Set Of Morphisms Research Articles

Wreaths, mixed wreaths and twisted coactions

Distributive laws between two monads in a 2-category $\mathscr K$, as defined by Jon Beck in the case $\mathscr{K} = \mathrm{Cat}$, were pointed out by the author to be monads in a 2-category $\mathrm{Mnd}\mathscr{K}$ of monads. Steve Lack and the author defined wreaths to be monads in a 2-category $\mathrm{EM}\mathscr{K}$ of monads with different 2-cells from $\mathrm{Mnd}\mathscr{K}$. Mixed distributive laws were also considered by Jon Beck, Mike Barr and, later, various others; they are comonads in $\mathrm{Mnd}\mathscr{K}$. Actually, as pointed out by John Power and Hiroshi Watanabe, there are a number of dual possibilities for mixed distributive laws. It is natural then to consider mixed wreaths as we do in this article; they are comonads in $\mathrm{EM}\mathscr{K}$. There are also mixed opwreaths: comonads in the Kleisli construction completion $\mathrm{Kl}\mathscr{K}$ of $\mathscr{K}$. The main example studied here arises from a twisted coaction of a bimonoid on a monoid. A wreath determines a monad structure on the composite of the two endomorphisms involved; this monad is called the wreath product. For mixed wreaths, corresponding to this wreath product, is a convolution operation analogous to the convolution monoid structure on the set of morphisms from a comonoid to a monoid. In fact, wreath convolution is composition in a Kleisli-like construction. Walter Moreira’s Heisenberg product of linear endomorphisms on a Hopf algebra, is an example of such convolution, actually involving merely a mixed distributive law. Monoidality of the Kleisli-like construction is also discussed.

Ideal regular languages and strongly connected synchronizing automata

We introduce the notion of a reset left regular decomposition of an ideal regular language, and we prove that the category formed by these decompositions with the adequate set of morphisms is equivalent to the category of strongly connected synchronizing automata. We show that every ideal regular language has at least one reset left regular decomposition. As a consequence, every ideal regular language is the set of synchronizing words of some strongly connected synchronizing automaton. Furthermore, this one-to-one correspondence allows us to introduce the notion of reset decomposition complexity of an ideal from which follows a reformulation of Černý's conjecture in purely language theoretic terms. Finally, we present and characterize a subclass of ideal regular languages for which a better upper bound for the reset decomposition complexity holds with respect to the general case.

HOMOTOPY THEORY OF MODULES AND ADAMS COCOMPLETION

AbstractIn this note, we have obtained a Whitehead-like tower of a module by considering a suitable set of morphisms and shown that the different stages of the tower are the Adams cocompletions of the module with respect to the suitably chosen set of morphisms.

Supergeometry in Locally Covariant Quantum Field Theory

In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a functor A : SLoc --> S*Alg to the category of super-*-algebras which can be interpreted as a non-interacting super-quantum field theory. This construction turns out to disregard supersymmetry transformations as the morphism sets in the above categories are too small. We then solve this problem by using techniques from enriched category theory, which allows us to replace the morphism sets by suitable morphism supersets that contain supersymmetry transformations as their higher superpoints. We construct super-quantum field theories in terms of enriched functors eA : eSLoc --> eS*Alg between the enriched categories and show that supersymmetry transformations are appropriately described within the enriched framework. As examples we analyze the superparticle in 1|1-dimensions and the free Wess-Zumino model in 3|2-dimensions.

PRIMARY DECOMPOSITION OF A 0-CONNECTED NILPOTENT SPACE

It is known that the dierent stages of the Cartan-Whitehead decomposition of a 0-connected space can be obtained as the Adams cocompletions of the space with respect to suitable sets of morphisms. In this note Cartan-Whitehead decomposition is obtained for a nilpotent space, in terms of Adams cocompletion, using the primary homotopy theory developed by Neisendonfer.

The hammock localization preserves homotopies

The hammock localization provides a model for a homotopy function complex in any Quillen model category. We prove that a homotopy between a pair of morphisms induces a homotopy between the maps induced by taking the hammock localization. We describe applications of this fact to the study of homotopy algebras over monads and homotopy idempotent functors. Among other things, we prove that, under Vop\v{e}nka's principle, every homotopy idempotent functor in a cofibrantly generated model category is determined by simplicial orthogonality with respect to a set of morphisms. We also give a new proof of the fact that left Bousfield localizations with respect to a class of morphisms always exist in any left proper combinatorial model category under Vop\v{e}nka's principle.

Computing path categories of finite directed cubical complexes

A new method is introduced for simplifying finite directed cubical complexes. This construction is effective in the sense that it can be computed relatively efficiently, and it induces a fully faithful inclusion on path categories. The resulting parallelized family of algorithms enables calculations of morphism sets of path categories of such complexes in many cases of practical interest, and working code is provided as a proof of concept.

Kan injectivity in order-enriched categories

Continuous lattices were characterised by Martín Escardó as precisely those objects that are Kan-injective with respect to a certain class of morphisms. In this paper we study Kan-injectivity in general categories enriched in posets. As an example, ω-CPO's are precisely the posets that are Kan-injective with respect to the embeddings ω ↪ ω + 1 and 0 ↪ 1.For every class$\mathcal{H}$of morphisms, we study the subcategory of all objects that are Kan-injective with respect to$\mathcal{H}$and all morphisms preserving Kan extensions. For categories such asTop0andPos, we prove that whenever$\mathcal{H}$is a set of morphisms, the above subcategory is monadic, and the monad it creates is a Kock–Zöberlein monad. However, this does not generalise to proper classes, and we present a class of continuous mappings inTop0for which Kan-injectivity does not yield a monadic category.

Geometric combinatorics of Weyl groupoids

We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of morphisms with fixed target object forms an ortho-complemented meet semilattice. We define the Coxeter complex of a Weyl groupoid with finite root system and show that it coincides with the triangulation of a sphere cut out by a simplicial hyperplane arrangement. As a consequence, one obtains an algebraic interpretation of many hyperplane arrangements that are not reflection arrangements.

On the geometry of the moduli spaces of semi-stable sheaves supported on plane quartics

We decompose each moduli space of semi-stable sheaves on the complex projective plane with support of dimension one and degree four into locally closed subvarieties, each subvariety being the good or geometric quotient of a set of morphisms of locally free sheaves modulo a reductive or a non-reductive group. We find locally free resolutions of length one for all these sheaves and describe them.

The category-theoretic solution of recursive metric-space equations

It is well known that one can use an adaptation of the inverse-limit construction to solve recursive equations in the category of complete ultrametric spaces. We show that this construction generalizes to a large class of categories with metric-space structure on each set of morphisms: the exact nature of the objects is less important. In particular, the construction immediately applies to categories where the objects are ultrametric spaces with ‘extra structure’, and where the morphisms preserve this extra structure. The generalization is inspired by classical domain-theoretic work by Smyth and Plotkin. For many of the categories we consider, there is a natural subcategory in which each set of morphisms is required to be a compact metric space. Our setting allows for a proof that such a subcategory always inherits solutions of recursive equations from the full category. As another application, we present a construction that relates solutions of generalized domain equations in the sense of Smyth and Plotkin to solutions of equations in our class of categories. Our primary motivation for solving generalized recursive metric-space equations comes from recent and ongoing work on Kripke-style models in which the sets of worlds must be recursively defined. We show a series of examples motivated by this line of work.

Epireflections and supercompact cardinals

We prove that the existence of arbitrarily large supercompact cardinals implies that every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class. In other words, if L is a localization functor on a balanced accessible category such that the unit morphism X → L X is an epimorphism for all X and the class of L -local objects is defined by an absolute formula, then the existence of a sufficiently large supercompact cardinal implies that L is a localization with respect to some set of morphisms.

On two theorems of Flavell

We extend two theorems due to P. Flavell [6] to arbitrary fusion systems. The fusion system of a finite group G at a prime p encodes the structure of a Sylow-p-subgroup S together with extra information on G-conjugacy within S in category theoretic terms. From work of Alperin and Broue [1] it emerged that fusion systems of finite groups are particular cases of fusion systems of p-blocks of finite groups. In the early 1990’s, Puig introduced the notion of an abstract fusion system on a finite p-group S as a category whose objects are the subgroups of S and whose morphism sets satisfy a list of properties modelled around what one observes in the case of fusion systems of finite groups and blocks. There are ‘exotic’ fusion systems which do not arise as fusion system of a block. Benson [2] suggested that nonetheless any fusion system should give rise to a p-complete topological space which should coincide with the p-completion of the classifying space BG in case the fusion system does arise as fusion system of a finite group G. Broto, Levi and Oliver laid in [4] the homotopy theoretic foundations of such spaces called p-local finite groups and gave in particular a cohomological criterion for the existence and uniqueness of a p-local finite group associated with a given fusion system. While the existence and uniqueness of p-local finite groups for arbitrary fusion systems is still an open problem, there has been in recent years a steadily growing body of work by many authors trying to add to the understanding of fusion systems by extending classical concepts and results on the p-local structure of finite groups (some of which were relevant for the classification of finite simple groups) to all fusion systems. This is also the underlying philosophy of the present note. The following two theorems are Flavell’s Theorem A and Theorem B in [6], extended to arbitrary fusion systems. Our general terminology on fusion systems follows [7]; in particular, by a fusion system we always mean a saturated fusion system. Theorem 1. Let p be an odd prime, S a finite p-group, F a fusion system on S and α an automorphism of S acting freely on S−{1}, which stabilises F and whose order is a prime number r which does not divide the orders of the automorphism groups AutF(R) for all F-centric radical subgroups R of S. Then F = NF(S). The hypothesis that α stablises F means that for any two subgroups Q, R of S and any morphism φ : Q → R in F the morphism α ◦ φ ◦ α|α(Q) : α(Q) → α(R) is again a morphism in F . Moreover, a subgroup Q of S is called F -centric if CS(Q ) = Z(Q) for all subgroups Q of S such that Q ∼= Q in F ; a subgroup Q of S is called F -radical if Op(AutF (Q)) = AutQ(Q), the group of inner automorphisms of Q. By Alperin’s fusion theorem, F is completely determined by the automorphism groups in F of F -centric radical subgroups of S.

Fiber categories for describing dynamic production systems

It is shown that in the extension of the category-theoretical approach to the domain of dynamic systems based on knowledge, specific structures called fiber categories arise. Morphisms of this category are sets of morphisms, and in each inference process in this category one and only one morphism of this set takes part. It is determined by the external conditions, formally given by a certain algebraic lattice what this morphism is precisely. In other words, the category for describing an expert system in the dynamic case is a superposition of similar categories discriminated only by morphisms employed.

Morphismes géométriquement plats et faisceaux dualisants

Let n be an integer, S a reduced pure dimensional complex analytic space. Let E ( S , n ) (resp. G ( S , n ) ) be the set of morphisms π : X → S from an countable at infinity analytic complex space X to S which are n-equidimensional and open (resp. the set of geometrically flat map that is the element of E ( S , n ) endowed with cycle of X × S which induce a good multiplicity on the fibers such that ( π −1 ( s ) ) s ∈ S become an analytic family of cycles in the sense of Barlet). After showing that each π of E ( S , n ) is endowed with a canonical O X -coherent analytic sheaf ω π n with many functorial properties, we state the main result of this Note which says that π is in G ( S , n ) if and only if there exists a canonical morphism (called a relative fundamental class of π) Ω X / S n → ω π n , compatible with any base change between reduced finite pure dimensional complex spaces, which satisfies the relative trace property and gives, in the local setting, the relative fundamental class of Barlet constructed in 1980. From this, we deduce that π belong to G ( S , n ) if and only if ω π n is stable by base change and is characterized by the relative trace property; that means precisely that the relative Kunz–Waldi (1988) sheaf of regular differential forms constructed for flat analytic algebra by Kersken (1983) exists in this setting. To cite this article: M. Kaddar, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

An algebraic model for the loop space homology of a homotopy fiber

Let F denote the homotopy fiber of a map f:K-->L of 2-reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of K and L, we construct a small, explicit chain algebra, the homology of which is isomorphic as a graded algebra to the homology of GF, the simplicial (Kan) loop group on F. To construct this model, we develop machinery for modeling the homotopy fiber of a morphism of chain Hopf algebras. Essential to our construction is a generalization of the operadic description of the category DCSH of chain coalgebras and of strongly homotopy coalgebra maps given in (math.AT/0505559) to strongly homotopy morphisms of comodules over Hopf algebras. This operadic description is expressed in terms of a general theory of monoidal structures in categories with morphism sets parametrized by co-rings, which we elaborate here.

$\lambda$ -presentable Morphisms, Injectivity and (Weak) Factorization Systems

We show that in a locally $\lambda$ -presentable category, every $\lambda_m$ -injectivity class (i.e., the class of all the objects injective with respect to some class of $\lambda$ -presentable morphisms) is a weakly reflective subcategory determined by a functorial weak factorization system cofibrantly generated by a class of $\lambda$ -presentable morphisms. This was known for small-injectivity classes, and referred to as the ‘small object argument.’ An analogous result is obtained for orthogonality classes and factorization systems, where $\lambda$ -filtered colimits play the role of the transfinite compositions in the injectivity case. $\lambda$ -presentable morphisms are also used to organize and clarify some related results (and their proofs), in particular on the existence of enough injectives (resp. pure-injectives). Finally, locally $\lambda$ -presentable categories are shown to be cellularly generated by the set of morphisms between $\lambda$ -presentable objects.

Twisted quiver bundles over almost complex manifolds

In this paper, we study twisted quiver bundle over general almost complex manifolds. A twisted quiver bundle is a set of J-holomorphic vector bundles over an almost complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of J-holomorphic vector bundles, labelled by the arrows. We prove a Hitchin–Kobayashi correspondence for twisted quiver bundles over a compact almost Hermitian regularized manifold, relating the existence of solutions to certain gauge equations to an appropriate notion of stability for the corresponding quivers. This result can be seen as a generalization of that in [2,9].

Hitchin–Kobayashi Correspondence, Quivers, and Vortices

A twisted quiver bundle is a set of holomorphic vector bundles over a complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles, labelled by the arrows. When the manifold is Kaelher, quiver bundles admit natural gauge-theoretic equations, which unify many known equations for bundles with extra structure. In this paper we prove a Hitchin--Kobayashi correspondence for twisted quiver bundles over a compact Kaehler manifold, relating the existence of solutions to the gauge equations to a stability criterion, and consider its application to a number of situations related to Higgs bundles and dimensional reductions of the Hermitian--Einstein equations.

Localic Galois theory

In Proposition I of “Mémoire sur les conditions de résolubilité des équations par radicaux” Galois considers the splitting field A of a polynomial with rational coefficients. Given any intermediate extension X, he proves that the action of the galois group Gal(A/ Q) on the set of morphisms [ X, A] is transitive, and that X is the fixed field of its galois group Gal( A/ X). In this article we first state and prove a (dual) categorical formulation of these statements, which turns out to be a theorem about atomic sites with a representable point. These abstract developments correspond exactly to Classical Galois Theory. We then consider the same situation, but with the point no longer representable. It determines a proobject and it becomes (tautologically) prorepresentable. Mutatis mutandis, we state and prove the prorepresentable version of the categorical formulation of Galois theorems. In this case the classical group of automorphisms has to be replaced by the localic group of automorphisms. These developments form the content of a theory that we call Localic Galois Theory. From our results it immediately follows the theorem: A topos with a point is connected atomic if and only if it is the classifying topos of a localic group, and this group can be taken to be the locale of automorphisms of the point. The hard implication in this equivalence was first proved in print in Joyal and Tierney (Mem. Amer. Math. Soc. 151 (1984)), Theorem 1, Section 3, Chapter VIII, and it follows from a characterization of atomic topoi in terms of open maps and from a theory of descent for morphisms of topoi and locales. Our proof of this theorem is completely independent of descent theory and of any other result in Joyal and Tierney (1984). The theorem follows as a straightforward consequence of our direct generalization of the fundamental results of Galois. We explain and give the necessary definitions to understand this paper without previous knowledge of the theory of locales. Other than this, our principal results (Theorems 2.2 and 2.3) require only elementary notions of category theory.