Applied Categorical Structures最新文献

Matrix properties are a type of property of categories which includes the ones of being Mal’tsev, arithmetical, majority, unital, strongly unital, and subtractive. Recently, an algorithm has been developed to determine implications (textrm{M}Rightarrow _{textrm{lex}_*}textrm{N}) between them. We show here that this algorithm reduces to constructing a partial term corresponding to (textrm{N}) from a partial term corresponding to (textrm{M}). Moreover, we prove that this is further equivalent to the corresponding implication between the weak versions of these properties, i.e., the one where only strong monomorphisms are considered instead of all monomorphisms.

We want to study the smash product of pointed topological spaces, in an organic way and full generality, without relying on some ‘convenient subcategory’. The n-ary smash product has a ‘colax’ form of associativity, which supplies a categorical framework for the properties of this operation and its connection with the function spaces. Various concrete computations of smash products are given, including a large class of cases where associativity fails. Lax and colax monoidal structures are unusual and interesting, in category theory. Some parts of this note will be obvious to a topologist and others to a categorist, in order to take into account both backgrounds.

We study various acyclicity conditions on higher-categorical pasting diagrams in the combinatorial framework of regular directed complexes. We present an apparently weakest acyclicity condition under which the (omega )-category presented by a diagram shape is freely generated in the sense of polygraphs. We then consider stronger conditions under which this (omega )-category is equivalent to one obtained from an augmented directed chain complex in the sense of Steiner, or consists only of subsets of cells in the diagram. Finally, we study the stability of these conditions under the operations of pasting, suspensions, Gray products, joins and duals.

Inspired by some properties of the (dual of the) category of 2-nilpotent groups, we introduce the notion of 2-unital and 2-Mal’tsev categories which, in some sense, generalises the notion of unital and Mal’tsev categories, and we characterise their varietal occurrences. This is actually the first step of an inductive process which we begin to unfold.

Categories enriched in the opposite poset of non-negative reals can be viewed as generalizations of metric spaces, known as Lawvere metric spaces. In this article, we develop model structures on the categories ({mathbb {R}_+text {-}textbf{Cat}}) and ({mathbb {R}_+text {-}textbf{Cat}}^textrm{sym}) of Lawvere metric spaces and symmetric Lawvere metric spaces, each of which captures different features pertinent to the study of metric spaces. More precisely, in the three model structures we construct, the fibrant–cofibrant objects are the extended metric spaces (in the usual sense), the Cauchy complete Lawvere metric spaces, and the Cauchy complete extended metric spaces, respectively. Finally, we show that two of these model structures are unique in a similar way to the canonical model structure on (textbf{Cat}).

After fixing a commutative ring with unit R, we present the definition of adequate category and consider the category of R-linear functors from an adequate category to the category of R-modules. We endow this category of functors with a monoidal structure and study monoids (generalized Green functors) over it. For one of these generalized Green functors, we define two new monoids, its commutant and its center, and study some of their properties and relations between them. This work generalizes the article [3].

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.