Weak Factorization Systems Research Articles

An algebraic weak factorisation system on 01-substitution sets: a constructive proof

We will construct an algebraic weak factorisation system on the category of 01 substitution sets such that the R-algebras are precisely the Kan fibrations together with a choice of Kan filling operation. The proof is based on Garner's small object argument for algebraic weak factorization systems. In order to ensure the proof is valid constructively, rather than applying the general small object argument, we give a direct proof based on the same ideas. We use this us to give an explanation why the J computation rule is absent from the original cubical set model and suggest a way to fix this.

Accessible Model Categories

We prove that a weak factorization system on a locally presentable category is accessible if and only if it is small generated in the sense of R. Garner. Moreover, we discuss an analogy of Smith’s theorem for accessible model categories.

Algebraic weak factorisation systems I: Accessible AWFS

Algebraic weak factorisation systems (awfs) refine weak factorisation systems by requiring that the assignations sending a map to its first and second factors should underlie an interacting comonad–monad pair on the arrow category. We provide a comprehensive treatment of the basic theory of awfs—drawing on work of previous authors—and complete the theory with two main new results. The first provides a characterisation of awfs and their morphisms in terms of their double categories of left or right maps. The second concerns a notion of cofibrant generation of an awfs by a small double category; it states that, over a locally presentable base, any small double category cofibrantly generates an awfs, and that the awfs so arising are precisely those with accessible monad and comonad. Besides the general theory, numerous applications of awfs are developed, emphasising particularly those aspects which go beyond the non-algebraic situation.

Algebraic weak factorisation systems II: Categories of weak maps

We investigate the categories of weak maps associated to an algebraic weak factorisation system (awfs) in the sense of Grandis–Tholen [14]. 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 spans with left leg an acyclic fibration. We moreover show that the 2-functor sending each awfs on a suitable category to its cofibrant replacement comonad has a fully faithful right adjoint: so exhibiting the theory of comonads, and dually of monads, as incorporated into the theory of awfs. We also describe various applications of the general theory: to the generalised sketches of Kinoshita–Power–Takeyama [22], to the two-dimensional monad theory of Blackwell–Kelly–Power [4], and to the theory of dg-categories [19].

Cellular categories

We study locally presentable categories equipped with a cofibrantly generated weak factorization system. Our main result is that these categories are closed under 2-limits, in particular under pseudopullbacks. We give applications to deconstructible classes in Grothendieck categories. We discuss pseudopullbacks of combinatorial model categories.

Weak factorization systems for S-acts

The concept of a weak factorization system has been studied extensively in homotopy theory and has recently found an application in one of the proofs of the celebrated flat cover conjecture, categorical versions of which have been presented by a number of authors including Rosický. One of the main aims of this paper is to draw attention to this interesting concept and to initiate a study of these systems in relation to flatness of S-acts and related concepts.

Retraction Note to: Understanding the Small Object Argument

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an algebraic refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.

Fibrewise injectivity and Kock–Zöberlein monads

Using Escardó’s characterization of injectivity via Kock–Zöberlein monads, we introduce suitable monads in comma categories of topological spaces that yield characterizations of fibrewise injectivity in topological T0-spaces, with respect to the class of embeddings, and of dense, of flat and of completely flat embeddings. Characterizations, in the category of topological spaces, of injective maps with respect to the same classes of embeddings follow easily from the results obtained for T0-spaces. Moreover, it is shown that, together with the corresponding embeddings, injective continuous maps form a weak factorization system in the category of topological (T0-)spaces and continuous maps.

Topological and Simplicial Models of Identity Types

In this paper we construct new categorical models for the identity types of Martin-Löf type theory, in the categories Top of topological spaces and SSet of simplicial sets. We do so building on earlier work of Awodey and Warren [2009], which has suggested that a suitable environment for the interpretation of identity types should be a category equipped with a weak factorization system in the sense of Bousfield--Quillen. It turns out that this is not quite enough for a sound model, due to some subtle coherence issues concerned with stability under substitution; and so our first task is to introduce a slightly richer structure, which we call a homotopy-theoretic model of identity types , and to prove that this is sufficient for a sound interpretation. Now, although both Top and SSet are categories endowed with a weak factorization system---and indeed, an entire Quillen model structure---exhibiting the additional structure required for a homotopy-theoretic model is quite hard to do. However, the categories we are interested in share a number of common features, and abstracting these leads us to introduce the notion of a path object category . This is a relatively simple axiomatic framework, which is nonetheless sufficiently strong to allow the construction of homotopy-theoretic models. Now by exhibiting suitable path object structures on Top and SSet , we endow those categories with the structure of a homotopy-theoretic model and, in this way, obtain the desired topological and simplicial models of identity types.

A homotopy-theoretic universal property of Leinster's operad for weak ω-categories

AbstractWe explain how any cofibrantly generated weak factorisation system on a category may be equipped with a universally and canonically determined choice of cofibrant replacement. We then apply this to the theory of weak ω-categories, showing that the universal and canonical cofibrant replacement of the operad for strict ω-categories is precisely Leinster's operad for weak ω-categories.

Weak Reflections and Weak Factorization Systems

We describe a one-to-one correspondence between saturated weak factorization systems and weak reflections in categories \(\mathcal{C}\) with finite products. This actually extends to an adjunction between the category of natural weak factorization systems on \(\mathcal{C}\) (in the sense of Grandis and Tholen, Arch Math 42:397–408, 2006, and Garner, arXiv preprint, 2007) and the category of monads on \(\mathcal{C}\). Explicit comparisons are made with the parallel result of Cassidy et al. (J Aust Math Soc 38:287–329, 1985), linking factorization systems and reflective subcategories.

The identity type weak factorisation system

We show that the classifying category C ( T ) of a dependent type theory T with axioms for identity types admits a non-trivial weak factorisation system. We provide an explicit characterisation of the elements of both the left class and the right class of the weak factorisation system. This characterisation is applied to relate identity types and the homotopy theory of groupoids.

Understanding the Small Object Argument

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an "algebraic" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.

RETRACTED ARTICLE: Understanding the Small Object Argument

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an “algebraic” refinement of the small object argument, cast in terms of Grandis and Tholen’s natural weak factorisation systems, which rectifies each of these three deficiencies.

$\lambda$ -presentable Morphisms, Injectivity and (Weak) Factorization Systems

We show that in a locally $\lambda$ -presentable category, every $\lambda_m$ -injectivity class (i.e., the class of all the objects injective with respect to some class of $\lambda$ -presentable morphisms) is a weakly reflective subcategory determined by a functorial weak factorization system cofibrantly generated by a class of $\lambda$ -presentable morphisms. This was known for small-injectivity classes, and referred to as the ‘small object argument.’ An analogous result is obtained for orthogonality classes and factorization systems, where $\lambda$ -filtered colimits play the role of the transfinite compositions in the injectivity case. $\lambda$ -presentable morphisms are also used to organize and clarify some related results (and their proofs), in particular on the existence of enough injectives (resp. pure-injectives). Finally, locally $\lambda$ -presentable categories are shown to be cellularly generated by the set of morphisms between $\lambda$ -presentable objects.

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 [J. Adámek, H. Herrlich, J. Rosický, W. Tholen, Weak factorization systems and topological functors, Appl. Categ. Structures 10 (3) (2002) 237–249 [2]; Papers in honour of the seventieth birthday of Professor Heinrich Kleisli (Fribourg, 2000); B. Chorny, The model category of maps of spaces is not cofibrantly generated, Proc. Amer. Math. Soc. 131 (2003) 2255–2259; J.D. Christensen, M. Hovey, Quillen model structures for relative homological algebra, Math. Proc. Cambridge Philos. Soc. 133 (2) (2002) 261–293; D.C. Isaksen, A model structure on the category of pro-simplicial sets, Trans. Amer. Math. Soc. 353 (2001) 2805–2841]. Our current approach allows for construction of functorial factorizations and localizations in the equivariant model structures on diagrams of spaces [E.D. Farjoun, Homotopy theories for diagrams of spaces, Proc. Amer. Math. Soc. 101 (1987) 181–189] and diagrams of chain complexes. We also formulate a non-functorial version of the argument, which applies in two different model structures on the category of pro-spaces [D.A. Edwards, H.M. Hastings, Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Mathematics, vol. 542, Springer, Berlin, 1976; D.C. Isaksen, A model structure on the category of pro-simplicial sets, Trans. Amer. Math. Soc. 353 (2001) 2805–2841]. The examples above suggest a natural extension of the framework of cofibrantly generated model categories. We introduce the concept of a class-cofibrantly generated model category, which is a model category generated by classes of cofibrations and trivial cofibrations satisfying some reasonable assumptions.

Weak Factorizations, Fractions and Homotopies

We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Winskel.

Lax factorization algebras

It is shown that many weak factorization systems appearing in functorial Quillen model categories, including all those that are cofibrantly generated, come with a rich computational structure, defined by a certain lax algebra with respect to the “squaring monad” on CAT. This structure largely facilitates natural choices for left or right liftings once certain basic natural choices have been made. The use of homomorphisms of such lax algebras is also discussed, with focus on “lax freeness”.

Flat covers and factorizations

The flat cover conjecture, saying that every module has a flat (pre)cover, has been recently proved by Bican, El Bashir, and Enochs. We relate flat precovers (and cotorsion preenvelopes) to weak factorizations and prove that flat monomorphisms form a left part of a weak factorization system.

Weak Factorization Systems and Topological Functors

Weak factorization systems, important in homotopy theory, are related to injective objects in comma-categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not cofibrantly generated. We also present a weak factorization system on the category of posets which is not cofibrantly generated. No such weak factorization systems were known until recently. This answers an open problem posed by M. Hovey.