A weak entwining structure in a 2-category K consists of a monad t and a comonad c, together with a 2-cell relating both structures in a way that generalizes a mixed distributive law.A weak entwining structure can be characterized as a compatible pair of a monad and a comonad, in 2-categories generalizing the 2-category of comonads and the 2-category of monads in K, respectively. This observation is used to define a 2-category Entw^w(K) of weak entwining structures in K. If the 2-category K admits Eilenberg-Moore constructions for both monads and comonads and idempotent 2-cells in K split, then there are pseudo-functors from Entw^w(K) to the 2-category of monads and to the 2-category of comonads in K, taking a weak entwining structure (t,c) to a `weak lifting' of t for c and a `weak lifting' of c for t, respectively. The Eilenberg-Moore objects of the lifted monad and the lifted comonad are shown to be equivalent. If K is the 2-category of functors induced by bimodules, then these Eilenberg-Moore objects are isomorphic to the usual category of weak entwined modules.
We define the secondary Hochschild complex for an entwining structure over a commutative $k$-algebra $B$. We show that this complex carries the structure of a weak comp algebra. We obtain two distinct cup product structures for the secondary cohomology groups. We also consider a subcomplex on which the two cup products coincide and which satisfies the axioms for being a comp algebra. The cohomology of this subcomplex then forms a Gerstenhaber algebra. We also construct a bicomplex that controls the deformations of the entwining structure over $B$.
{\em Galois comodules} over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of {\em Galois functors} over some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injectives (projectives) in this context. Then we generalise the notion of corings (derived from an entwining of an algebra and a coalgebra) to the entwining of a monad and a comonad. Hereby a key role is played by the notion of a {\em grouplike natural transformation} $g:I\to G$ generalising the groupli…
We construct a `weak' version EM^w(K) of Lack & Street's 2-category of monads in a 2-category K, by replacing their compatibility constraint of 1-cells with the units of monads by an additional condition on the 2-cells. A relation between monads in EM^w(K) and composite pre-monads in K is discussed. If K admits Eilenberg-Moore constructions for monads, we define two symmetrical notions of `weak liftings' for monads in K. If moreover idempotent 2-cells in K split, we describe both kinds of a weak lifting via an appropriate pseudo-functor EM^w(K) --> K. Weak entwining structures and partial entw…
The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to {\em monoidal} categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal structure of the category of endofunctors on any category $\A$ and by this we retain some of the combinatorial complexity which makes the theory so inter…
A weak mixed distributive law (also called weak entwining structure) in a 2-category consists of a monad and a comonad, together with a 2-cell relating them in a way which generalizes a mixed distributive law due to Beck. We show that a weak mixed distributive law can be described as a compatible pair of a monad and a comonad, in 2-categories extending, respectively, the 2-category of comonads and the 2-category of monads. Based on this observation, we define a 2-category whose 0-cells are weak mixed distributive laws. In a 2-category K which admits Eilenberg-Moore constructions both for monad…
We provide a complete description of the category of pseudo-categories (including pseudo-functors, natural and pseudo-natural transformations and pseudo modifications). A pseudo-category is a non strict version of an internal category. It was called a weak category and weak double category in some earlier papers. When internal to Cat it is at the same time a generalization of a bicategory and a double category.
Batanin defines a weak $ω$-category as an algebra for a certain operad. Leinster refines this idea and defines the weak $ω$-category operad as the initial object of a category of "operads with contraction". We demonstrate how a higher category structure arises from this definition by explicitly constructing various composites, associativity and coherence laws, and an Eckmann-Hilton braiding.
Classical definitions of weak higher-dimensional categories are given inductively; for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak $n$-categories for all natural numbers $n$, or of weak $ω$-categories, take more sophisticated approaches, and the nature of the "hom" is often not immediate from the definitions. In this paper, we focus on Leinster's definition of weak $ω$-category based on an earlier definition by Batanin, and construct for each weak $ω$-category $\mathcal{A}$,…
It has long been known that every weak monoidal category A is equivalent via monoidal functors and monoidal natural transformations to a strict monoidal category st(A). We generalise the definition of weak monoidal category to give a definition of weak P-category for any strongly regular (operadic) theory P, and show that every weak P-category is equivalent via P-functors and P-transformations to a strict P-category. This strictification functor is then shown to have an interesting universal property.
Let $\mathcal{C}$ be a finitely complete small category. In this paper, first we construct two weak (Lawvere-Tierney) topologies on the category of presheaves. One of them is established by means of a subfunctor of the Yoneda functor and the other one, is constructed by an admissible class on $\mathcal{C}$ and the internal existential quantifier in the presheaf topos $\widehat{\mathcal{C}}$. Moreover, by using an admissible class on $\mathcal{C},$ we are able to define an action on the subobject classifier $Ω$ of $\widehat{\mathcal{C}}$. Then we find some necessary conditions for that the two …
In [BaSc2] the authors introduced a much weaker homotopical structure than a model category, called a "weak cofibration category". We further showed that a small weak cofibration category induces in a natural way a model category structure on its ind-category, provided the ind-category satisfies a certain two out of three property. The purpose of this paper is to serve as a companion to the papers above, proving results which say that if a certain property or structure exists in the weak cofibration category, then the same property or structure also holds in the induced model structure on the …
In this paper we introduce the notion of weak operator and the theory of Yetter-Drinfeld modules over a weak braided Hopf algebra with invertible antipode in a strict monoidal category. We prove that the class of such objects constitutes a non strict monoidal category. It is also shown that this category is not trivial, that is to say that it admits objects generated by the adjoint action (coaction) associated to the weak braided Hopf algebra.
We define a homotopy relation between arrows of a category with weak equivalences, and give a condition under which the quotient by the homotopy relation yields the homotopy category. In the case of the fibrant-cofibrant objects of a model category this condition holds, and we show that our notion of homotopy coincides with the classical one. We also show that Quillen's construction of the homotopy category of a model category, in which the arrows are homotopical classes of a single arrow between fibrant-cofibrant objects, can be made as well for categories with weak equivalences using this no…
This paper continues the development of a simplicial theory of weak omega-categories, by studying categories which are enriched in weak complicial sets. These complicial Gray-categories generalise both the Kan complex enriched categories of homotopy theory and the Gray-categories of weak 3-category theory. We derive a simplicial nerve construction, which is closely related to Cordier and Porter's homotopy coherent nerve, and show that this faithfully represents complicial Gray-categories as weak complicial sets. The category of weak complicial sets may itself be canonically enriched to a compl…
In a recent paper we introduced a much weaker and easy to verify structure than a model category, which we called a "weak fibration category". We further showed that a small weak fibration category can be "completed" into a full model category structure on its pro-category, provided the pro-category satisfies a certain two out of three property. In the present paper we give sufficient intrinsic conditions on a weak fibration category for this two out of three property to hold. We apply these results to prove theorems giving sufficient conditions for the finite accessibility of the category of …
Gambino and Garner proved that the syntactic category of a dependent type theory with identity types can be endowed with a weak factorization system structure, called identity type weak factorization system. In this paper we consider an enrichment of Joyal's notion of tribe, which we call a tribe with weakly stable path objects, and prove a purely category-theoretic version of the identity type weak factorization system, thus generalizing Gambino and Garner's result. We conclude showing that this structure subsumes also the weak factorization systems coming from the topological and simplicial …
We generalise the concepts introduced by Baez and Dolan to define opetopes constructed from symmetric operads with a category, rather than a set, of objects. We describe the category of 1-level generalised multicategories, a special case of the concept introduced by Hermida, Makkai and Power, and exhibit a full embedding of this category in the category of symmetric operads with a category of objects. As an analogy to the Baez-Dolan slice construction, we exhibit a certain multicategory of function replacement as a slice construction in the multitopic setting, and use it to construct multitope…
Model categories have long been a useful tool in homotopy theory, allowing many generalizations of results in topological spaces to other categories. Giving a localization of a model category provides an additional model category structure on the same base category, which alters what objects are being considered equivalent by increasing the class of weak equivalences. In some situations, a model category where the class of weak equivalences is restricted from the original one could be more desirable. In this situation we need the notion of a delocalization. In this paper, right Bousfield deloc…
Tannaka Duality describes the relationship between algebraic objects in a given category and their representations; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful "fibre functors" to the category of vector spaces. We simultaneously generalize the theory of Tannaka duality in two ways: first, we replace Hopf algebras with weak Hopf algebras and strong monoidal functors with separable Frobenius monoidal functors; second, we replace the category of vector spaces with an arbitrary braided monoidal category. To accomplish thi…
We define Euler characteristic of a category enriched by a monoidal model category. If a monoidal model category V is equipped with Euler characteristic that is compatible with weak equivalences and fibrations in V, then our Euler characteristic is also compatible with weak equivalences and fibrations in the model structure induced by that of V. In particular, we focus on the case of topological categories; that is, categories enriched by the category of topological spaces. As its application, we obtain the ordinary Euler characteristic of a cellular stratified space X by computing the Euler c…
We prove that the category of flows cannot be the underlying category of a model category whose corresponding homotopy types are the flows up to weak dihomotopy. Some hints are given to overcome this problem. In particular, a new approach of dihomotopy involving simplicial presheaves over an appropriate small category is proposed. This small category is obtained by taking a full subcategory of a locally presentable version of the category of flows.
Many definitions of weak n-category have been proposed. It has been widely observed that each of these definitions is of one of two types: algebraic definitions, in which composites and coherence cells are explicitly specified, and non-algebraic definitions, in which a coherent choice of composites and constraint cells is merely required to exist. Relatively few comparisons have been made between definitions, and most of those that have concern the relationship between definitions of just one type. The aim of this thesis is to establish more comparisons, including a comparison between an algeb…
We introduce a notion of "weak model category" which is a weakening of the notion of Quillen model category, still sufficient to define a homotopy category, Quillen adjunctions, Quillen equivalences and most of the usual construction of categorical homotopy theory. Both left and right semi-model categories are weak model categories, and the opposite of a weak model category is again a weak model category. The main advantages of weak model categories is that they are easier to construct than Quillen model categories. In particular we give some simple criteria on two weak factorization systems f…
We study semi-strict tricategories in which the only weakness is in vertical composition. We construct these as categories enriched in the category of bicategories with strict functors, with respect to the cartesian monoidal structure. As these are a form of tricategory it follows that doubly-degenerate ones are braided monoidal categories. We show that this form of semi-strict tricategory is weak enough to produce all braided monoidal categories. That is, given any braided monoidal category $B$ there is a doubly-degenerate ``vertically weak'' semi-strict tricategory whose associated braided m…
In this paper, which is subsequent to our previous paper [PS] (but can be read independently from it), we continue our study of the closed model structure on the category $\mathrm{Cat}_{\mathrm{dgwu}}(\Bbbk)$ of small weakly unital dg categories (in the sense of Kontsevich-Soibelman [KS]) over a field $\Bbbk$. In [PS], we constructed a closed model structure on the category of weakly unital dg categories, imposing a technical condition on the weakly unital dg categories, saying that $\mathrm{id}_x\cdot \mathrm{id}_x=\mathrm{id}_x$ for any object $x$. Although this condition led us to a great s…
We consider four categories: the category of diagrams of small categories indexed by a given small category O, the (comma) category of small categories over O, the category of diagrams of simplicial sets indexed by O, and the category of simplicial sets over the nerve of O. Fritsch and Golasinski claimed that these four categories have equivalent homotopy categories but, in fact, their proof contains an error and the homotopy categories are not equivalent with the weak equivalences they use in the comma categories. We show here that the correct weak equivalences are the ``weak fibre homotopy e…
We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal's category Theta_n. This notion is a generalization of that of complete Segal spaces (which are precisely the (infty,1)-Theta-spaces). Our main result is that the above model category is cartesian.
This paper develops the foundations of a simplicial theory of weak omega-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) omega-categories, Kan complexes and Joyal's quasi-categories. We generalise a number of results due to the current author with regard to complicial sets and strict omega-categories to provide an armoury of well behaved technical devices, such as joins and Gray tensor products, which will be used to s…
Based on the novel notion of `weakly counital fusion morphism', regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields and multiplier bimonoids in braided monoidal categories. Under some assumptions the so-called base object of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid structure; hence to possess a monoidal category of bicomodules. In this case, appropriately defined modules over a regular weak multiplier bimonoid are proven to constitute a monoidal category with a strict monoidal…
Weak bimonoids in duoidal categories are introduced. They provide a common generalization of bimonoids in duoidal categories and of weak bimonoids in braided monoidal categories. Under the assumption that idempotent morphisms in the base category split, they are shown to induce weak bimonads (in four symmetric ways). As a consequence, they have four separable Frobenius base (co)monoids, two in each of the underlying monoidal categories. Hopf modules over weak bimonoids are defined by weakly lifting the induced comonad to the Eilenberg-Moore category of the induced monad. Making appropriate ass…
In his book on model categories, Hovey asked whether the 2-category $\mathbf{Mod}$ of model categories admits a "model 2-category structure" whose weak equivalences are the Quillen equivalences. We show that $\mathbf{Mod}$ does not have pullbacks and so cannot form a model 2-category. This lack of pullbacks can be traced to the two-out-of-three axiom on the weak equivalences of a model category. Accordingly, we define a premodel category to be a complete and cocomplete category equipped with two nested weak factorization systems. Combinatorial premodel categories form a complete and cocomplete…
In [BaSc2], the author and Tomer Schlank introduced a much weaker homotopical structure than a model category, which we called a "weak cofibration category". We further showed that a small weak cofibration category induces in a natural way a model category structure on its indcategory, provided the ind-category satisfies a certain two out of three property. The main purpose of this paper is to give sufficient intrinsic conditions on a weak cofibration category for this two out of three property to hold. We consider an application to the category of compact metrizable spaces, and thus obtain a …
A weak asynchronous system is a trace monoid with a partial action on a set. A polygonal morphism between weak asynchronous systems commutes with the actions and preserves the independence of events. We prove that the category of weak asynchronous systems and polygonal morphisms has all limits and colimits.
In a previous work, we have introduced a weakening of Quillen model categories called weak model categories. They still allow all the usual constructions of model category theory, but are easier to construct and are in some sense better behaved. In this paper we continue to develop their general theory by introducing combinatorial and accessible weak model categories. We give simple necessary and sufficient conditions under which such a weak model category can be extended into a left and/or right semi-model category. As an application, we recover Cisinski-Olschok theory and generalize it to we…
In a previous paper we lifted Charles Rezk's complete Segal model structure on the category of simplicial spaces to a Quillen equivalent one on the category of "relative categories," and our aim in this successor paper is to obtain a more explicit description of the weak equivalences in this model structure by showing that these weak equivalences are exactly the DK-equivalences, i.e. those maps between relative categories which induce a weak equivalence between their simplicial localizations.
Given a 2-category $\mathcal{A}$, a $2$-functor $\mathcal{A} \overset {F} {\longrightarrow} \mathcal{C}at$ and a distinguished 1-subcategory $Σ\subset \mathcal{A}$ containing all the objects, a $σ$-cone for $F$ (with respect to $Σ$) is a lax cone such that the structural $2$-cells corresponding to the arrows of $Σ$ are invertible. The conical $σ$-limit is the universal (up to isomorphism) $σ$-cone. The notion of $σ$-limit generalises the well known notions of pseudo and lax limit. We consider the fundamental notion of $σ$-filtered} pair $(\mathcal{A}, \, Σ)$ which generalises the notion of 2-f…
We introduce the category HG, whose objects are topological groupoids endowed with compatible measure theoretic data: a Haar system and a measure on the unit space. We then define and study the notion of weak-pullback in the category of topological groupoids, and subsequently in HG. The category HG is the setting for topological groupoidification, which we present in separate papers, and in which the weak pullback is a key ingredient.
Vopěnka's Principle says that the category of graphs has no large discrete full subcategory, or equivalently that the category of ordinals cannot be fully embedded into it. Weak Vopěnka's Principle is the dual statement, which says that the opposite category of ordinals cannot be fully embedded into the category of graphs. It was introduced in 1988 by Adámek, Rosický, and Trnková, who showed that it follows from Vopěnka's Principle and asked whether the two statements are equivalent. We show that they are not. However, we show that Weak Vopěnka's Principle is equivalent to the generalization o…
In this paper, we study weakly unital dg categories as they were defined by Kontsevich and Soibelman [KS, Sect.4]. We construct a cofibrantly generated Quillen model structure on the category $\mathrm{Cat}_{\mathrm{dgwu}}(\Bbbk)$ of small weakly unital dg categories over a field $\Bbbk$. Our model structure can be thought of as an extension of the model structure on the category $\mathrm{Cat}_{\mathrm{dg}}(\Bbbk)$ of (strictly unital) small dg categories over $\Bbbk$, due to Tabuada [Tab]. More precisely, we show that the imbedding of $\mathrm{Cat}_{\mathrm{dg}}(\Bbbk)$ to $\mathrm{Cat}_{\math…
We investigate the categories of weak maps associated to an algebraic weak factorisation system (AWFS) in the sense of Grandis-Tholen. For any AWFS on a category with an initial object, cofibrant replacement forms a comonad, and the category of (left) weak maps associated to the AWFS is by definition the Kleisli category of this comonad. We exhibit categories of weak maps as a kind of "homotopy category", that freely adjoins a section for every "acyclic fibration" (=right map) of the AWFS; and using this characterisation, we give an alternate description of categories of weak maps in terms of …
We introduce the groupoidal analogue \tildeΘto Joyal's cell category Θand we prove that \tildeΘis a strict test category in the sense of Grothendieck. This implies that presheaves on \tildeΘmodel homotopy types in a canonical way. We also prove that the canonical functor from Θto \tildeΘis aspherical, again in the sense of Grothendieck. This allows us to compare weak equivalences of presheaves on \tildeΘto weak equivalences of presheaves on Θ. Our proofs apply to other categories analogous to Θ.
We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and classify module categories over the fusion category of $\hat{sl}(2)$ at a positive integer level where we meet once again ADE classification pattern.
We develop a model structure on presheaves of small simplicially enriched categories on a site $\mathscr{C}$, for which the weak equivalences are 'stalkwise' weak equivalences for the Bergner model structure. This model structure is right proper, and can be connected via a zig-zag of Quillen equivalences to local analogues of the Joyal and Rezk model structures. Because the local Bergner model structure is right proper, we can apply Jardine's cocycle categories to study its homotopy category. As an application of the cocycle theory, we describe the maps $[*, X]$ in the homotopy category of the…
We present a weak form of a recognition principle for Quillen model categories due to J.H. Smith. We use it to put a model category structure on the category of small categories enriched over a suitable monoidal simplicial model category. The proof uses a part of the model structure on small simplicial categories due to J. Bergner. We give an application of the weak form of Smith's result to left Bousfield localizations of categories of monoids in a suitable monoidal model category.
Lack described a Quillen model structure on the category GrayCat of Gray-categories and Gray-functors, for which the weak equivalences are the weak 3-equivalences. In this note, we adapt the technique of Gurski, Johnson, and Osorno to show the localization of GrayCat at the weak equivalences is equivalent to the category of algebraic tricategories and pseudo-natural equivalence classes of weak 3-functors.
The set-theoretic axiom WISC states that for every set there is a set of surjections to it cofinal in all such surjections. By constructing an unbounded topos over the category of sets and using an extension of the internal logic of a topos due to Shulman, we show that WISC is independent of the rest of the axioms of the set theory given by a well-pointed topos. This also gives an example of a topos that is not a predicative topos as defined by van den Berg.
In these notes we describe models of globular weak $(\infty,m)$-categories ($m\in\mathbb{N}$) in the Grothendieck style, i.e for each $m\in\mathbb{N}$ we define a globular coherator $Θ^{\infty}_{\mathbb{M}^m}$ whose set-models are globular weak $(\infty,m)$-categories. Then we describe the combinatorics of the small category $Θ_0$ whose objects are cubical pasting diagrams and whose morphisms are morphisms of cubical sets. This provides an accurate description of the monad, on the category of cubical sets (without degeneracies and connections), of cubical strict $\infty$-categories with connec…
Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the "homotopy theory" of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a "homotopy theory of homotopy theories." In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to …
In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result -- the lifting theorem for multitensors -- enables us to see the Gray tensor product of 2-categories and the Crans tensor product of Gray categories as part of this framework. We define weak n-categories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category …
In this paper we put a cofibrantly generated model category structure on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories.
