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In the spirit of Morse homology initiated by Witten and Floer, we construct two ∞-categories A and B . The weak one A comes out of the Morse–Smale pairs and their higher homotopies, and the strict one B concerns the chain complexes of the Morse functions. Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters, we build up a weak ∞-functor F : A → B . Higher algebraic structures behind Morse homology are revealed with the perspective of defects in topological quantum field theory. [ABSTRACT FROM AUTHOR]

We study the problem of classification of simple transitive 2-representations for the (non-finitary) 2-category of bimodules over the dual numbers. We show that simple transitive 2-representations with finitary apex are necessarily of rank 1 or 2, and those of rank 2 are exactly the cell 2-representations. For 2-representations of rank 1, we show that they cannot be constructed using the approach of (co)algebra 1-morphisms. We also propose an alternative definition of (co-)Duflo 1-morphisms for finitary 2-categories and describe them in the case of bimodules over the dual numbers. [ABSTRACT FROM AUTHOR]

We introduce the notion of a distributive law between a relative monad and a monad. We call this a relative distributive law and define it in any 2-category K . In order to do that, we introduce the 2-category of relative monads in a 2-category K with relative monad morphisms and relative monad transformations as 1- and 2-cells, respectively. We relate our definition to the 2-category of monads in K defined by Street. Using this perspective, we prove two Beck-type theorems regarding relative distributive laws. We also describe what does it mean to have Eilenberg–Moore and Kleisli objects in this context and give examples in the 2-category of locally small categories. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the generalization of finitary 2-representation theory of finitary 2-categories to finitary birepresentation theory of finitary bicategories. In previous papers on the subject, the classification of simple transitive 2-representations of a given 2-category was reduced to that for certain subquotients. These reduction results were all formulated as bijections between equivalence classes of 2-representations. In this paper, we generalize them to biequivalences between certain 2-categories of birepresentations. Furthermore, we prove an analog of the double centralizer theorem in finitary birepresentation theory. [ABSTRACT FROM AUTHOR]

Throughout this article, we fix a group G and a commutative ring k . This is an exposition on 2-equivalences between a 2-category of small k -categories with pseudoG-actions and a 2-category of G-graded small k -categories, which generalizes (Asashiba in Appl Categor Sturct 25(8):3278–3296, 2017). This article is a translation of some parts of Ch. 4 and 5 in the book (Asashiba in Categories and representation theory, focused on 2-categorical covering theory, SGC library 155, Saiensu-sha, 2019) in Japanese. [ABSTRACT FROM AUTHOR]

We show that the category of graphs has the structure of a 2-category with homotopy as the 2-cells. We then develop an explicit description of homotopies for finite graphs, in terms of what we call 'spider moves.' We then create a category by modding out by the 2-cells of our 2-category and use the spider moves to show that for finite graphs, this category is a homotopy category in the sense that it satisfies the universal property for localizing homotopy equivalences. We then show that finite stiff graphs form a skeleton of this homotopy category. [ABSTRACT FROM AUTHOR]

We focus on the transfer of some known orthogonal factorization systems from Cat to the 2-category Fib (B) of fibrations over a fixed base category B: the internal version of the comprehensive factorization, and the factorization systems given by (sequence of coidentifiers, discrete morphism) and (sequence of coinverters, conservative morphism) respectively. For the class of fibrewise opfibrations in Fib (B) , the construction of the latter two simplify to a single coidentifier (respectively coinverter) followed by an internal discrete opfibration (resp. fibrewise opfibration in groupoids). We show how these results follow from their analogues in Cat , providing suitable conditions on a 2-category C , that allow the transfer of the construction of coinverters and coidentifiers from C to Fib C (B) . [ABSTRACT FROM AUTHOR]

We prove a decomposition formula for the dimensional reduction of an extended topological field theory that arises as an orbifold of an equivariant topological field theory. Our decomposition formula can be expressed in terms of a categorification of the integral with respect to groupoid cardinality. The application of our result to topological field theories of Dijkgraaf–Witten type proves a recent conjecture of Qiu–Wang. [ABSTRACT FROM AUTHOR]

Assume that a basic algebra A over an algebraically closed field k with a basic set A0 of primitive idempotents has the property that eAe = k for all e ∈ A 0. Let n be a nonzero integer, and ϕ and ψ two automorphisms of the repetitive category A ̂ of A with jump n (namely, they send A [ 0 ] to A [ n ] , where A [ i ] is the i-th copy of A in A ̂ for all i ∈ ℤ ). If ϕ and ψ coincide on the objects and if there exists a map ρ : A 0 → k such that ρ 0 (y) ϕ 0 (a) = ψ 0 (a) ρ 0 (x) for all morphisms a ∈ A (x , y) , then the orbit categories A ̂ / 〈 ϕ 〉 and A ̂ / 〈 ψ 〉 are isomorphic as ℤ -graded categories. [ABSTRACT FROM AUTHOR]

In this paper we complete a chain of explicit Quillen equivalences between the model category for Θn+1‐spaces and the model category of small categories enriched in Θn‐spaces. The Quillen equivalences given here connect Segal category objects in Θn‐spaces, complete Segal objects in Θn‐spaces, and Θn+1‐spaces. [ABSTRACT FROM AUTHOR]

We address the (pointed) homotopy of crossed module morphisms in modified categories of interest that unify the notions of groups and various algebraic structures. We prove that the homotopy relation gives rise to an equivalence relation as well as to a groupoid structure with no restriction on either domain or co-domain of the corresponding crossed module morphisms. Furthermore, we also consider particular cases such as crossed modules in the categories of associative algebras, Leibniz algebras, Lie algebras and dialgebras of the unified homotopy definition. Finally, as one of the major objectives of this paper, we prove that the functor from simplicial objects to crossed modules in modified categories of interest preserves the homotopy as well as the homotopy equivalence. [ABSTRACT FROM AUTHOR]

In category theory, the existence of free objects is very important, especially free modules that play an important role in homological algebra. Although algebraic hyperstructures are a natural extension of algebraic structures, due to the major difference between them, study-free objects in algebraic hyperstructures become very difficult. In this article, we provide a categorical approach for the construction of free hypermodules. In fact, by considering appropriate morphisms between hypermodules, we characterize free hypermodules from three different perspectives. [ABSTRACT FROM AUTHOR]

We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common classes of higher-categorical cell shapes, including globes, cubes, oriented simplices, and a large sub-class of opetopes, and are closed under lax Gray products and joins. We define constructible polygraphs to be presheaves on a category of atoms and inclusions, and extend the monoidal structures. We show that constructible directed complexes are a well-behaved subclass of Steiner's directed complexes, which we use to define a realisation functor from constructible polygraphs to ω -categories. We prove that the realisation of a constructible polygraph is a polygraph in restricted cases, and in all cases conditionally to a conjecture. Finally, we define the geometric realisation of a constructible polygraph, and prove that it is a CW complex with one cell for each of its elements. [ABSTRACT FROM AUTHOR]

In this paper, we describe the combinatorics of the cell structure of the tensor category of bimodules over a radical square zero Nakayama algebra. This accounts to an explicit description of left, right and two-sided cells. [ABSTRACT FROM AUTHOR]

To the memory of Trevor Evans (1925–1991), the pioneer of interchange laws in universal algebra We establish a combinatorial model for the Boardman–Vogt tensor product of several absolutely free operads, that is, free symmetric operads that are also free as -modules. Our results imply that such a tensor product is always a free -module, in contrast with the results of Kock and Bremner–Madariaga on hidden commutativity for the Boardman–Vogt tensor square of the operad of non-unital associative algebras. [ABSTRACT FROM AUTHOR]

This paper concerns the problem of lifting a KZ doctrine P to the 2-category of pseudo T-algebras for some pseudomonad T. Here we show that this problem is equivalent to giving a pseudo-distributive law (meaning that the lifted pseudomonad is automatically KZ), and that such distributive laws may be simply described algebraically and are essentially unique [as known to be the case in the (co)KZ over KZ setting]. Moreover, we give a simple description of these distributive laws using Bunge and Funk's notion of admissible morphisms for a KZ doctrine (the principal goal of this paper). We then go on to show that the 2-category of KZ doctrines on a 2-category is biequivalent to a poset. We will also discuss here the problem of lifting a locally fully faithful KZ doctrine, which we noted earlier enjoys most of the axioms of a Yoneda structure, and show that a bijection between oplax and lax structures is exhibited on the lifted "Yoneda structure" similar to Kelly's doctrinal adjunction. We also briefly discuss how this bijection may be viewed as a coherence result for oplax functors out of the bicategories of spans and polynomials, but leave the details for a future paper. [ABSTRACT FROM AUTHOR]

We prove a generalization of Quillen's Theorem B to strict ∞‐categories. More generally, we show that under similar hypothesis as for Theorem B, the comma construction for strict ∞‐categories, that we introduced with Maltsiniotis in a previous paper, is the homotopy pullback with respect to Thomason equivalences. We give several applications of these results including the construction of new models for certain Eilenberg–Mac Lane spaces. [ABSTRACT FROM AUTHOR]

Several constructive homological methods based on noncommutative Gröbner bases are known to compute free resolutions of associative algebras. In particular, these methods relate the Koszul property for an associative algebra to the existence of a quadratic Gröbner basis of its ideal of relations. In this article, using a higher-dimensional rewriting theory approach, we give several improvements of these methods. We define polygraphs for associative algebras as higher-dimensional linear rewriting systems that generalise the notion of noncommutative Gröbner bases, and allow more possibilities of termination orders than those associated to monomial orders. We introduce polygraphic resolutions of associative algebras, giving a categorical description of higher-dimensional syzygies for presentations of algebras. We show how to compute polygraphic resolutions starting from a convergent presentation, and how these resolutions can be linked with the Koszul property. [ABSTRACT FROM AUTHOR]

We suggest a geometric approach to quantisation of the twisted Poisson structure underlying the dynamics of charged particles in fields of generic smooth distributions of magnetic charge, and dually of closed strings in locally non-geometric flux backgrounds, which naturally allows for representations of nonassociative magnetic translation operators. We show how one can use the 2-Hilbert space of sections of a bundle gerbe in a putative framework for canonical quantisation. We define a parallel transport on bundle gerbes on R d and show that it naturally furnishes weak projective 2-representations of the translation group on this 2-Hilbert space. We obtain a notion of covariant derivative on a bundle gerbe and a novel perspective on the fake curvature condition. [ABSTRACT FROM AUTHOR]

The category-valued trace assigns to a bimodule category over a linear monoidal category a linear category. It generalizes Drinfeld centers of monoidal categories and the relative Deligne product of bimodule categories. In this article, we study bimodule categories that are given as categories of bicomodules over a Hopf algebra. Our main result is a representation-theoretic realization of the category-valued trace as a category of generalized Hopf bimodules. [ABSTRACT FROM AUTHOR]

We study actions of discrete groups on 2-categories. The motivating examples are actions on the 2-category of representations of finite tensor categories and their relation with the extension theory of tensor categories by groups. Associated to a group action on a 2-category, we construct the 2-category of equivariant objects. We also introduce the G-equivariant notions of pseudofunctor, pseudonatural transformation and modification. Our first main result is a coherence theorem for 2-categories with an action of a group. For a 2-category B with an action of a group G, we construct a braided G-crossed monoidal category Z G (B) with trivial component the Drinfeld center of B . We prove that, in the case of a G-action on the 2-category of representation of a tensor category C , the 2-category of equivariant objects is biequivalent to the module categories over an associated G-extension of C . Finally, we prove that the center of the equivariant 2-category is monoidally equivalent to the equivariantization of a relative center, generalizing results obtained in Gelaki et al. (Algebra Number Theory 3(8):959–990, 2009). [ABSTRACT FROM AUTHOR]

In this article, we study simple transitive 2-representations of certain 2-subcategories of the 2-category of projective functors over a star algebra. We show that in the simplest case, which is associated to the Dynkin type A2, simple transitive 2-representations are classified by cell 2-representations. In the general case we conjecture that there exist many simple transitive 2-representations which are not cell 2-representations and provide some evidence for our conjecture. [ABSTRACT FROM AUTHOR]

Given a pseudomonad T, we prove that a lax T-morphism between pseudoalgebras is a T-pseudomorphism if and only if there is a suitable (possibly non-canonical) invertible T-transformation. This result encompasses several results on non-canonical isomorphisms, including Lack's result on normal monoidal functors between braided monoidal categories, since it is applicable in any 2-category of pseudoalgebras, such as the 2-categories of monoidal categories, cocomplete categories, bicategories, pseudofunctors and so on. [ABSTRACT FROM AUTHOR]

Let be a field, G a group, and (Q, I) a bound quiver. A map is called a G-weight on Q, which defines a G-graded -category , and W is called homogeneous if I is a homogeneous ideal of the G-graded -category. Then we have a G-graded -category. We can then form a smash product of and G, which canonically defines a Galois covering with group G [we will see that all such Galois coverings to have this form for some W]. First we give a quiver presentation of the smash product. Next if (Q, I, W) is defined by a Brauer graph with an admissible weight, then the smash product is again defined by a Brauer graph, which will be computed explicitly. The computation is simplified by introducing a concept of Brauer permutations as an intermediate one between Brauer graphs and Brauer bound quivers. This extends and simplifies the result by Green–Schroll–Snashall on the computation of coverings of Brauer graphs, which dealt with the case that G is a finite abelian group, while in our case G is an arbitrary group. In particular, it enables us to delete all cycles in Brauer graphs to transform it to an infinite Brauer tree. [ABSTRACT FROM AUTHOR]

A triple vector bundle is a cube of vector bundle structures which commute in the (strict) categorical sense. A grid in a triple vector bundle is a collection of sections of each bundle structure with certain linearity properties. A grid provides two routes around each face of the triple vector bundle, and six routes from the base manifold to the total manifold; the warps measure the lack of commutativity of these routes. In this paper we first prove that the sum of the warps in a triple vector bundle is zero. The proof we give is intrinsic and, we believe, clearer than the proof using decompositions given earlier by one of us. We apply this result to the triple tangent bundle T3M of a manifold and deduce (as earlier) the Jacobi identity. We further apply the result to the triple vector bundle T2A for a vector bundle A using a connection in A to define a grid in T2A. In this case the curvature emerges from the warp theorem. [ABSTRACT FROM AUTHOR]

We extend the work of Kock (J Homot Relat Struct 2(2):217-228, 2007) and Bremner and Madariaga (Semigroup Forum 92:335-360,  2016) on commutativity in double interchange semigroups (DIS) to relations with 10 arguments. Our methods involve the free symmetric operad generated by two binary operations with no symmetry, its quotient by the two associative laws, its quotient by the interchange law, and its quotient by all three laws. We also consider the geometric realization of free double interchange magmas by rectangular partitions of the unit square I2. We define morphisms between these operads which allow us to represent elements of free DIS both algebraically as tree monomials and geometrically as rectangular partitions. With these morphisms we reason diagrammatically about free DIS and prove our new commutativity relations. [ABSTRACT FROM AUTHOR]

We introduce the notion of a relative pseudomonad, which generalizes the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient method to define pseudomonads on the Kleisli bicategory of a relative pseudomonad. The results are applied to define several pseudomonads on the bicategory of profunctors in an homogeneous way and provide a uniform approach to the definition of bicategories that are of interest in operad theory, mathematical logic, and theoretical computer science. [ABSTRACT FROM AUTHOR]

We show that a double Lie algebroid, together with a chosen decomposition, is equivalent to a pair of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the double Lie algebroids arising from the tangent bundle of a Lie algebroid and the cotangent bundle of a Lie bialgebroid. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict ∞-categories. This result is central to the homotopy theory of strict ∞-categories developed by the authors. The proof presented here is of a simplicial nature and uses Steiner's theory of augmented directed complexes. In a subsequent paper, we will prove the same result by purely ∞-categorical methods. [ABSTRACT FROM AUTHOR]

Let S be a site. First we define the 3-category of torsors under a Picard S-2-stack and we compute its homotopy groups. Using calculus of fractions, we define also a pure algebraic analogue of the 3-category of torsors under a Picard S-2-stack. Then we describe extensions of Picard S-2-stacks as torsors endowed with a group law on the fibers. As a consequence of such a description, we show that any Picard S-2-stack admits a canonical free partial left resolution that we compute explicitly. Moreover, we get an explicit right resolution of the 3-category of extensions of Picard S-2-stacks in terms of 3-categories of torsors. Using the homological interpretation of Picard S-2-stacks, we rewrite this three categorical dimensions higher right resolution in the derived category D(S) of abelian sheaves on S. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of the n -globular operad used in Batanin's definition of weak n -category. This operad is initial in the category of n -globular operads equipped with two extra pieces of structure: a system of compositions and a contraction. Our approach closely follows a proof by Leinster of the existence of a similar n -globular operad used in his definition of weak n -category (itself a variant of Batanin's definition) – we show that there is a functor giving the free operad equipped with a contraction and system of compositions on an n -globular collection, and applying this functor to the initial collection gives the desired initial operad. Since there is no interaction between the contraction and operad structures we are able to treat their free constructions separately. This is not true of the system of compositions structure, which cannot exist separately from the operad structure, so we use an interleaving-style construction to describe the free operad with system of compositions. [ABSTRACT FROM AUTHOR]

We show that every simple transitive 2-representation of the 2-category of projective functors for a certain quotient of the quadratic dual of the preprojective algebra associated with a tree is equivalent to a cell 2-representation. [ABSTRACT FROM AUTHOR]

It is known by results of Dyckerhoff–Kapranov and of Gálvez-Carrillo–Kock–Tonks that the output of the Waldhausen S • -construction has a unital 2-Segal structure. Here, we prove that a certain S • -functor defines an equivalence between the category of augmented stable double categories and the category of unital 2-Segal sets. The inverse equivalence is described explicitly by a path construction. We illustrate the equivalence for the known examples of partial monoids, cobordism categories with genus constraints and graph coalgebras. [ABSTRACT FROM AUTHOR]

A detailed description of a normalized internal bicategory in the category of groups is derived from the general description of internal bicategories in weakly Mal'tsev categories endowed with a V-Mal'tsev operation in the sense of Pedicchio. The example of bicategory of paths in a topological abelian group is presented. [ABSTRACT FROM AUTHOR]

We prove that if a finite tensor category $${\mathcal C}$$ is symmetric, then the monoidal category of one-sided $${\mathcal C}$$ -bimodule categories is symmetric. Consequently, the Picard group of $${\mathcal C}$$ (the subgroup of the Brauer-Picard group introduced by Etingov-Nikshych-Gelaki) is abelian in this case. We then introduce a cohomology over such $${\mathcal C}$$ . An important piece of tool for this construction is the computation of dual objects for bimodule categories and the fact that for invertible one-sided $${\mathcal C}$$ -bimodule categories the evaluation functor involved is an equivalence, being the coevaluation functor its quasi-inverse, as we show. Finally, we construct an infinite exact sequence à la Villamayor-Zelinsky for $${\mathcal C}$$ . It consists of the corresponding cohomology groups evaluated at three types of coefficients which repeat periodically in the sequence. We compute some subgroups of the groups appearing in the sequence. [ABSTRACT FROM AUTHOR]

We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations amongst the relations. Such extended presentations are used for representations of monoids, in particular, it is a way to describe actions of monoids on categories. Moreover, a coherent presentation provides the first step in the computation of a categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method introduced by Squier that computes a coherent presentation from a convergent one. We compute a finite coherent presentation of a plactic monoid from its column presentation and we reduce it to a Tietze equivalent one having Knuth's generators. [ABSTRACT FROM AUTHOR]

This article is concerned with 2-functors defined on the path 2-groupoid of a smooth manifold. We set up a procedure to extract local data of such 2-functors, similar to the extraction of transition functions of a fibre bundle. The main result of this paper establishes an equivalence between the globally defined 2-functors and their local data. This is a contribution to a project that provides an axiomatic formulation of connections on (possibly non-abelian) gerbes in terms of 2-functors, of which the present paper provides the first part. The second part provides equivalences between the local data, on one side, and various existing versions of gerbes with connection on the other side. [ABSTRACT FROM AUTHOR]

We discuss the folklore construction of the Gray tensor product of 2-categories as obtained by factoring the map from the funny tensor product to the cartesian product. We show that this factorisation can be obtained without using a concrete presentation of the Gray tensor product, but merely its defining universal property, and use it to give another proof that the Gray tensor product forms part of a symmetric monoidal structure. The main technical tool is a method of producing new algebra structures over Lawvere 2-theories from old ones via a factorisation system. [ABSTRACT FROM AUTHOR]

We study right quasi-representable differential graded bimodules as quasi-functors between dg-categories. We prove that a quasi-functor has a left adjoint if and only if it is left quasi-representable. With this characterisation, we prove an existence result of adjoints, under suitable hypotheses on the dg-categories. [ABSTRACT FROM AUTHOR]

Motivated by the appearance of multisemigroups in the study of additive 2-categories, we define and investigate the notion of a multisemigroup with multiplicities. This notion seems to be well suited for applications in higher representation theory. [ABSTRACT FROM AUTHOR]

We show how the notion of intercategory encompasses a wide variety of three-dimensional structures from the literature, notably duoidal categories, monoidal double categories, cubical bicategories, double bicategories and Gray categories. Variations on the notion of span provide further examples of interest, an important one being the intercategory of sets. We consider the three kinds of morphism of intercategory as well as the cells binding them with applications to the above structures. In particular hom functors are studied. [ABSTRACT FROM AUTHOR]

Given a group G, we define suitable 2-categorical structures on the class of all small categories with G-actions and on the class of all small G-graded categories, and prove that 2-categorical extensions of the orbit category construction and of the smash product construction turn out to be 2-equivalences (2-quasi-inverses to each other), which extends the Cohen-Montgomery duality. Further we characterize equivalences in both 2-categories. [ABSTRACT FROM AUTHOR]

We propose a categorification of the Chern character that refines earlier work of Toën and Vezzosi and of Ganter and Kapranov. If X is an algebraic stack, our categorified Chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over X , which we introduce, to S 1 -equivariant perfect complexes on the derived free loop stack L X . As an application of the theory, we show that Toën and Vezzosi's secondary Chern character factors through secondary K -theory. Our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal ( ∞ , n ) -categories, which is of independent interest. [ABSTRACT FROM AUTHOR]

Gluing of two pseudo functors has been studied by Deligne, Ayoub, and others in the construction of extraordinary direct image functors in étale cohomology, stable homotopy, and mixed motives of schemes. In this article, we study more generally the gluing of finitely many pseudo functors. Given pseudo functors $$F_i:\mathcal {A}_i\rightarrow \mathcal {D}$$ defined on sub-2-categories $$\mathcal {A}_i$$ of a 2-category $$\mathcal {C}$$ , we are concerned with the problem of finding pseudo functors $$\mathcal {C}\rightarrow \mathcal {D}$$ extending $$F_i$$ up to pseudo natural equivalences. With the help of n-fold categories, we organize gluing data for n pseudo functors into 2-categories. We establish general criteria for equivalence between such 2-categories for n pseudo functors and for $$n-1$$ pseudo functors, which can be applied inductively to the gluing problem. Results of this article are used in Zheng (Sci China Math 58(3):565-632, 2015) to construct extraordinary direct image functors in étale cohomology of Deligne-Mumford stacks. [ABSTRACT FROM AUTHOR]

We identify the trace, or zeroth Hochschild homology, of categorified quantum groups of ADE type with the corresponding current algebra of the same type. To prove this, we show that 2-representations defined using categories of modules over cyclotomic (or deformed cyclotomic) quotients of Khovanov-Lauda-Rouquier algebras correspond to local (or global) Weyl modules. We also investigate the implications for centers of categories in 2-representations of categorified quantum groups. [ABSTRACT FROM AUTHOR]

The fundamental bigroupoid of a topological space is one way of capturing its homotopy 2-type. When the space is semilocally 2-connected, one can lift the construction to a bigroupoid internal to the category of topological spaces, as Brown and Danesh-Naruie lifted the fundamental groupoid to a topological groupoid. For locally relatively contractible spaces the resulting topological bigroupoid is locally trivial in a way analogous to the case of the topologised fundamental groupoid. This is the published version of . [ABSTRACT FROM AUTHOR]