Antonio Martinez Cegarra - Profile on Academia.edu (original) (raw)

Papers by Antonio Martinez Cegarra

Research paper thumbnail of Classifying spaces for braided monoidal categories and lax diagrams of bicategories

arXiv (Cornell University), Jul 6, 2009

This work contributes to clarifying several relationships between certain higher categorical stru... more This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be thought of as the homotopy colimit of the classifying spaces of the bicategories that arise from the initial input data given by the lax diagram. This result is applied to produce bicategories whose classifying space has a double loop space with the same homotopy type, up to group completion, as the underlying category of any given (non-necessarily strict) braided monoidal category. Specifically, it is proven that these double delooping spaces, for categories enriched with a braided monoidal structure, can be explicitly realized by means of certain genuine simplicial sets characteristically associated to any braided monoidal categories, which we refer to as their (Street's) geometric nerves.

Research paper thumbnail of Computability of the (co)homology of cyclic monoids

arXiv: K-Theory and Homology, 2016

Leech's (co)homology groups of finite cyclic monoids are computed.

Research paper thumbnail of A Dold-Kan Theorem for Simplicial Lie Algebras

We introduce and study hypercrossed complexes of Lie algebras, that is, non-negatively graded cha... more We introduce and study hypercrossed complexes of Lie algebras, that is, non-negatively graded chain complexes of Lie algebras L = (Ln, ∂n) endowed with an additional structure by means of a suitable set of bilinear maps Lr × Ls → Ln. The Moore complex of any simplicial Lie algebra acquires such a structure and, in this way, we prove a Dold-Kan type equivalence between the category of simplicial Lie algebras and the category of hypercrossed complexes of Lie algebras. Several consequences of examining particular classes of hypercrossed complexes of Lie algebras are presented.

Research paper thumbnail of Cohomology of Homotopy Colimits of Simplicial Sets and Small Categories

Mathematics, 2020

This paper deals with well-known weak homotopy equivalences that relate homotopy colimits of smal... more This paper deals with well-known weak homotopy equivalences that relate homotopy colimits of small categories and simplicial sets. We show that these weak homotopy equivalences have stronger cohomology-preserving properties than for local coefficients.

Research paper thumbnail of Cohomology of Presheaves of Monoids

Mathematics, 2020

The purpose of this work is to extend Leech cohomology for monoids (and so Eilenberg-Mac Lane coh... more The purpose of this work is to extend Leech cohomology for monoids (and so Eilenberg-Mac Lane cohomology of groups) to presheaves of monoids on an arbitrary small category. The main result states and proves a cohomological classification of monoidal prestacks on a category with values in groupoids with abelian isotropy groups. The paper also includes a cohomological classification for extensions of presheaves of monoids, which is useful to the study of H -extensions of presheaves of regular monoids. The results apply directly in several settings such as presheaves of monoids on a topological space, simplicial monoids, presheaves of simplicial monoids on a topological space, monoids or simplicial monoids on which a fixed monoid or group acts, and so forth.

Research paper thumbnail of Nonabelian cocycles and the spectrum of a symmetric monoidal category

TURKISH JOURNAL OF MATHEMATICS, 2019

We present an Eilenberg-MacLane-type description for the first, second and third spaces of the sp... more We present an Eilenberg-MacLane-type description for the first, second and third spaces of the spectrum defined by a symmetric monoidal category.

Research paper thumbnail of Higher cohomologies of commutative monoids

Journal of Pure and Applied Algebra, 2019

Extending Eilenberg-Mac Lane's methods, higher level cohomologies for commutative monoids are int... more Extending Eilenberg-Mac Lane's methods, higher level cohomologies for commutative monoids are introduced and studied. Relationships with pre-existing theories (Leech, Grillet, etc.) are stated. The paper includes a cohomological classification for symmetric monoidal groupoids and explicit computations for cyclic monoids.

Research paper thumbnail of A Cohomology Theory for Commutative Monoids

Mathematics, 2015

Extending Eilenberg-Mac Lane's cohomology of abelian groups, a cohomology theory is introduced fo... more Extending Eilenberg-Mac Lane's cohomology of abelian groups, a cohomology theory is introduced for commutative monoids. The cohomology groups in this theory agree with the pre-existing ones by Grillet in low dimensions, but they differ beyond dimension two. A natural interpretation is given for the three-cohomology classes in terms of braided monoidal groupoids.

Research paper thumbnail of Comparing geometric realizations of tricategories

Algebraic & Geometric Topology, 2014

This paper contains some contributions to the study of classifying spaces for tricategories, with... more This paper contains some contributions to the study of classifying spaces for tricategories, with applications to the homotopy theory of monoidal categories, bicategories, braided monoidal categories and monoidal bicategories. Any small tricategory has various associated simplicial or pseudosimplicial objects and we explore the relationship between three of them: the pseudosimplicial bicategory (so-called Grothendieck nerve) of the tricategory, the simplicial bicategory termed its Segal nerve and the simplicial set called its Street geometric nerve. We prove that the geometric realizations of all of these 'nerves of the tricategory' are homotopy equivalent. By using Grothendieck nerves we state the precise form in which the process of taking classifying spaces transports tricategorical coherence to homotopy coherence. Segal nerves allow us to prove that, under natural requirements, the classifying space of a monoidal bicategory is, in a precise way, a loop space. With the use of geometric nerves, we obtain simplicial sets whose simplices have a pleasing geometrical description in terms of the cells of the tricategory and we prove that, via the classifying space construction, bicategorical groups are a convenient algebraic model for connected homotopy 3-types.

Research paper thumbnail of Homotopy classification of categorical torsors

Applied Categorical Structures, 2001

The long-known results of Schreier on group extensions are here raised to a categorical level by ... more The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group (G, ⊗) over a small category B. We show a natural bijection between the set of equivalence classes of such torsors and [B(B), B(G, ⊗)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These results are applied to algebraically interpret the set of homotopy classes of maps from a CW-complex X to a path-connected CW-complex Y with π i (Y) = 0 for all i = 1, 2.

Research paper thumbnail of The relationship between the diagonal and the bar constructions on a bisimplicial set

Topology and its Applications, 2005

The aim of this paper is to prove that the homotopy type of any bisimplicial set X is modelled by... more The aim of this paper is to prove that the homotopy type of any bisimplicial set X is modelled by the simplicial set W X, the bar construction on X. We stress the interest of this result by showing two relevant theorems which now become simple instances of it; namely, the Homotopy colimit theorem of Thomason, for diagrams of small categories, and the generalized Eilenberg-Zilber theorem of Dold-Puppe for bisimplicial Abelian groups. Among other applications, we give an algebraic model for the homotopy theory of (not necessarily path-connected) spaces whose homotopy groups vanish in degree 4 and higher.

Research paper thumbnail of Bicategorical homotopy fiber sequences

Journal of Homotopy and Related Structures, 2013

Small Bénabou's bicategories and, in particular, Mac Lane's monoidal categories, have well-unders... more Small Bénabou's bicategories and, in particular, Mac Lane's monoidal categories, have well-understood classifying spaces, which give geometric meaning to their cells. This paper contains some contributions to the study of the relationship between bicategories and the homotopy types of their classifying spaces. Mainly, generalizations are given of Quillen's Theorems A and B to lax functors between bicategories.

Research paper thumbnail of Structure and classification of monoidal groupoids

Semigroup Forum, 2013

The structure of monoidal categories in which every arrow is invertible is analyzed in this paper... more The structure of monoidal categories in which every arrow is invertible is analyzed in this paper, where we develop a 3-dimensional Schreier-Grothendieck theory of non-abelian factor sets for their classification. In particular, we state and prove precise classification theorems for those monoidal groupoids whose isotropy groups are all abelian, as well as for their homomorphisms, by means of Leech's cohomology groups of monoids.

Research paper thumbnail of On the Geometry of 2-Categories and their Classifying Spaces

K-Theory, 2003

In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categor... more In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillen's Theorem A.

Research paper thumbnail of An exact sequence in the first variable for torsor cohomology: The 2-dimensional theory of obstructions

Journal of Pure and Applied Algebra, 1986

Research paper thumbnail of Higher dimensional obstruction theory in algebraic categories

Journal of Pure and Applied Algebra, 1987

In this paper we generalize Duskin's low dimensional obstruction theory, established for the Barr... more In this paper we generalize Duskin's low dimensional obstruction theory, established for the Barr-Beck's cotriple cohomology H~, to higher dimensions by giving a new interpretation of H~+ 1 in terms of obstructions to the existence of non-singular n-extensions or realizations to n-dimensional abstract kernels. We find a surjective map Obs from the set of all n-dimensional abstract kernels with center a fixed S-module A to H~+l(S, A) in such a way that an abstract kernel has a realization if and only if its obstruction vanishes, the set of equivalence classes of such realizations being in this case a principal homogeneous space over H'::,(S, A).

Research paper thumbnail of Homotopy fiber sequences induced by 2-functors

Journal of Pure and Applied Algebra, 2011

This paper contains some contributions to the study of the relationship between 2-categories and ... more This paper contains some contributions to the study of the relationship between 2-categories and the homotopy types of their classifying spaces. Mainly, generalizations are given of both Quillen's Theorem B and Thomason's Homotopy Colimit Theorem to 2-functors.

Research paper thumbnail of Homotopy classification of braided graded categorical groups

Journal of Pure and Applied Algebra, 2007

For any group G, a certain cohomology theory of G-modules is developed. This cohomology arises fr... more For any group G, a certain cohomology theory of G-modules is developed. This cohomology arises from the homotopy theory of G-spaces and it is called the "abelian cohomology of G-modules". Then, as the main results of this paper, natural one-toone correspondences between elements of the 3 rd cohomology groups of G-modules, G-equivariant pointed simply-connected homotopy 3-types and equivalence classes of braided G-graded categorical groups are established. The relationship among all these objects with equivariant quadratic functions between G-modules is also discussed.

Research paper thumbnail of Graded extensions of categories

Journal of Pure and Applied Algebra, 2000

The problem of extending categories by groups, including theory of obstructions, is studied by me... more The problem of extending categories by groups, including theory of obstructions, is studied by means of factor systems and various homological invariants, generalized from Schreier-Eilenberg-Mac Lane group extension theory. Explicit applications are then given to the classiÿcation of several algebraic constructions long known as crossed products, appearing in many di erent contexts such as monoids, Cli ord systems or twisted group rings.

Research paper thumbnail of Graded Extensions of Monoidal Categories

Journal of Algebra, 2001

The long-known results of Schreier᎐Eilenberg᎐Mac Lane on group extensions are raised to a categor... more The long-known results of Schreier᎐Eilenberg᎐Mac Lane on group extensions are raised to a categorical level, for the classification and construction of the manifold of all graded monoidal categories, the type being given group ⌫ with 1-component a given monoidal category. Explicit application is made to the classification of strongly graded bialgebras over commutative rings.

Research paper thumbnail of Classifying spaces for braided monoidal categories and lax diagrams of bicategories

arXiv (Cornell University), Jul 6, 2009

This work contributes to clarifying several relationships between certain higher categorical stru... more This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be thought of as the homotopy colimit of the classifying spaces of the bicategories that arise from the initial input data given by the lax diagram. This result is applied to produce bicategories whose classifying space has a double loop space with the same homotopy type, up to group completion, as the underlying category of any given (non-necessarily strict) braided monoidal category. Specifically, it is proven that these double delooping spaces, for categories enriched with a braided monoidal structure, can be explicitly realized by means of certain genuine simplicial sets characteristically associated to any braided monoidal categories, which we refer to as their (Street's) geometric nerves.

Research paper thumbnail of Computability of the (co)homology of cyclic monoids

arXiv: K-Theory and Homology, 2016

Leech's (co)homology groups of finite cyclic monoids are computed.

Research paper thumbnail of A Dold-Kan Theorem for Simplicial Lie Algebras

We introduce and study hypercrossed complexes of Lie algebras, that is, non-negatively graded cha... more We introduce and study hypercrossed complexes of Lie algebras, that is, non-negatively graded chain complexes of Lie algebras L = (Ln, ∂n) endowed with an additional structure by means of a suitable set of bilinear maps Lr × Ls → Ln. The Moore complex of any simplicial Lie algebra acquires such a structure and, in this way, we prove a Dold-Kan type equivalence between the category of simplicial Lie algebras and the category of hypercrossed complexes of Lie algebras. Several consequences of examining particular classes of hypercrossed complexes of Lie algebras are presented.

Research paper thumbnail of Cohomology of Homotopy Colimits of Simplicial Sets and Small Categories

Mathematics, 2020

This paper deals with well-known weak homotopy equivalences that relate homotopy colimits of smal... more This paper deals with well-known weak homotopy equivalences that relate homotopy colimits of small categories and simplicial sets. We show that these weak homotopy equivalences have stronger cohomology-preserving properties than for local coefficients.

Research paper thumbnail of Cohomology of Presheaves of Monoids

Mathematics, 2020

The purpose of this work is to extend Leech cohomology for monoids (and so Eilenberg-Mac Lane coh... more The purpose of this work is to extend Leech cohomology for monoids (and so Eilenberg-Mac Lane cohomology of groups) to presheaves of monoids on an arbitrary small category. The main result states and proves a cohomological classification of monoidal prestacks on a category with values in groupoids with abelian isotropy groups. The paper also includes a cohomological classification for extensions of presheaves of monoids, which is useful to the study of H -extensions of presheaves of regular monoids. The results apply directly in several settings such as presheaves of monoids on a topological space, simplicial monoids, presheaves of simplicial monoids on a topological space, monoids or simplicial monoids on which a fixed monoid or group acts, and so forth.

Research paper thumbnail of Nonabelian cocycles and the spectrum of a symmetric monoidal category

TURKISH JOURNAL OF MATHEMATICS, 2019

We present an Eilenberg-MacLane-type description for the first, second and third spaces of the sp... more We present an Eilenberg-MacLane-type description for the first, second and third spaces of the spectrum defined by a symmetric monoidal category.

Research paper thumbnail of Higher cohomologies of commutative monoids

Journal of Pure and Applied Algebra, 2019

Extending Eilenberg-Mac Lane's methods, higher level cohomologies for commutative monoids are int... more Extending Eilenberg-Mac Lane's methods, higher level cohomologies for commutative monoids are introduced and studied. Relationships with pre-existing theories (Leech, Grillet, etc.) are stated. The paper includes a cohomological classification for symmetric monoidal groupoids and explicit computations for cyclic monoids.

Research paper thumbnail of A Cohomology Theory for Commutative Monoids

Mathematics, 2015

Extending Eilenberg-Mac Lane's cohomology of abelian groups, a cohomology theory is introduced fo... more Extending Eilenberg-Mac Lane's cohomology of abelian groups, a cohomology theory is introduced for commutative monoids. The cohomology groups in this theory agree with the pre-existing ones by Grillet in low dimensions, but they differ beyond dimension two. A natural interpretation is given for the three-cohomology classes in terms of braided monoidal groupoids.

Research paper thumbnail of Comparing geometric realizations of tricategories

Algebraic & Geometric Topology, 2014

This paper contains some contributions to the study of classifying spaces for tricategories, with... more This paper contains some contributions to the study of classifying spaces for tricategories, with applications to the homotopy theory of monoidal categories, bicategories, braided monoidal categories and monoidal bicategories. Any small tricategory has various associated simplicial or pseudosimplicial objects and we explore the relationship between three of them: the pseudosimplicial bicategory (so-called Grothendieck nerve) of the tricategory, the simplicial bicategory termed its Segal nerve and the simplicial set called its Street geometric nerve. We prove that the geometric realizations of all of these 'nerves of the tricategory' are homotopy equivalent. By using Grothendieck nerves we state the precise form in which the process of taking classifying spaces transports tricategorical coherence to homotopy coherence. Segal nerves allow us to prove that, under natural requirements, the classifying space of a monoidal bicategory is, in a precise way, a loop space. With the use of geometric nerves, we obtain simplicial sets whose simplices have a pleasing geometrical description in terms of the cells of the tricategory and we prove that, via the classifying space construction, bicategorical groups are a convenient algebraic model for connected homotopy 3-types.

Research paper thumbnail of Homotopy classification of categorical torsors

Applied Categorical Structures, 2001

The long-known results of Schreier on group extensions are here raised to a categorical level by ... more The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group (G, ⊗) over a small category B. We show a natural bijection between the set of equivalence classes of such torsors and [B(B), B(G, ⊗)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These results are applied to algebraically interpret the set of homotopy classes of maps from a CW-complex X to a path-connected CW-complex Y with π i (Y) = 0 for all i = 1, 2.

Research paper thumbnail of The relationship between the diagonal and the bar constructions on a bisimplicial set

Topology and its Applications, 2005

The aim of this paper is to prove that the homotopy type of any bisimplicial set X is modelled by... more The aim of this paper is to prove that the homotopy type of any bisimplicial set X is modelled by the simplicial set W X, the bar construction on X. We stress the interest of this result by showing two relevant theorems which now become simple instances of it; namely, the Homotopy colimit theorem of Thomason, for diagrams of small categories, and the generalized Eilenberg-Zilber theorem of Dold-Puppe for bisimplicial Abelian groups. Among other applications, we give an algebraic model for the homotopy theory of (not necessarily path-connected) spaces whose homotopy groups vanish in degree 4 and higher.

Research paper thumbnail of Bicategorical homotopy fiber sequences

Journal of Homotopy and Related Structures, 2013

Small Bénabou's bicategories and, in particular, Mac Lane's monoidal categories, have well-unders... more Small Bénabou's bicategories and, in particular, Mac Lane's monoidal categories, have well-understood classifying spaces, which give geometric meaning to their cells. This paper contains some contributions to the study of the relationship between bicategories and the homotopy types of their classifying spaces. Mainly, generalizations are given of Quillen's Theorems A and B to lax functors between bicategories.

Research paper thumbnail of Structure and classification of monoidal groupoids

Semigroup Forum, 2013

The structure of monoidal categories in which every arrow is invertible is analyzed in this paper... more The structure of monoidal categories in which every arrow is invertible is analyzed in this paper, where we develop a 3-dimensional Schreier-Grothendieck theory of non-abelian factor sets for their classification. In particular, we state and prove precise classification theorems for those monoidal groupoids whose isotropy groups are all abelian, as well as for their homomorphisms, by means of Leech's cohomology groups of monoids.

Research paper thumbnail of On the Geometry of 2-Categories and their Classifying Spaces

K-Theory, 2003

In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categor... more In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillen's Theorem A.

Research paper thumbnail of An exact sequence in the first variable for torsor cohomology: The 2-dimensional theory of obstructions

Journal of Pure and Applied Algebra, 1986

Research paper thumbnail of Higher dimensional obstruction theory in algebraic categories

Journal of Pure and Applied Algebra, 1987

In this paper we generalize Duskin's low dimensional obstruction theory, established for the Barr... more In this paper we generalize Duskin's low dimensional obstruction theory, established for the Barr-Beck's cotriple cohomology H~, to higher dimensions by giving a new interpretation of H~+ 1 in terms of obstructions to the existence of non-singular n-extensions or realizations to n-dimensional abstract kernels. We find a surjective map Obs from the set of all n-dimensional abstract kernels with center a fixed S-module A to H~+l(S, A) in such a way that an abstract kernel has a realization if and only if its obstruction vanishes, the set of equivalence classes of such realizations being in this case a principal homogeneous space over H'::,(S, A).

Research paper thumbnail of Homotopy fiber sequences induced by 2-functors

Journal of Pure and Applied Algebra, 2011

This paper contains some contributions to the study of the relationship between 2-categories and ... more This paper contains some contributions to the study of the relationship between 2-categories and the homotopy types of their classifying spaces. Mainly, generalizations are given of both Quillen's Theorem B and Thomason's Homotopy Colimit Theorem to 2-functors.

Research paper thumbnail of Homotopy classification of braided graded categorical groups

Journal of Pure and Applied Algebra, 2007

For any group G, a certain cohomology theory of G-modules is developed. This cohomology arises fr... more For any group G, a certain cohomology theory of G-modules is developed. This cohomology arises from the homotopy theory of G-spaces and it is called the "abelian cohomology of G-modules". Then, as the main results of this paper, natural one-toone correspondences between elements of the 3 rd cohomology groups of G-modules, G-equivariant pointed simply-connected homotopy 3-types and equivalence classes of braided G-graded categorical groups are established. The relationship among all these objects with equivariant quadratic functions between G-modules is also discussed.

Research paper thumbnail of Graded extensions of categories

Journal of Pure and Applied Algebra, 2000

The problem of extending categories by groups, including theory of obstructions, is studied by me... more The problem of extending categories by groups, including theory of obstructions, is studied by means of factor systems and various homological invariants, generalized from Schreier-Eilenberg-Mac Lane group extension theory. Explicit applications are then given to the classiÿcation of several algebraic constructions long known as crossed products, appearing in many di erent contexts such as monoids, Cli ord systems or twisted group rings.

Research paper thumbnail of Graded Extensions of Monoidal Categories

Journal of Algebra, 2001

The long-known results of Schreier᎐Eilenberg᎐Mac Lane on group extensions are raised to a categor... more The long-known results of Schreier᎐Eilenberg᎐Mac Lane on group extensions are raised to a categorical level, for the classification and construction of the manifold of all graded monoidal categories, the type being given group ⌫ with 1-component a given monoidal category. Explicit application is made to the classification of strongly graded bialgebras over commutative rings.