The notion of pseudocategory, as considered in [11], is extended from the context of a 2-category to the more general one of a sesquicategory, which is considered as a category equipped with a 2-cell structure. Some particular examples of 2-cells arising form internal transformations in internal categories, conjugations in groups, derivations in crossed-modules or homotopies in abelian chain complexes are studied in this context, namely their behaviour as abstract 2-cells in a 2-cell structure. Issues such as naturality of a 2-cell structure are investigated. This article is intended as a preliminary starting work towards the study of the geometrical aspects of the 2-cell structures from an algebraic point of view.
We define bicategories internal to 2-categories. When the ambient 2-category is symmetric monoidal categories, this provides a convenient framework for encoding the structures of a symmetric monoidal 3-category. This framework is well suited to examples arising in geometry and algebra, such as the 3-category of bordisms or the 3-category of conformal nets.
Let $\mathsf{PreOrd}(\mathbb C)$ be the category of internal preorders in an exact category $\mathbb C$. We show that the pair $(\mathsf{Eq}(\mathbb C), \mathsf{ParOrd}(\mathbb C))$ is a pretorsion theory in $\mathsf{PreOrd}(\mathbb C)$, where $\mathsf{Eq}(\mathbb C)$ and $\mathsf{ParOrd}(\mathbb C)$) are the full subcategories of internal equivalence relations and of internal partial orders in $\mathbb C$, respectively. We observe that $\mathsf{ParOrd}(\mathbb C)$ is a reflective subcategory of $\mathsf{PreOrd}(\mathbb C)$ such that each component of the unit of the adjunction is a pullback-s…
Enrichment and internal categories are two different way to generalize the notion of category. As such, enriching double categories (which are categories internal to Cat) is not a clear concepts. One can look at the internal categories of the category of C-enriched categories or the internal C-category in Cat. This concepts, unfortunately, are not powerful enough, even thin (posetal) double categories can't be seen this way. However, an orthogonal investigation, that of the enrichment by a double category, will help relaxing the internal C-category to a incorporate the internal categories of t…
A new construction to associate an internal category to an enriched one is presented. The key concept is that of extensive ambient category, and the construction follows the one that associates a category whose idempotents split to a given one. The association turns out to be functorial and left adjoint to an already known one when we restrict to a particular class of internal categories whose idempotents split in some canonical way and impose a size restriction.
In this note, we review a construction of category with families (CwF) in a presheaf category. When the base category of a presheaf category is a CwF, we internalize this CwF structure in the CwF of the presheaf category. This note assumes working knowledge on category theory.
The main aim of this paper is to provide a description of neighbourhood operators in finitely complete categories with finite coproducts and a proper factorisation system such that the semilattice of admissible subobjects make a distributive complete lattice. The equivalence between neighbourhoods, Kuratowski interior operators and pseudo-frame sets is proved. Furthermore the categories of internal neighbourhoods is shown to be topological. Regular epimorphisms of categories of neighbourhoods are described and conditions ensuring hereditary regular epimorphisms are probed. It is shown the cate…
The aim of this paper is to characterize the notion of internal category (groupoid) in the category of Leibniz algebras and investigate the properties of well-known notions such as covering groupoid and groupoid operations (actions) in this category. Further, for a fixed internal groupoid $G$, we prove that the category of covering groupoids of $G$ and the category of internal groupoid actions of $G$ on Leibniz algebras are equivalent. Finally we interpret the corresponding notion of covering groupoids in the category of crossed modules of Leibniz algebras.
In order to deduce the internal version of the Brown exact sequence from the internal version of the Gabriel-Zisman exact sequence, we characterize fibrations and $\ast$-fibrations in the 2-category of internal groupoids in terms of the comparison functor from certain pullbacks to the corresponding strong homotopy pullbacks. A similar analysis in the category of arrows allows us to give a characterization of protomodular categories using strong homotopy kernels.
The category of internal coalgebras in a cocomplete category $\mathcal{C}$ with respect to a variety $\mathcal{V}$ is equivalent to the category of left adjoint functors from $\mathcal{V}$ into $\mathcal{C}$. This can be seen best when considering such coalgebras as finite coproduct preserving functors from $\mathcal{T}_\mathcal{V}^\mathsf{op}$, the dual of the Lawvere theory of $\mathcal{V}$, into $\mathcal{C}$: coalgebras are restrictions of left adjoints and any such left adjoint is the left Kan extension of a coalgebra along the embedding of $\mathcal{T}_\mathcal{V}^\mathsf{op}$ into $\mat…
In this paper, we define the notion of Hopf crossed square for cocommutative Hopf algebras extending the notions of crossed squares of groups and of Lie algebras. We prove the equivalence between the category of Hopf crossed squares and the category of double internal groupoids in the category of cocommutative Hopf algebras. The Hopf crossed squares turn out to be the internal crossed modules in the category of crossed modules in the category of cocommutative Hopf algebras.
We propose a new model for multicategories with symmetries with respect to Zhang's group operads. The fully faithful embedding of the category of group operads into that of crossed interval groups is made use of, and it is shown that every multicategory gives rise to a fibration, in a sense, over a quotient of the total category of group operads. The symmetric structures can be presented as structures of internal presheaves over a category internal to the category of small categories, in other words a double category.
In this paper we prove some results on the covering morphisms of internal groupoids. We also give a result on the coverings of the crossed modules of groups with operations.
We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital $A_\infty$ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. Moreover we show that the last two localizations are equivalent to the corresponding quotients by the relation of being isomorphic in the cohomology of the $A_\infty$ category of $A_\infty$ functors. As an application we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly un…
In this study, internal categories in the category of the crossed modules are characterized and it has been shown that there is a natural equivalence between the category of the crossed modules over crossed modules, i.e. crossed squares, and the category of the internal categories within the category of crossed modules. Finally, we obtain examples of crossed squares using this equivalence.
Lenses may be characterised as objects in the category of algebras over a monad, however they are often understood instead as morphisms, which propagate updates between systems. Working internally to a category with pullbacks, we define lenses as simultaneously functors and cofunctors between categories. We show that lenses may be canonically represented as a particular commuting triangle of functors, and unify the classical state-based lenses with both c-lenses and d-lenses in this framework. This new treatment of lenses leads to considerable simplifications that are important in applications…
An answer to the question investigated in this paper brings a new characterization of internal groupoids such that: (a) it holds even when finite limits are not assumed to exist; (b) it is a full subcategory of the category of involutive-2-links, that is, a category whose objects are morphisms equipped with a pair of interlinked involutions. This result highlights the fact that even thought internal groupoids are internal categories equipped with an involution, they can equivalently be seen as tri-graphs with an involution. Moreover, the structure of a tri-graph with an involution can be furth…
When $\mathbb C$ is a semi-abelian category, it is well known that the category $\mathsf{Grpd}(\mathbb C)$ of internal groupoids in $\mathbb C$ is again semi-abelian. The problem of determining whether the same kind of phenomenon occurs when the property of being semi-abelian is replaced by the one of being action representable (in the sense of Borceux, Janelidze and Kelly) turns out to be rather subtle. In the present article we give a sufficient condition for this to be true: in fact we prove that the category $\mathsf{Grpd}(\mathbb C)$ is a semi-abelian action representable algebraically co…
We focus on two factorization systems for opfibrations in the 2-category Fib(B) of fibrations over a fixed base category B. The first one is the internal version of the so called comprehensive factorization, where the right orthogonal class is given by internal discrete opfibrations. The second one has as its right orthogonal class internal opfibrations in groupoids, i.e. with groupoidal fibres. These factorizations can be obtained by means of a single step 2-colimit. Namely, their left orthogonal parts are nothing but suitable coidentifiers and coinverters respectively. We will show how these…
In this note we prove that distributors between groupoids in a Barr-exact category E form the bicategory of relations relative to the comprehensive factorization system in Gpd(E). Revised version.
This paper emerged as a result of tackling the following three issues. Firstly, we would like the well known embedding of bicategories into pseudo double categories to be monoidal, which it is not if one uses the usual notion of a monoidal pseudo double category. Secondly, in \cite{Gabi} the question was raised: which would be an alternative notion to intercategories of Grandis and Paré, so that monoids in Böhm's monoidal category $(Dbl,\ot)$ of strict double categories and strict double functors with a Gray type monoidal product be an example of it? We obtain and prove that precisely the mono…
Stellenbosch University（斯泰伦博斯大学）；National Institute for Theoretical and Computational Sciences, South Africa（南非理论与计算科学国家研究所）；University of Minho（米尼奥大学）
We introduce the theory of enrichment over an internal monoidal category as a common generalization of both the standard theories of enriched and internal categories. The aim of the paper is to justify and contextualize the new notion by comparing it to other known generalizations of enrichment: namely, those for indexed categories and for generalized multicategories. It turns out that both of these notions are closely related to internal enrichment and, as a corollary, to each other.
We explain how, in the context of a semi-abelian category, the concept of an internal crossed square may be used to set up an intrinsic approach to the Brown-Loday non-abelian tensor product.
Fibrations over a category $B$, introduced to category theory by Grothendieck, encode pseudo-functors $B^{op} \rightsquigarrow {\bf Cat}$, while the special case of discrete fibrations encode presheaves $B^{op} \to {\bf Set}$. A two-sided discrete variation encodes functors $B^{op} \times A \to {\bf Set}$, which are also known as profunctors from $A$ to $B$. By work of Street, all of these fibration notions can be defined internally to an arbitrary 2-category or bicategory. While the two-sided discrete fibrations model profunctors internally to ${\bf Cat}$, unexpectedly, the dual two-sided cod…
A partial combinatory algebra (PCA) is a set equipped with a partial binary operation that models a notion of computability. This paper studies a generalization of PCAs, introduced by W. Stekelenburg, where a PCA is not a set but an object in a given regular category. The corresponding class of categories of assemblies is closed both under taking small products and under slicing, which is to be contrasted with the situation for ordinary PCAs. We describe these two constructions explicitly at the level of PCAs, allowing us to compute a number of examples of products and slices of PCAs. Moreover…
For M and N finite module categories over a finite tensor category C, the category Rex_C(M,N) of right exact module functors is a finite module category over the Drinfeld center Z(C). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map C-C-bimodule functors to objects of Z(C), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if M and N are exact C-modules and C is pivotal, then the Z(C)-module Rex_C(M,N) is exact. We compute its relative…
Given an exact category $\mathcal{C}$, it is well known that the connected component reflector $π_0\colon\mathsf{Gpd}(\mathcal{C})\to\mathcal{C}$ from the category $\mathsf{Gpd}(\mathcal{C})$ of internal groupoids in $\mathcal{C}$ to the base category $\mathcal{C}$ is semi-left-exact. In this article we investigate the existence of a monotone-light factorisation system associated with this reflector. We show that, in general, there is no monotone-light factorisation system $(\mathcal{E}',\mathcal{M}^*)$ in $\mathsf{Gpd}(\mathcal{C})$, where $\mathcal{M}^*$ is the class of coverings in the sens…
In this paper, for given an algebraic theory $T$ whose category $C$ of models is semi-abelian, we consider the topological models of $T$ called topological $T$-algebras and obtain some results related to the fundamental groups of topological $T$-algebras. We also deal with the internal groupoid structure in the category of models providing that the fundamental groupoid deduces a functor from topological $T$-algebras to the internal groupoids in $C$ and prove a criterion for the lifting of such an internal groupoid structure to the covering groupoids.
In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while we make use of an endofunctor $\mathcal{U}$ on a topos of sets $S$ together with a natural transformation $\upsilon$, instead of the terms as "standard", "internal" or "external". Moreover, we propose a general notion of a space called $\mathcal{U}$-space, and the category $\mathcal{U}space$ whose objects are $\mathcal{U}$-spaces and morphisms are functions …
The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category $\mathbb A$, the corresponding internal hom functor $|[ \mathbb A,-]|$ sends a double category $\mathbb B$ to the double category whose 0-cells are the double functors $\mathbb A \to \mathbb B$, whose horizontal and vertical 1-cells are the horizontal and vertical pseudotransformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.
