Applied Categorical Structures最新文献

Let ({{mathcal {O}}}rightarrow {text {BM}}) be a ({text {BM}})-operad that exhibits an (infty )-category ({{mathcal {D}}}) as weakly bitensored over non-symmetric (infty )-operads ({{mathcal {V}}}rightarrow text {Ass }, {{mathcal {W}}}rightarrow text {Ass }) and ({{mathcal {C}}}) a ({{mathcal {V}}})-enriched (infty )-precategory. We construct an equivalence

of (infty )-categories weakly right tensored over ({{mathcal {W}}}) between Hinich’s construction of ({{mathcal {V}}})-enriched functors of Hinich (Adv Math 367:107129, 2020) and our construction of ({{mathcal {V}}})-enriched functors of Heine (Adv Math 417:108941, 2023).

We give an explicit description of two operad structures on the species composition (textbf{p}circ textbf{q}), where (textbf{q}) is any given positive operad, and where (textbf{p}) is the ({text{ NAP } }) operad, or a shuffle version of the magmatic operad ({text{ Mag } }). No distributive law between (textbf{p}) and (textbf{q}) is assumed.

The approximation classes of modules that arise as components of cotorsion pairs are tied up by Salce’s duality. Here we consider general approximation classes of modules and investigate possibilities of dualization in dependence on closure properties of these classes. While some proofs are easily dualized, other dualizations require large cardinal principles, and some fail in ZFC, with counterexamples provided by classes of (aleph _1)-projective modules over non-perfect rings. For example, we show that the statement “each covering class of modules closed under homomorphic images is of the form ({mathrm{Gen,}}(M)) for a module M” is equivalent to Vopěnka’s Principle.

Via the adjunction ( - *mathbbm {1} dashv mathcal V(mathbbm {1},-) :textsf {Span}({mathcal {V}}) rightarrow {mathcal {V}} text {-} textsf {Mat} ) and a cartesian monad T on an extensive category ( {mathcal {V}} ) with finite limits, we construct an adjunction ( - *mathbbm {1} dashv {mathcal {V}}(mathbbm {1},-) :textsf {Cat}(T,{mathcal {V}}) rightarrow ({overline{T}}, mathcal V)text{- }textsf{Cat} ) between categories of generalized enriched multicategories and generalized internal multicategories, provided the monad T satisfies a suitable property, which holds for several examples. We verify, moreover, that the left adjoint is fully faithful, and preserves pullbacks, provided that the copower functor ( - *mathbbm {1} :textsf {Set} rightarrow {mathcal {V}} ) is fully faithful. We also apply this result to study descent theory of generalized enriched multicategorical structures. These results are built upon the study of base-change for generalized multicategories, which, in turn, was carried out in the context of categories of horizontal lax algebras arising out of a monad in a suitable 2-category of pseudodouble categories.

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].