Embedding Functor Research Articles

Goodwillie’s cosimplicial model for the space of long knots and its applications

We work out the details of a correspondence observed by Goodwillie between cosimplicial spaces and good functors from a category of open subsets of the interval to the category of compactly generated weak Hausdorff spaces. Using this, we compute the first page of the integral Bousfield–Kan homotopy spectral sequence of the tower of fibrations, given by the Taylor tower of the embedding functor associated to the space of long knots. Based on the methods in Conant (Am J Math 130(2):341–357. https://doi.org/10.1353/ajm.2008.0020, 2008), we give a combinatorial interpretation of the differentials d^1 mapping into the diagonal terms, by introducing the notion of (i, n)-marked unitrivalent graphs.

The homotopy theory of operad subcategories

We study the subcategory of topological operads $P$ such that $P(0) = *$ (the category of unitary operads in our terminology). We use that this category inherits a model structure, like the category of all operads in topological spaces, and that the embedding functor of this subcategory of unitary operads into the category of all operads admits a left Quillen adjoint. We prove that the derived functor of this left Quillen adjoint functor induces a left inverse of the derived functor of our category embedding at the homotopy category level. We deduce from this result that the derived mapping spaces associated to our model category of unitary operads are homotopy equivalent to the standard derived operad mapping spaces, which we form in the model category of all operads in topological spaces. We prove that analogous statements hold for the subcategory of $k$-truncated unitary operads within the model category of all $k$-truncated operads, for any fixed arity bound $k\geq 1$, where a $k$-truncated operad denotes an operad that is defined up to arity $k$.

On the relation between hyperrings and fuzzy rings

We construct a full embedding of the category of hyperfields into Dress’s category of fuzzy rings and explicitly characterize the essential image—it fails to be essentially surjective in a very minor way. This embedding provides an identification of Baker and Bowler’s theory of strong matroids over hyperfields with Dress’s theory of matroids over fuzzy rings (provided one restricts to those fuzzy rings in the essential image). The embedding functor extends from hyperfields to hyperrings, and we study this extension in detail. We also analyze the relation between hyperfields and Baker and Bowler’s partial demifields.

Embedding functor for classical groups and Brauer–Manin obstruction

Embedding functor for classical groups and Brauer–Manin obstruction

Abelian model structures and Ding homological dimensions

Let $R$ be an $n$-FC ring. For $0<t\leq n$, we construct a new abelian model structure on $R$-Mod, called the Ding $t$-projective ($t$-injective) model structure. Based on this, we establish a bijective correspondence between $dg$-$t$-projective ($dg$-$t$-injective) $R$-complexes and Ding $t$-projective ($t$-injective) $A$-modules under some additional conditions, where $A=R[x]/(x^2)$. This gives a generalized version of the bijective correspondence established in [14] between $dg$-projective ($dg$-injective) $R$-complexes and Gorenstein projective (injective) $A$-modules. Finally, we show that the embedding functors $K(\mathcal{D} \mathcal{P})\rightarrow K$ ($R$-Mod) and $K(\mathcal{D} \mathcal{J})\rightarrow K$ ($R$-Mod) have right and left adjoints respectively, where $K(\mathcal{D} \mathcal{P})$ ($K(\mathcal{D} \mathcal{J})$) is the homotopy category of complexes of Ding projective (injective) modules, and $K$ ($R$-Mod) denotes the homotopy category.

Presheaves of Superselection Structures in Curved Spacetimes

We show that superselection structures on curved spacetimes that are expected to describe quantum charges affected by the underlying geometry are categories of sections of presheaves of symmetric tensor categories. When an embedding functor is given, the superselection structure is a Tannaka-type dual of a locally constant group bundle, which hence becomes a natural candidate for the role of the gauge group. Indeed, we show that any locally constant group bundle (with suitable structure group) acts on a net of C* algebras fulfilling normal commutation relations on an arbitrary spacetime. We also give examples of gerbes of C* algebras, defined by Wightman fields and constructed using projective representations of the fundamental group of the spacetime, which we propose as solutions for the problem that existence and uniqueness of the embedding functor are not guaranteed.

Embedding functors and their arithmetic properties

In this article, we focus on how to embed a torus \mathrm{T} into a reductive group \mathrm{G} with respect to a given root datum \Psi over a scheme \mathrm{S} . This problem is also related to embedding an étale algebra with involution into a central simple algebra with involution (cf. [PR10]). We approach this problem by defining the embedding functor, which is representable and is a left homogeneous space over \mathrm{S} under the automorphism group of \mathrm{G} . In order to fix a connected component of the embedding functor, we define an orientation u of \Psi with respect to \mathrm{G} . We show that the oriented embedding functor is also representable and is a homogeneous space under the adjoint action of \mathrm{G} . Over a local field, the orientation u and the Tits index of \mathrm{G} determine the existence of an embedding of \mathrm{T} into \mathrm{G} with respect to the given root datum \Psi . We also use the techniques developed in Borovoi’s paper [Bo99] to prove that the local-global principle holds for oriented embedding functors in certain cases. Actually, the Brauer–Manin obstruction is the only obstruction to the local-global principle for the oriented embedding functor. Finally, we apply the results on oriented embedding functors to give an alternative proof of Prasad and Rapinchuk’s Theorem, and to improve Theorem 7.3 in [PR10].

ON GENERALISED PRONORMAL SUBGROUPS OF FINITE GROUPS

AbstractFor a formation $\mathfrak F$, a subgroup M of a finite group G is said to be $\mathfrak F$-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug〉$\mathfrak F$ such that Ux = Ug. Let f be a subgroup embedding functor such that f(G) contains the set of normal subgroups of G and is contained in the set of Sylow-permutable subgroups of G for every finite group G. Given such an f, let fT denote the class of finite groups in which f(G) is the set of subnormal subgroups of G; this is the class of all finite groups G in which to be in f(G) is a transitive relation in G. A subgroup M of a finite group G is said to be $\mathfrak F$-normal in G if G/CoreG(M) belongs to $\mathfrak F$. A subgroup U of a finite group G is called K-$\mathfrak F$-subnormal in G if either U = G or there exist subgroups U = U0 ≤ U1 ≤ . . . ≤ Un = G such that Ui–1 is either normal or $\mathfrak F$-normal in Ui, for i = 1,2, …, n. We call a finite group G an $fT_{\mathfrak F}$-group if every K-$\mathfrak F$-subnormal subgroup of G is in f(G). In this paper, we analyse for certain formations $\mathfrak F$ the structure of $fT_{\mathfrak F}$-groups. We pay special attention to the $\mathfrak F$-pronormal subgroups in this analysis.

A theory of induction and classification of tensor C*-categories

This paper addresses the problem of describing the structure of tensor C*-categories \mathcal{M} with conjugates and irreducible tensor unit. No assumption on the existence of a braided symmetry or on amenability is made. Our assumptions are motivated by the remark that these categories often contain non-full tensor C*-subcategories with conjugates and the same objects admitting an embedding into the Hilbert spaces. Such an embedding defines a compact quantum group by Woronowicz duality. An important example is the Temperley–Lieb category canonically contained in a tensor C*-category generated by a single real or pseudoreal object of dimension ≥ 2 . The associated quantum groups are the universal orthogonal quantum groups of Wang and Van Daele. Our main result asserts that there is a full and faithful tensor functor from \mathcal{M} to a category of Hilbert bimodule representations of the compact quantum group. In the classical case, these bimodule representations reduce to the G -equivariant Hermitian bundles over compact homogeneous G -spaces, with G a compact group. Our structural results shed light on the problem of whether there is an embedding functor of \mathcal{M} into the Hilbert spaces. We show that this is related to the problem of whether a classical compact Lie group can act ergodically on a non-type I von Neumann algebra. In particular, combining this with a result of Wassermann shows that an embedding exists if \mathcal{M} is generated by a pseudoreal object of dimension 2.

Amalgamations of Categories

AbstractWe consider the pushout of embedding functors in Cat, the category of small categories. We show that if the embedding functors satisfy a 3-for-2 property, then the induced functors to the pushout category are also embeddings. The result follows from the connectedness of certain associated slice categories. The condition is motivated by a similar result for maps of semigroups. We show that our theorem can be applied to groupoids and to inclusions of full subcategories. We also give an example to show that the theorem does not hold when the property only holds for one of the inclusion functors, or when it is weakened to a one-sided condition.

Derivatives of embedding functors I: the stable case

For smooth manifolds $M$ and $N$, let $\Ebar(M, N)$ be the homotopy fiber of the map $\Emb(M, N)\longrightarrow \Imm(M, N)$. Consider the functor from the category of Euclidean spaces to the category of spectra, defined by the formula $V\mapsto \Sigma^\infty\Ebar(M, N\times V)$. In this paper, we describe the Taylor polynomials of this functor, in the sense of M. Weiss' orthogonal calculus, in the case when $N$ is a nice open submanifold of a Euclidean space. This leads to a description of the derivatives of this functor when $N$ is a tame stably parallelizable manifold (we believe that the parallelizability assumption is not essential). Our construction involves a certain space of rooted forests (or, equivalently, a space of partitions) with leaves marked by points in $M$, and a certain ``homotopy bundle of spectra'' over this space of trees. The $n$-th derivative is then described as the ``spectrum of restricted sections'' of this bundle. This is the first in a series of two papers. In the second part, we will give an analogous description of the derivatives of the functor $\Ebar(M, N\times V)$, involving a similar construction with certain spaces of connected graphs (instead of forests) with points marked in $M$.

MONOIDS, EMBEDDING FUNCTORS AND QUANTUM GROUPS

We show that the left regular representation πl of a discrete quantum group (A, Δ) has the absorbing property and forms a monoid [Formula: see text] in the representation category Rep (A, Δ). Next we show that an absorbing monoid in an abstract tensor *-category [Formula: see text] gives rise to an embedding functor (or fiber functor) [Formula: see text], and we identify conditions on the monoid, satisfied by [Formula: see text], implying that E is *-preserving. As is well-known, from an embedding functor [Formula: see text] the generalized Tannaka theorem produces a discrete quantum group (A, Δ) such that [Formula: see text]. Thus, for a C*-tensor category [Formula: see text] with conjugates and irreducible unit the following are equivalent: (1) [Formula: see text] is equivalent to the representation category of a discrete quantum group (A, Δ), (2) [Formula: see text] admits an absorbing monoid, (3) there exists a *-preserving embedding functor [Formula: see text].

A New Proof on Embedding the Category of Proximity Spaces into the Category of Nearness Spaces

This paper establishes an explicit embedding functor from the category of the proximity spaces into the category of nearness spaces. This is done by defining methods to constructively inducing a pr...

Configuration space integrals and Taylor towers for spaces of knots

We describe Taylor towers for spaces of knots arising from Goodwillie–Weiss calculus of the embedding functor and extend the configuration space integrals of Bott and Taubes from spaces of knots to the stages of the towers. We show that certain combinations of integrals, indexed by trivalent diagrams, yield cohomology classes of the stages of the tower, just as they do for ordinary knots.

Finite type knot invariants and the calculus of functors

We associate a Taylor tower supplied by the calculus of the embedding functor to the space of long knots and study its cohomology spectral sequence. The combinatorics of the spectral sequence along the line of total degree zero leads to chord diagrams with relations as in finite type knot theory. We show that the spectral sequence collapses along this line and that the Taylor tower represents a universal finite type knot invariant.

Fuzzy convergence: open and proper maps

The notions of open and proper maps are introduced in the category FCS of fuzzy convergence spaces and each is shown to be an extension of the same notion with respect to the embedding functor from the category PCS of probabilistic convergence spaces. Proper maps in FCS are shown to be productive and, moreover, preserve closures of fuzzy sets, regularity and local precompactness. All projection maps in FCS are open and open maps are finitely productive.

Representations of Algebraic Quantum Groups and Reconstruction Theorems for Tensor Categories

We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the Tannaka–Krein reconstruction problem. We show that every concrete semisimple tensor *-category with conjugates is equivalent to the category of finite-dimensional nondegenerate *-representations of a discrete algebraic quantum group. Working in the self-dual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and R-matrices in this context. Our approach emphasizes the role of the natural transformations of the embedding functor. Thanks to the semisimplicity of our categories and the emphasis on representations rather than corepresentations, our proof is more direct and conceptual than previous reconstructions. As a special case, we reprove the classical Tannaka–Krein result for compact groups. It is only here that analytic aspects enter, otherwise we proceed in a purely algebraic way. In particular, the existence of a Haar functional is reduced to a well-known general result concerning discrete multiplier Hopf *-algebras.

Regularity in fuzzy convergence spaces

The notions of regularity and weak regularity are introduced in the category of fuzzy convergence spaces, and each is shown to be an extension of regularity with respect to the embedding functor from the category of convergence spaces. Moreover, the category whose objects are weakly regular fuzzy spaces is topological and thus initial and final regular structures exist. The relationship between regularity and a diagonal condition is given.

A categorical version of the Lefschets-Nöbeling-Pontryagin theorem on embedding compacta in R n

For a category K we use Ob(K) to denote the class of all objects of K; if X, Y ϵ Ob( K), then Mor K ( X, Y) is the set of all K-morphisms from X into Y. Let A and B be subcategories of the category of all topological spaces and their continuous maps. We say that a covariant functor F: A → B is an embedding functor if there exists a class { i x : X ϵ Ob( A)} satisfying the following conditions: 1. (i) i x : X → F( X) is a homeomorphic embedding for every X ϵ Ob( A), and 2. (ii) if X, Y ϵ Ob( A) and f ϵ Mor K ( X, Y), then F( f) o i x = i y o f. For a natural number n let C( n) denote the category of all n-dimensional compact metric spaces and their continuous maps. Let G(< ∞) be the category of all Hausdorff finite-dimensional topological groups and their continuous group homomorphisms. We prove that there is no embedding covariant functor F: C(1) → G(< ∞), but there exists a covariant embedding functor F: C(0) → G(0), where G(0) is the category consisting of the single (zero-dimensional) compact metric group Z 2 ω and all its continuous group homomorphisms into itself, i.e., Ob( G(0)) = { Z 2 ω } and Mor G(0)(Z 2 ω, Z 2 ω) is the set of all continuous group homomorphisms from Z 2 ω into Z 2 ω.

Submodule lattice quasivarieties and exact embedding functors for rings with prime power characteristic

Given ringsR with prime power characteristicpk, quasivarieties ℒ(R) of lattices generated by lattices of submodules ofR-modules are studied. An algebra of expressionsd not dependent onR is developed, such that each suchd uniquely determines a two-sides idealdR ofR. The main technical result is that ℒ(R)\( \subseteq\)ℒ(S) makes all implications of the formds=S⇒ dR=R true, for any such expressiond. The proof makes use of the known equivalence between ℒ(R)\( \subseteq\)ℒ(S) and existence of an exact embedding functorR-Mod →S-Mod. Fork ≥ 2, the ordered setW(pk) of all lattice quasivarieties ℒ(R),R having characteristic pK, is shown to be large and complicated, with ascending and descending chains and antichains having continuously many elements. More precisely,W(pk) has a subset which is order isomorphic to the Boolean algebra of all subsets of a denumerably infinite set. Also, given any prime powerpk,k ≥ 2, a ringR can be constructed so that ℒ(R) and ℒ(Rop) for the opposite ringRop are distinct elements ofW(pk).