Applied Categorical Structures最新文献

In this article, we expand upon the concepts introduced in Spivak (Metric realization of fuzzy simplicial sets, 2009. http://www.dspivak.net/metric_realization090922.pdf) about the relationship between the category (textbf{UM}) of uber metric spaces and the category (textbf{sFuz}) of fuzzy simplicial sets. We show that fuzzy simplicial sets can be regarded as natural combinatorial generalizations of metric relations. Furthermore, we take inspiration from UMAP (McInnes et al, in: Umap: Uniform manifold approximation and projection for dimension reduction, 2018) to apply the theory to manifold learning, dimension reduction and data visualization, while refining some of their constructions to put the corresponding theory on a more solid footing. A generalization of the adjunction between (textbf{UM}) and (textbf{sFuz}) allows us to view the adjunctions used in both publications as special cases. Moreover, we derive an explicit description of colimits in (textbf{UM}) and the realization functor (text {Re}:textbf{sFuz}rightarrow textbf{UM}), and show that (textbf{UM}) can be embedded into (textbf{sFuz}). Furthermore, we prove analogous results for the category of extended-pseudo metric spaces (textbf{EPMet}). We also provide rigorous definitions of functors that make it possible to recursively merge sets of fuzzy simplicial sets and provide a description of the adjunctions between the category of truncated fuzzy simplicial sets and (textbf{sFuz}), which we relate to persistent homology. Combining those constructions, we can show a surprising connection between the well-known dimension reduction methods UMAP and Isomap (Tenenbaum et al. in Science 290(5500):2319–2323, 2000) and derive an alternative algorithm, which we call IsUMap, that combines some of the strengths of both methods. Additionally, we developed a new embedding method that allows to preserve clusters detected in the original metric space that we construct from the data. The visualization of the optimization process gives the user information, both about the inner-cluster distributions in the original metric space and their inter-cluster relations. We compare our new method with UMAP, Isomap and t-SNE on a series of low- and high-dimensional datasets and provide explanations for observed differences and improvements.

In a previous work on quadratic algebras (Pirashvili in Glas Math J 61: 151–167, 2018), I constructed an internal category in the category of monoids, recalled in Sect. 3.2.1. Based on this, we introduce the notion of a crossed semi-bimodule in this paper. This new construction generalises the notion of a crossed semi-module, introduced independently by R. Street and A. Patchkoria, see Joyal (Macquarie Math Reports 860081, 1986) and Patchkoria (Georg Math J 5: 575–581, 1986) respectively. We also show that there is a one to one correspondence between crossed semi-bimodules and strict monoidal category structures on transformation categories satisfying the cc-condition, see Sects. 4 and 5.

We prove that the Leibniz PROP is isomorphic as (Bbbk )-linear categories (not as monoidal categories) to the symmetric crossed presimplicial algebra (Bbbk [(Delta ^+)^{op} mathbb {S}]) where (Delta ^+) is the skeletal category of finite well-ordered sets with surjections, but the distributive law between ((Delta ^+)^{op}) and the symmetric groups (mathbb {S} = bigsqcup _{nge 1} S_n) is not the standard one.

We provide details of the proof of Lurie’s theorem on operadic Kan extensions. Along the way, we generalize the construction of monoidal envelopes of (infty )-operads to families of (infty )-operads and use it to construct the fiberwise direct sum functor, both of which we characterize by certain universal properties. Aside from their use in elaborating the proof of Lurie’s theorem, these results and constructions have their independent interest.

We prove that the category 2-(textrm{Grpd}(mathscr {C})) of internal 2-groupoids is a Birkhoff subcategory of the category (textrm{Grpd}^2(mathscr {C})) of double groupoids in a regular Mal’tsev category (mathscr {C}) with finite colimits, and we provide a simple description of the reflector. In particular, when (mathscr {C}) is a Mal’tsev variety of universal algebras, the category 2-(textrm{Grpd}(mathscr {C})) is also a Mal’tsev variety, of which we describe the corresponding algebraic theory. When (mathscr {C}) is a naturally Mal’tsev category, the reflector from (textrm{Grpd}^2(mathscr {C})) to 2-(textrm{Grpd}(mathscr {C})) has an additional property related to the commutator of equivalence relations. We prove that the category 2-(textrm{Grpd}(mathscr {C})) is semi-abelian when (mathscr {C}) is semi-abelian, and then provide sufficient conditions for 2-(textrm{Grpd}(mathscr {C})) to be action representable.

It is known that the construction of the frame of ideals from a distributive lattice induces a monad whose algebras are precisely the frames and frame homomorphisms. Using the Fakir construction of an idempotent approximation of a monad, we extend B. Jacobs’ results on lax idempotent monads and show that the sequence of monads and comonads generated by successive iterations of this ideal functor on its algebras and coalgebras do not strictly lead to a new category. We further extend this result and provide a new proof of the equivalence between distributive lattices and coherent frames by showing that when the first inductive step in the Fakir construction is the identity monad, then the ambient category is equivalent to the category of free algebras.

We introduce (omega )-catoids as generalisations of (strict) (omega )-categories and in particular the higher path categories generated by computads or polygraphs in higher-dimensional rewriting. We also introduce (omega )-quantales that generalise the (omega )-Kleene algebras recently proposed for algebraic coherence proofs in higher-dimensional rewriting. We then establish correspondences between (omega )-catoids and convolution (omega )-quantales. These are related to Jónsson-Tarski-style dualisms between relational structures and lattices with operators. We extend these correspondences to ((omega,p))-catoids, catoids with a groupoid structure above some dimension, and convolution ((omega,p))-quantales, using Dedekind quantales above some dimension to capture homotopic constructions and proofs in higher-dimensional rewriting. We also specialise them to finitely decomposable ((omega, p))-catoids, an appropriate setting for defining ((omega, p))-semirings and ((omega, p))-Kleene algebras. These constructions support the systematic development and justification of (omega )-Kleene algebra and (omega )-quantale axioms, improving on the recent approach mentioned, where axioms for (omega )-Kleene algebras have been introduced in an ad hoc fashion.

In [N. Bezhanishvili and W. H. Holliday. Choice-free Stone duality. J. Symb. Log., 85(1):109–148, 2020.], the authors develop a choice-free topological duality for the algebraic category of Boolean algebras. We adapt the techniques and constructions given by Bezhanishvili and Holliday to develop a topological duality for the algebraic category of Tarski algebras without using the Axiom of Choice. Then, we show that the duality presented here for Tarski algebras is in fact a generalization of the duality given by Bezhanishvili and Holliday for Boolean algebras. We also obtain a choice-free topological duality for the algebraic category of generalized Boolean algebra.

A pseudomonad on a 2-category whose underlying endomorphism is a 2-functor can be seen as a diagram (textbf{Psmnd} rightarrow textbf{Gray}) for which weighted limits and colimits can be considered. The 2-category of pseudoalgebras, pseudomorphisms and 2-cells is such a Gray-enriched weighted limit [21], however neither the Kleisli bicategory nor the 2-category of free pseudoalgebras are the analogous weighted colimit [13]. In this paper we describe the actual weighted colimit via a presentation, and show that the comparison 2-functor induced by any other pseudoadjunction splitting the original pseudomonad is bi-fully faithful. As a consequence, we see that biessential surjectivity on objects characterises left pseudoadjoints whose codomains have an ‘up to biequivalence’ version of the universal property for Kleisli objects. This motivates a homotopical study of Kleisli objects for pseudomonads, and to this end we show that the weight for Kleisli objects is cofibrant in the projective model structure on the enriched functor category ([textbf{Psmnd}^text {op} , textbf{Gray}]).

We follow the work of Aguiar (Internal categories and quantum groups. PhD Thesis, Cornell University, 1997) on internal categories and introduce simplicial objects internal to a monoidal category as certain colax monoidal functors. Then we compare three approaches to equipping them with a discrete set of vertices. We introduce based colax monoidal functors and show that under suitable conditions they are equivalent to the templicial objects defined by Lowen and Mertens (Algebr Geom Topol, 2024). We also compare templicial objects to the enriched Segal precategories appearing in Lurie ((Infinity,2)-categories and the Goodwillie calculus I. Preprint at https://arxiv.org/abs/0905.0462v2), Simpson (Homotopy theory of higher categories. New mathematical monographs, Cambridge University Press, Cambridge, vol 19, p 634, 2012, https://doi.org/10.1017/CBO9780511978111) and Bacard (Theory Appl Categ 35:1227–1267, 2020), and show that they are equivalent for cartesian monoidal categories, but not in general.