oh-Ohta-Ono.

@article{zbMATH06342603,
     author = {Muro, Fernando and Tonks, Andrew},
     title = {{Unital associahedra}},
     fjournal = {Forum Mathematicum},
     journal = {Forum Math.},
     issn = {0933-7741},
     volume = {26},
     number = {2},
     pages = {593--620},
     year = {2014},
     language = {English},
     doi = {10.1515/forum-2011-0130},
     keywords = {18D50,18G55},
     url = {idus.us.es/handle/11441/41889},
     }

In this paper we study spaces of algebras over an operad (non-symmetric) in symmetric monoidal model categories. We first compute the homotopy fiber of the forgetful functor sending an algebra to its underlying object, extending a result of Rezk. We then apply this computation to the construction of geometric moduli stacks of algebras over an operad in a homotopical algebraic geometry context in the sense of To\"en and Vezzosi. We show under mild hypotheses that the moduli stack of unital associative algebras is a Zariski open substack of the moduli stack of non-necessarily unital associative algebras. The classical analogue for finite-dimensional vector spaces was noticed by Gabriel.

@article{zbMATH06281920,
     author = {Muro, Fernando},
     title = {{Moduli spaces of algebras over nonsymmetric operads}},
     fjournal = {Algebraic \& Geometric Topology},
     journal = {Algebr. Geom. Topol.},
     issn = {1472-2747},
     volume = {14},
     number = {3},
     pages = {1489--1539},
     year = {2014},
     language = {English},
     doi = {10.2140/agt.2014.14.1489},
     keywords = {18D50,14K10,55U35},
     }

We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of algebras over operads. We also show left properness results on model categories of operads and algebras over operads. As an application, we prove homotopy invariance for (unital) associative operads.

@article{zbMATH06243329,
     author = {Muro, Fernando},
     title = {{Homotopy theory of non-symmetric operads. {II}: {Change} of base category and left properness}},
     fjournal = {Algebraic \& Geometric Topology},
     journal = {Algebr. Geom. Topol.},
     issn = {1472-2747},
     volume = {14},
     number = {1},
     pages = {229--281},
     year = {2014},
     language = {English},
     doi = {10.2140/agt.2014.14.229},
     keywords = {18D50,18D10,18G55,55U35},
     }

In this paper we show an example of two differential graded algebras that have the same derivator K-theory but non-isomorphic Waldhausen K-theory. We also prove that Maltsiniotis's comparison and localization conjectures for derivator K-theory cannot be simultaneously true.

@article{zbMATH05918250,
     author = {Muro, Fernando and Raptis, George},
     title = {{A note on {{\(K\)}}-theory and triangulated derivators}},
     fjournal = {Advances in Mathematics},
     journal = {Adv. Math.},
     issn = {0001-8708},
     volume = {227},
     number = {5},
     pages = {1827--1845},
     year = {2011},
     language = {English},
     doi = {10.1016/j.aim.2011.04.005},
     keywords = {19D99,18E30,16E45},
     }

We endow categories of non-symmetric operads with natural model structures. We work with no restriction on our operads and only assume the usual hypotheses for model categories with a symmetric monoidal structure. We also study categories of algebras over these operads in enriched non-symmetric monoidal model categories.

@article{zbMATH05904608,
     author = {Muro, Fernando},
     title = {{Homotopy theory of nonsymmetric operads}},
     fjournal = {Algebraic \& Geometric Topology},
     journal = {Algebr. Geom. Topol.},
     issn = {1472-2747},
     volume = {11},
     number = {3},
     pages = {1541--1599},
     year = {2011},
     language = {English},
     doi = {10.2140/agt.2011.11.1541},
     keywords = {18D50,18G55,18D10,55U35},
     }

The primary algebraic model of a ring spectrum is the ring of homotopy groups. We introduce the secondary model which has the structure of a secondary analogue of a ring. This new algebraic model determines Massey products and cup-one squares. As an application we obtain new derivations of the homotopy ring.

@article{zbMATH05882112,
     author = {Baues, Hans-Joachim and Muro, Fernando},
     title = {{The algebra of secondary homotopy operations in ring spectra}},
     fjournal = {Proceedings of the London Mathematical Society. Third Series},
     journal = {Proc. Lond. Math. Soc. (3)},
     issn = {0024-6115},
     volume = {102},
     number = {4},
     pages = {637--696},
     year = {2011},
     language = {English},
     doi = {10.1112/plms/pdq034},
     keywords = {55P42,55P43,55Q35,55S20,55S30},
     url = {idus.us.es/xmlui/handle/11441/41904},
     }

In this paper we prove the laws of Toda brackets on the homotopy groups of a connective ring spectrum and the laws of the cup-one square in the homotopy groups of a commutative connective ring spectrum.

@article{zbMATH05538788,
     author = {Baues, Hans-Joachim and Muro, Fernando},
     title = {{Toda brackets and cup-one squares for ring spectra}},
     fjournal = {Communications in Algebra},
     journal = {Commun. Algebra},
     issn = {0092-7872},
     volume = {37},
     number = {1},
     pages = {56--82},
     year = {2009},
     language = {English},
     doi = {10.1080/00927870802241188},
     keywords = {18G50,55P42,55Q35},
     url = {idus.us.es/xmlui/handle/11441/41881},
     }

@article{zbMATH05528664,
     author = {Baues, H.-J. and Muro, F.},
     title = {{Cohomologically triangulated categories. {II}}},
     fjournal = {Journal of \(K\)-Theory},
     journal = {J. \(K\)-Theory},
     issn = {1865-2433},
     volume = {3},
     number = {1},
     pages = {1--52},
     year = {2009},
     language = {English},
     doi = {10.1017/is008007021jkt061},
     keywords = {18E30,18G60},
     }

We show that the symmetric track group, which is an extension of the symmetric group associated to the second Stiefel- Withney class, acts as a crossed module on the secondary homotopy group of a pointed space. An application is given to cup-one products in unstable homotopy groups of spheres, generalizing a formula of Barratt-Jones-Mahowald.

@article{zbMATH05496973,
     author = {Baues, Hans-Joachim and Muro, Fernando},
     title = {{The symmetric action on secondary homotopy groups}},
     fjournal = {Bulletin of the Belgian Mathematical Society - Simon Stevin},
     journal = {Bull. Belg. Math. Soc. - Simon Stevin},
     issn = {1370-1444},
     volume = {15},
     number = {4},
     pages = {733--768},
     year = {2008},
     language = {English},
     keywords = {55Q05,55Q25,55S45},
     }

We give a simple representation of all elements in K_1 of a Waldhausen category and prove relations between these representatives which hold in K_1.

@incollection{zbMATH05381934,
     author = {Muro, Fernando and Tonks, Andrew},
     title = {{On {{\(K_1\)}} of a {Waldhausen} category}},
     booktitle = {{\(K\)-theory and noncommutative geometry. Proceedings of the ICM 2006 satellite conference, Valladolid, Spain, August 31--September 6, 2006}},
     isbn = {978-3-03719-060-9},
     pages = {91--115},
     year = {2008},
     publisher = {Z{\"u}rich: European Mathematical Society (EMS)},
     language = {English},
     keywords = {18F25,19D10},
     }

We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered. In particular we prove a formula for the crossed effect of the cup-one product operation on unstable homotopy groups of spheres which was claimed by Barratt-Jones-Mahowald.

@article{zbMATH05367270,
     author = {Baues, Hans-Joachim and Muro, Fernando},
     title = {{Smash products for secondary homotopy groups}},
     fjournal = {Applied Categorical Structures},
     journal = {Appl. Categ. Struct.},
     issn = {0927-2852},
     volume = {16},
     number = {5},
     pages = {551--616},
     year = {2008},
     language = {English},
     doi = {10.1007/s10485-007-9071-x},
     keywords = {55Q35,55Q15,55S45},
     }

Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of homotopy types with homotopy groups concentrated in two consecutive dimensions.

@article{zbMATH05351182,
     author = {Baues, Hans-Joachim and Muro, Fernando},
     title = {{Secondary homotopy groups}},
     fjournal = {Forum Mathematicum},
     journal = {Forum Math.},
     issn = {0933-7741},
     volume = {20},
     number = {4},
     pages = {631--677},
     year = {2008},
     language = {English},
     doi = {10.1515/FORUM.2008.032},
     keywords = {18D05,55Q25,55S45},
     url = {idus.us.es/xmlui/handle/11441/41883},
     }

@article{zbMATH05347000,
     author = {Baues, H.-J. and Muro, F.},
     title = {{Cohomologically triangulated categories. {I}}},
     fjournal = {Journal of \(K\)-Theory},
     journal = {J. \(K\)-Theory},
     issn = {1865-2433},
     volume = {1},
     number = {1},
     pages = {3--48},
     year = {2008},
     language = {English},
     doi = {10.1017/is007011018jkt019},
     keywords = {18E30,18D05},
     }

We show that K_1 of an exact category agrees with K_1 of the associated triangulated derivator. More generally we show that K_1 of a Waldhausen category with cylinders and a saturated class of weak equivalences coincides with K_1 of the associated right pointed derivator.

@article{zbMATH05306025,
     author = {Muro, Fernando},
     title = {{Maltsiniotis's first conjecture for {{\(K_{1}\)}}}},
     fjournal = {IMRN. International Mathematics Research Notices},
     journal = {Int. Math. Res. Not.},
     issn = {1073-7928},
     volume = {2008},
     pages = {31},
     note = {Id/No rnm153},
     year = {2008},
     language = {English},
     doi = {10.1093/imrn/rnm153},
     keywords = {19D99,18E10,18E30,19B99,55P42},
     url = {idus.us.es/handle/11441/41891},
     }

We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories.

@article{zbMATH05208165,
     author = {Muro, Fernando and Schwede, Stefan and Strickland, Neil},
     title = {{Triangulated categories without models}},
     fjournal = {Inventiones Mathematicae},
     journal = {Invent. Math.},
     issn = {0020-9910},
     volume = {170},
     number = {2},
     pages = {231--241},
     year = {2007},
     language = {English},
     doi = {10.1007/s00222-007-0061-2},
     keywords = {18E30,55P42,16E45,13D25},
     url = {idus.us.es/xmlui/handle/11441/41893},
     }

We give a small functorial algebraic model for the 2-stage Postnikov section of the K-theory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors.

@article{zbMATH05201720,
     author = {Muro, Fernando and Tonks, Andrew},
     title = {{The 1-type of a {Waldhausen} {{\(K\)}}-theory spectrum}},
     fjournal = {Advances in Mathematics},
     journal = {Adv. Math.},
     issn = {0001-8708},
     volume = {216},
     number = {1},
     pages = {178--211},
     year = {2007},
     language = {English},
     doi = {10.1016/j.aim.2007.05.008},
     keywords = {19B99,16E20,18G50,18G55},
     }

@article{zbMATH05186664,
     author = {Baues, Hans-Joachim and Muro, Fernando},
     title = {{The homotopy category of pseudofunctors and translation cohomology}},
     fjournal = {Journal of Pure and Applied Algebra},
     journal = {J. Pure Appl. Algebra},
     issn = {0022-4049},
     volume = {211},
     number = {3},
     pages = {821--850},
     year = {2007},
     language = {English},
     doi = {10.1016/j.jpaa.2007.04.008},
     keywords = {18D05,18G60,55S35},
     }

In this paper we show that the Baues-Wirsching complex used to define cohomology of categories is a 2-functor from a certain 2-category of natural systems of abelian groups to the 2-category of chain complexes, chain homomorphism and relative homotopy classes of chain homotopies. As a consequence we derive (co)localization theorems for this cohomology.

@article{zbMATH02244508,
     author = {Muro, Fernando},
     title = {{On the functoriality of cohomology of categories}},
     fjournal = {Journal of Pure and Applied Algebra},
     journal = {J. Pure Appl. Algebra},
     issn = {0022-4049},
     volume = {204},
     number = {3},
     pages = {455--472},
     year = {2006},
     language = {English},
     doi = {10.1016/j.jpaa.2005.05.004},
     keywords = {18G60,18D05,18E35},
     url = {idus.us.es/xmlui/handle/11441/41897},
     }

@article{zbMATH05012978,
     author = {C{\'a}rdenas, M. and Lasheras, F. F. and Muro, F. and Quintero, A.},
     title = {{Proper {L}--{S} category, fundamental pro-groups and 2-dimensional proper co-{H}-spaces}},
     fjournal = {Topology and its Applications},
     journal = {Topology Appl.},
     issn = {0166-8641},
     volume = {153},
     number = {4},
     pages = {580--604},
     year = {2005},
     language = {English},
     doi = {10.1016/j.topol.2005.01.032},
     keywords = {55M30,55P57,55P45},
     }

@article{zbMATH05012977,
     author = {C{\'a}rdenas, M. and Muro, F. and Quintero, A.},
     title = {{The proper {L}-{S} category of {Whitehead} manifolds}},
     fjournal = {Topology and its Applications},
     journal = {Topology Appl.},
     issn = {0166-8641},
     volume = {153},
     number = {4},
     pages = {557--579},
     year = {2005},
     language = {English},
     doi = {10.1016/j.topol.2005.01.031},
     keywords = {55M30,55P57,57N65},
     }

@article{zbMATH02172214,
     author = {Muro, Fernando},
     title = {{Suspensions of crossed and quadratic complexes, co-{H}-structures and applications}},
     fjournal = {Transactions of the American Mathematical Society},
     journal = {Trans. Am. Math. Soc.},
     issn = {0002-9947},
     volume = {357},
     number = {9},
     pages = {3623--3653},
     year = {2005},
     language = {English},
     doi = {10.1090/S0002-9947-04-03597-4},
     keywords = {55P40,55P45,55P65,55U15,20F14,20F18,20J05},
     }

In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame cases. The algebra of row-column-finite (or locally finite) matrices over an arbitrary field is one of the algebras considered in this paper, its representation type is shown to be finite.

@article{zbMATH05068816,
     author = {Muro, Fernando},
     title = {{Representation theory of some infinite dimensional algebras arising in continuously controlled algebra and topology.}},
     fjournal = {\(K\)-Theory},
     journal = {\(K\)-Theory},
     issn = {0920-3036},
     volume = {33},
     number = {1},
     pages = {23--65},
     year = {2004},
     language = {English},
     doi = {10.1007/s10977-004-1837-4},
     keywords = {16G60,16S50,19J25,57N99},
     url = {idus.us.es/xmlui/handle/11441/41899},
     }

@article{zbMATH02010229,
     author = {Ayala, R. and C{\'a}rdenas, M. and Muro, F. and Quintero, A.},
     title = {{An elementary approach to the projective dimension in proper homotopy theory}},
     fjournal = {Communications in Algebra},
     journal = {Commun. Algebra},
     issn = {0092-7872},
     volume = {31},
     number = {12},
     pages = {5995--6017},
     year = {2003},
     language = {English},
     doi = {10.1081/AGB-120024863},
     keywords = {18G20,55P57,18G10,55P99},
     }