Applied Categorical Structures最新文献

A tangent category is a categorical abstraction of the tangent bundle construction for smooth manifolds. In that context, Cockett and Cruttwell develop the notion of differential bundle which, by work of MacAdam, generalizes the notion of smooth vector bundle to the abstract setting. Here we provide a new characterization of differential bundles and show that, up to isomorphism, a differential bundle is determined by its projection map and zero section. We show how these results can be used to quickly identify differential bundles in various tangent categories.

We extend Willerton’s [24] graphical calculus for bimonads to comodule monads, a monadic interpretation of module categories over a monoidal category. As an application, we prove a version of Tannaka–Krein duality for these structures.

(mathfrak {KNJ}) is the category of compact normal joinfit frames and frame homomorphisms. (mathcal {P}F) is the complete boolean algebra of polars of the frame F. A function (mathfrak {X}) that assigns to each (F in mathfrak {KNJ}) a subalgebra (mathfrak {X}(F)) of (mathcal {P}F) that contains the complemented elements of F is a polar function. A polar function (mathfrak {X}) is invariant (resp., functorial) if whenever (phi : F longrightarrow H in mathfrak {KNJ}) is (mathcal {P})-essential (resp., skeletal) and (p in mathfrak {X}(F)), then (phi (p)^{perp perp } in mathfrak {X}(H)). (phi : F longrightarrow H in mathfrak {KNJ}) is (mathfrak {X})-splitting if (phi ) is (mathcal {P})-essential and whenever (p in mathfrak {X}(F)), then (phi (p)^{perp perp }) is complemented in H. (F in mathfrak {KNJ}) is (mathfrak {X})-projectable means that every (p in mathfrak {X}(F)) is complemented. For a polar function (mathfrak {X}) and (F in mathfrak {KNJ}), we construct the least (mathfrak {X})-splitting frame of F. Moreover, we prove that if (mathfrak {X}) is a functorial polar function, then the class of (mathfrak {X})-projectable frames is a (mathcal {P})-essential monoreflective subcategory of (mathfrak {KNJS}), the category of (mathfrak {KNJ})-objects and skeletal maps (the case (mathfrak {X}= mathcal {P}) is the result from Martínez and Zenk, which states that the class of strongly projectable (mathfrak {KNJ})-objects is a reflective subcategory of (mathfrak {KNJS})).

Affine schemes can be understood as objects of the opposite of the category of commutative and unital algebras. Similarly, (mathscr {P})-affine schemes can be defined as objects of the opposite of the category of algebras over an operad (mathscr {P}). An example is the opposite of the category of associative algebras. The category of operadic schemes of an operad carries a canonical tangent structure. This paper aims to initiate the study of the geometry of operadic affine schemes via this tangent category. For example, we expect the tangent structure over the opposite of the category of associative algebras to describe algebraic non-commutative geometry. In order to initiate such a program, the first step is to classify differential bundles, which are the analogs of vector bundles for differential geometry. In this paper, we prove that the tangent category of affine schemes of the enveloping operad (mathscr {P}^{(A)}) over a (mathscr {P})-affine scheme A is precisely the slice tangent category over A of (mathscr {P})-affine schemes. We are going to employ this result to show that differential bundles over a (mathscr {P})-affine scheme A are precisely A-modules in the operadic sense.

In a previous paper we introduced the concept of semiseparable functor. Here we continue our study of these functors in connection with idempotent (Cauchy) completion. To this aim, we introduce and investigate the notions of (co)reflection and bireflection up to retracts. We show that the (co)comparison functor attached to an adjunction whose associated (co)monad is separable is a coreflection (reflection) up to retracts. This fact allows us to prove that a right (left) adjoint functor is semiseparable if and only if the associated (co)monad is separable and the (co)comparison functor is a bireflection up to retracts, extending a characterization pursued by X.-W. Chen in the separable case. Finally, we provide a semi-analogue of a result obtained by P. Balmer in the framework of pre-triangulated categories.

We study model structures on the category of comodules of a supercommutative Hopf algebra A over fields of characteristic 0. Given a graded Hopf algebra quotient (A rightarrow B) satisfying some finiteness conditions, the Frobenius tensor category ({mathcal {D}}) of graded B-comodules with its stable model structure induces a monoidal model structure on ({mathcal {C}}). We consider the corresponding homotopy quotient (gamma : {mathcal {C}} rightarrow Ho {mathcal {C}}) and the induced quotient ({mathcal {T}} rightarrow Ho {mathcal {T}}) for the tensor category ({mathcal {T}}) of finite dimensional A-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in (Ho {mathcal {T}}). We apply these results in the Rep(GL(m|n))-case and study its homotopy category (Ho {mathcal {T}}) associated to the parabolic subgroup of upper triangular block matrices. We construct cofibrant replacements and show that the quotient of (Ho{mathcal {T}}) by the negligible morphisms is again the representation category of a supergroup scheme.

We construct a category ({textrm{HomCob}}) whose objects are homotopically 1-finitely generated topological spaces, and whose morphisms are cofibrant cospans. Given a manifold submanifold pair (M, A), we prove that there exists functors into ({textrm{HomCob}}) from the full subgroupoid of the mapping class groupoid (textrm{MCG}_{M}^{A}), and from the full subgroupoid of the motion groupoid (textrm{Mot}_{M}^{A}), whose objects are homotopically 1-finitely generated. We also construct a family of functors ({textsf{Z}}_G:{textrm{HomCob}}rightarrow {textbf{Vect}}), one for each finite group G. These generalise topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf–Witten. Given a space X, we prove that ({textsf{Z}}_G(X)) can be expressed as the ({mathbb {C}})-vector space with basis natural transformation classes of maps from (pi (X,X_0)) to G for some finite representative set of points (X_0subset X), demonstrating that ({textsf{Z}}_G) is explicitly calculable.

This paper studies presentations of the Sierpinski gasket as a final coalgebra for a functor on three categories of metric spaces with additional designated points. The three categories which we study differ on their morphisms: one uses short (non-expanding) maps, the second uses continuous maps, and the third uses Lipschitz maps. The functor in all cases is very similar to what we find in the standard presentation of the gasket as an attractor. It was previously known that the Sierpinski gasket is bilipschitz equivalent (though not isomorhpic) to the final coalgebra of this functor in the category with short maps, and that final coalgebra is obtained by taking the completion of the initial algebra. In this paper, we prove that the Sierpiniski gasket itself is the final coalgebra in the category with continuous maps, though it does not occur as the completion of the initial algebra. In the Lipschitz setting, we show that the final coalgebra for this functor does not exist.

A locally coherent exact category is a finitely accessible additive category endowed with an exact structure in which the admissible short exact sequences are the directed colimits of admissible short exact sequences of finitely presentable objects. We show that any exact structure on a small idempotent-complete additive category extends uniquely to a locally coherent exact structure on the category of ind-objects; in particular, any finitely accessible category has the unique maximal and the unique minimal locally coherent exact category structures. All locally coherent exact categories are of Grothendieck type in the sense of Št’ovíček. We also discuss the canonical embedding of a small exact category into the abelian category of additive sheaves in connection with the locally coherent exact structure on the ind-objects, and deduce two periodicity theorems as applications.

Functorial semi-norms on singular homology measure the “size” of homology classes. A geometrically meaningful example is the (ell ^1)-semi-norm. However, the (ell ^1)-semi-norm is not universal in the sense that it does not vanish on as few classes as possible. We show that universal finite functorial semi-norms do exist on singular homology on the category of topological spaces that are homotopy equivalent to finite CW-complexes. Our arguments also apply to more general settings of functorial semi-norms.