A Quillen model category structure on some categories of comonoids

Generalization of Algebraic Operations via Enrichment

In this dissertation we examine enrichment relations between categories of dual structure and we sketch an abstract framework where the theory of fibrations and enriched category theory are appropriately united. We initially work in the context of a monoidal category, where we study an enrichment of the category of monoids in the category of comonoids under certain assumptions. This is induced by the existence of the universal measuring comonoid, a notion originally defined by Sweedler in vector spaces. We then consider the fibred category of modules over arbitrary monoids, and we establish it…

Quillen model structures on the category of graphs

We present different ways of endowing a particular category of graphs with Quillen model structures. We show, among other things, that the core of a graph can be seen as its homotopy type in an appropriate Quillen model structure, and that an infinity of Quillen model structures exist for our particular category of graphs.

Map monoidales and duoidal $\infty$-categories

In this paper we give an example of duoidal $\infty$-categories. We introduce map $\mathcal{O}$-monoidales in an $\mathcal{O}$-monoidal $(\infty,2)$-category for an $\infty$-operad $\mathcal{O}^{\otimes}$. We show that the endomorphism mapping $\infty$-category of a map $\mathcal{O}$-monoidale is a coCartesian $(Δ^{\rm op},\mathcal{O})$-duoidal $\infty$-category. After that, we introduce a convolution product on the mapping $\infty$-category from an $\mathcal{O}$-comonoidale to an $\mathcal{O}$-monoidale. We show that the $\mathcal{O}$-monoidal structure on the duoidal endomorphism mapping $\i…

Composing PROBs

A PROB is a "product and braid" category. Such categories can be used to encode the structure borne by an object in a braided monoidal category. In this paper we provide PROBs whose categories of algebras in a braided monoidal category are equivalent to the categories of monoids and comonoids using the category associated to the braid crossed simplicial group of Fiedorowicz and Loday. We show that PROBs can be composed by generalizing the machinery introduced by Lack for PROPs. We use this to define a PROB for bimonoids in a braided monoidal category as a composite of the PROBs for monoids and…

Cofree Objects in The Centralizer and The Center Categories

We study cocompleteness, co-wellpoweredness and generators in the centralizer category of an object or morphism in a monoidal category, and the center or the weak center of a monoidal category. We explicitly give some answers for when colimits, cocompleteness, co-wellpoweredness and generators in these monoidal categories can be inherited from their base monidal categories. Most importantly, we investigate cofree objects of comonoids in these monoidal categories.

Quillen model categories without equalisers or coequalisers

Quillen defined a {\em model category} to be a category with finite limits and colimits carrying a certain extra structure. In this paper, we show that only finite products and coproducts (in addition to the certain extra structure alluded to above) are really necessary to construct the homotopy category. This leads to the interesting observation that the homotopy category construction could feasibly be iterated.1

A simplicial approach to multiplier bimonoids

Although multiplier bimonoids in general are not known to correspond to comonoids in any monoidal category, we classify them in terms of maps from the Catalan simplicial set to another suitable simplicial set; thus they can be regarded as (co)monoids in something more general than a monoidal category (namely, the simplicial set itself). We analyze the particular simplicial maps corresponding to that class of multiplier bimonoids which can be regarded as comonoids.

再论代数上余纤维与纤维对象

Algebraically cofibrant and fibrant objects revisited

本研究在弱模型范畴框架下拓展代数余纤维与纤维对象的转移模型结构结果，证明可达弱模型范畴对应两类对象上存在Quillen等价的转移结构，且组合弱模型范畴可通过Quillen等价链连接到所有对象都纤维的组合Quillen模型范畴。

Faithfulness of a functor of Quillen

There exists a canonical functor from the category of fibrant objects of a model category modulo cylinder homotopy to its homotopy category. We show that this functor is faithful under certain conditions, but not in general.

A Quillen model structure on the category of Kontsevich-Soibelman weakly unital dg categories

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…

Normal and conormal maps in homotopy theory

Let M be a monoidal category endowed with a distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids. We define and study homotopy-invariant notions of normality for maps of monoids and of conormality for maps of comonoids in M. These notions generalize both principal bundles and crossed modules and are preserved by nice enough monoidal functors, such as the normaliized chain complex functor. We provide several explicit classes of examples of homotopy-normal and of homotopy-conormal maps, when M is the category of simplicial sets …

Implications of a Quillen Model Structures-Based Framework for Locality under Logical Equivalence

In [15] a homotopic variation for locality of logics was presented, namely a Quillen model category-based framework for locality under logical equivalence, for every primitive-positive sentence of quantifier-rank $k$. In this paper, we will present some of the implications and possible themes for investigations that arise from the aforementioned framework.

Stacks and sheaves of categories as fibrant objects

We show that the category of categories fibred over a site is a generalized Quillen model category in which the weak equivalences are the local equivalences and the fibrant objects are the stacks, as they were defined by J. Giraud. The generalized model category restricts to one on the full subcategory whose objects are the categories fibred in groupoids. We show that the category of sheaves of categories is a model category that is Quillen equivalent to the generalized model category for stacks and to the model category for strong stacks due to A. Joyal and M. Tierney.

A folk Quillen model structure for operads

We establish, by elementary means, the existence of a cofibrantly generated monoidal model structure on the category of operads. By slicing over a suitable operad the classical Rezk model structure on the category of small categories is recovered.

An Algebraic Proof of Quillen's Resolution Theorem for K_1

In his 1973 paper Quillen proved a resolution theorem for the K-Theory of an exact category; his proof was homotopic in nature. By using the main result of a paper by Nenashev, we are able to give an algebraic proof of Quillen's Resolution Theorem for K_1 of an exact category.

A model category structure on the category of simplicial multicategories

We establish a Quillen model category structure on the category of symmetric simplicial multicategories. This model structure extends the model structure on simplicial categories due to J. Bergner.

Sheafifiable homotopy model categories

If a Quillen model category can be specified using a certain logical syntax (intuitively, ``is algebraic/combinatorial enough''), so that it can be defined in any category of sheaves, then the satisfaction of Quillen's axioms over any site is a purely formal consequence of their being satisfied over the category of sets. Such data give rise to a functor from the category of topoi and geometric morphisms to Quillen model categories and Quillen adjunctions.

Comparison Between Different Topological Models of Concurrency

In this note, we provide an explicit non-Quillen equivalence between the category of precubical sets and Gaucher's category of flows via a class of "realization functors" (with mild assumptions on the cofibrations of the category of precubical sets). In addition, we demonstrate a Quillen equivalence between simplicial semicategories and flows before proving that simplicial semicategories satisfy many of the same properties as flows. Finally, we introduce the category of boxed symmetric trees, presheaves on which may provide a slightly more flexible setting for concurrent computing than (pre)cu…

BiHom Hopf algebras viewed as Hopf monoids

We introduce monoidal categories whose monoidal products of any positive number of factors are lax coherent and whose nullary products are oplax coherent. We call them $\mathsf{Lax}^+\mathsf{Oplax}^0$-monoidal. Dually, we consider $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$-duoidal categories with compatible $\mathsf{Lax}^+\mathsf{Oplax}^0$- and $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal structures. We introduce comonoids in $\mathsf{La…

小闭合笛卡儿范畴的上同调

Cohomology of Small Cartesian Closed Categories

该研究证明了笛卡儿闭合范畴的奎伦上同调与巴乌斯-维尔斯钦格上同调的同构性，并扩展了此前关于小范畴和具有有限积的小范畴的结果。

Left fibrations and homotopy colimits II

For a small simplicial category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the homotopy-coherent nerve of A provides a left Quillen equivalence between the projective model structure on the former category and the covariant model structure on the latter. We compare this Quillen equivalence to the straightening-unstraightening equivalence previously established by Lurie, where the left adjoint goes in the opposite direction. The existence of left Quillen functors in both directions considerably simplifies t…

A Linear Category of Polynomial Diagrams

We present a categorical model for intuitionistic linear logic where objects are polynomial diagrams and morphisms are simulation diagrams. The multiplicative structure (tensor product and its adjoint) can be defined in any locally cartesian closed category, whereas the additive (product and coproduct) and exponential Tensor-comonoid comonad) structures require additional properties and are only developed in the category Set, where the objects and morphisms have natural interpretations in terms of games, simulation and strategies.

Transfer of algebras over operads along derived Quillen adjunctions

Let V be a cofibrantly generated monoidal model category and let M be a monoidal V-model category. Given a cofibrant C-coloured operad P in V, we give sufficient conditions for the fibrant replacement and cofibrant replacement functors in M^C to preserve P-algebra structures. In particular, we show how P-algebra structures can be transferred along derived Quillen adjunctions of monoidal V-model categories, and we apply this result to the Quillen adjunctions defined by enriched Bousfield localizations and colocalizations on M. As an application, we prove that in the category of symmetric spectr…

Weak model categories in classical and constructive mathematics

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…

通过 Dold-Kan 对应建立单纯代数的模型范畴结构

Model Category Structure on Simplicial Algebras via Dold-Kan Correspondence

本文通过 Dold-Kan 对应和 Quillen-Kan 转移机器，给出单纯交换代数范畴具有模型范畴结构的纯代数证明，并借助 shuffle 积与 Eilenberg-Zilber 定理变体解决无环性条件问题。

Quillen-Segal algebras and Stable homotopy theory

Let $\mathscr{M}$ be a monoidal model category that is also combinatorial and left proper. If $\mathscr{O}$ is a monad, operad, properad, or a PROP; following Segal's ideas we develop a theory of Quillen-Segal $\mathscr{O}$-algebras and show that we have a Quillen equivalence between usual $\mathscr{O}$-algebras and Quillen-Segal algebras. We use this theory to get the stable homotopy category by a similar method as Hovey.

Dendroidal sets and simplicial operads

We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model category of simplicial operads to the model category structure for infinity-operads on the category of dendroidal sets. By slicing over the monoidal unit, this also gives the Quillen equivalence between Segal categories and simplicial categories proved by J. Bergner, as well as the Quillen equivalence between quasi-categories and simplicial categories proved by …

A Quillen model structure for Gray-categories

A Quillen model structure on the category Gray-Cat of Gray-categories is described, for which the weak equivalences are the triequivalences. It is shown to restrict to the full subcategory Gray-Gpd of Gray-groupoids. This is used to provide a functorial and model-theoretic proof of the unpublished theorem of Joyal and Tierney that Gray-groupoids model homotopy 3-types. The model structure on Gray-Cat is conjectured to be Quillen equivalent to a model structure on the category Tricat of tricategories and strict homomorphisms of tricategories.

When stable short exact sequences define an exact structure on any additive category

Rump has recently showed the existence of a unique maximal Quillen exact structure on any additive category. We study when this is given by the stable short exact sequences, i.e. kernel-cokernel pairs consisting of a semi-stable kernel and a semi-stable cokernel.

Hopf monads on monoidal categories

We define Hopf monads on an arbitrary monoidal category, extending the definition given previously for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode. Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a cat…

Quillen equivalences for stable categories

For an abelian category $\mathcal{A}$ we investigate when the stable categories $\underline{\mathrm{GPro}}\mathrm{j}(\mathcal{A})$ and $\underline{\mathrm{GIn}}\mathrm{j}(\mathcal{A})$ are triangulated equivalent. To this end, we realize these stable categories as homotopy categories of certain (non-trivial) model categories and give conditions on $\mathcal{A}$ that ensure the existence of a Quillen equivalence between the model categories in question. We also study when such a Quillen equivalence transfers from $\mathcal{A}$ to categories naturally associated to $\mathcal{A}$, such as $\mathr…

范畴的 Quillen 否定幺半群及其在态射类上作用的 Schreier 图

The Quillen negation monoid of a category, and Schreier graphs of its action on classes of morphisms

本文定义了范畴的 Quillen 否定幺半群，通过提升性质作用于态射类，计算了拓扑空间范畴中最简单态射类轨道的 Schreier 图，发现其包含 21 个具有标准拓扑意义的态射类。

A model 2-category of enriched combinatorial premodel categories

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…

Monads on higher monoidal categories

We study the action of monads on categories equipped with several monoidal structures. We identify the structure and conditions that guarantee that the higher monoidal structure is inherited by the category of algebras over the monad. Monoidal monads and comonoidal monads appear as the base cases in this hierarchy. Monads acting on duoidal categories constitute the next case. We cover the general case of $n$-monoidal categories and discuss several naturally occurring examples in which $n\leq 3$.

A Thomason Model Structure on the Category of Small n-fold Categories

We construct a cofibrantly generated Quillen model structure on the category of small n-fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n-fold functor is a weak equivalence if and only if the diagonal of its n-fold nerve is a weak equivalence of simplicial sets. This is an n-fold analogue to Thomason's Quillen model structure on Cat. We introduce an n-fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the n-fold nerve. As a consequence, we completely prove that the un…

A category of multiplier bimonoids

The central object studied in this paper is a multiplier bimonoid in a braided monoidal category C. Adapting the philosophy of Janssen and Vercruysse, and making some mild assumptions on the category C, we consider a category M whose objects are certain semigroups in C and whose morphisms from A to B can be regarded as suitable multiplicative morphisms from A to the multiplier monoid of B. We equip this category M with a monoidal structure and describe multiplier bimonoids in C (whose structure morphisms belong to a distinguished class of regular epimorphisms) as certain comonoids in M. This p…

丰富的Koszul对偶

Enriched Koszul Duality

本文为非余单位余幂零dg-余代数与非单位dg-代数建立相容的模型结构，并将二者间的Quillen等价推广到非单位模范畴层面，从而给出Koszul对偶的丰富化形式。

Groupoidal 2-quasi-categories and homotopy 2-types

We define a notion of groupoidal 2-quasi-categories and show that they are the fibrant objects of a model structure on the category of $Θ_2$-sets. We show that this model category is Quillen equivalent to the Kan-Quillen model category of simplicial sets and that 2-truncated groupoidal 2-quasi-categories are models for homotopy 2-types.

One-sided exact categories

One-sided exact categories appear naturally as instances of Grothendieck pretopologies. In an additive setting they are given by considering the one-sided part of Keller's axioms defining Quillen exact categories. We study one-sided exact additive categories and a stronger version defined by adding the one-sided part of Quillen "obscure axiom". We show that some homological results, such as the Short Five Lemma and the 3 X 3 Lemma, can be proved in our context. We also note that the derived category of a one-sided exact additive category can be constructed.

Quillen Model Structures-Based Notions of Locality of Logics over Finite Models

Locality is a property of logics, based on Hanf's and Gaifman's theorems, and that was shown to be very useful in the context of finite model theory. In this paper I present a homotopic variation for locality, namely a Quillen model category-based framework for locality under k-logical equivalence, for every primitive-positive sentence of quantifier-rank k.

A Quillen model structure for bigroupoids and pseudofunctors

A model structure on the category of (small) bigroupoids and pseudofunctors is constructed. In this model structure, every object is cofibrant. In order to keep certain calculations of manageable size, a coherence theorem for bigroupoids and a coherence theorem for pseudofunctors are proven, which may be of independent interest as well.

A generalization of Quillen's small object argument

We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen's small object argument). The necessity of such a generalization arose with appearance of several important examples of model categories which were proven to be non-cofibrantly generated. Our current approach allows for construction of functorial factorizations and localizations in the equivariant model structures on diagrams of spaces and diagrams of chain complexes. We also formulate a non-functorial version of the argument, which applies in two diffe…

Homotopy Theory of Labelled Symmetric Precubical Sets

This paper is the third paper of a series devoted to higher dimensional transition systems. The preceding paper proved the existence of a left determined model structure on the category of cubical transition systems. In this sequel, it is proved that there exists a model category of labelled symmetric precubical sets which is Quillen equivalent to the Bousfield localization of this left determined model category by the cubification functor. The realization functor from labelled symmetric precubical sets to cubical transition systems which was introduced in the first paper of this series is use…

Comparison of models for $(\infty, n)$-categories, II

In this paper we complete a chain of explicit Quillen equivalences between the model category for $Θ_{n+1}$-spaces and the model category of small categories enriched in $Θ_n$-spaces. The Quillen equivalences given here connect Segal category objects in $Θ_n$-spaces, complete Segal objects in $Θ_n$-spaces, and $Θ_{n+1}$-spaces.

拓扑空间上的Quillen模型结构的合成方法

Synthetic approach to the Quillen model structure on topological spaces

本文通过公理化方法，将Quillen对拓扑空间的模型结构扩展到Δ生成空间和伪拓扑空间，并在局部域范畴中构建了新的模型结构。

Left fibrations and homotopy colimits

For a small category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the nerve of A establishes a left Quillen equivalence between the projective (or Reedy) model structure on the former category and the covariant model structure on the latter. We compare this equivalence to a Quillen equivalence in the opposite direction previously established by Lurie. From our results we deduce that a categorical equivalence of simplicial sets induces a Quillen equivalence on the corresponding over-categories, equipped with …

Some functorial factorizations for Quillen functors

We prove that any right Quillen functor between arbitrary model categories admits non trivial functorial factorizations that are similar to those of a model structure. We also prove that these factorizations can be made for lax monoidal right Quillen functors. Given a monad, operad or a PROP(erad) $\mathcal{O}$, if we apply one of the factorizations to the forgetful functor $\mathcal{U} : \mathcal{O}-Alg(\mathcal{M}) \rightarrow \mathcal{M}$, we extend the theory of Quillen-Segal $\mathcal{O}$-algebras without the hypothesis of $\mathcal{M}$ being a combinatorial model category.

A Quillen Model Structure Approach to the Finitistic Dimension Conjectures

We explore the interlacing between model category structures attained to classes of modules of finite $\mathcal{X}$-dimension, for certain classes of modules $\mathcal{X}$. As an application we give a model structure approach to the Finitistic Dimension Conjectures and present a new conceptual framework in which these conjectures can be studied.

一个用于扭曲导出范畴的注入模型及其弯曲柯西三重性

An injective Model for Twisted Derived Categories and Curved Koszul Triality

本文提出一种新的模型结构，将扭曲导出范畴与弯曲柯西三重性理论连接起来，并证明张量积是Quillen双函子。

Homotopy theory in a quasi-abelian category

We prove that the category of dg-modules and dg-algebras in a Grothendieck quasi-abelian category are endowed with a Quillen model structure. This allows some flexibility in setting up a theory of derived algebraic geometry in the infinite dimensional setting. For example, the category of complete bornological vector spaces, or equivalently, convenient vector spaces, is a Grothendieck quasi-abelian category. Closely related is the Grothendieck quasi-abelian category of ind-Banach spaces whose associated model category is shown to be Quillen equivalent. Applications include the Chevalley-Eilenb…

双范畴的便捷模型范畴

A convenient model category for bicategories

本文引入并研究柔性2-范畴的模型范畴，它与Lack的2-范畴模型范畴Quillen等价，但具有所有对象都是余纤维的、关于笛卡尔闭结构是幺半模型范畴等优良性质，其纤维对象全子范畴等价于双范畴与正规伪函子的范畴。

A Quillen model for classical Morita theory and a tensor categorification of the Brauer group

Let K be a commutative ring. In this article we construct a symmetric monoidal Quillen model structure on the category of small K-categories which enhances classical Morita theory. We then use it in order to obtain a natural tensor categorification of the Brauer group and of its functoriality.

Coalgebraic models for combinatorial model categories

We show that the category of algebraically cofibrant objects in a combinatorial and simplicial model category A has a model structure that is left-induced from that on A. In particular it follows that any presentable model category is Quillen equivalent (via a single Quillen equivalence) to one in which all objects are cofibrant.

形式弱充实范畴论

Formal weakly enriched category theory

本文构造了编码弱充实的形式范畴论，证明其与拟范畴的无穷宇宙等价，并导出充实版Quillen定理A。