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.

On a fat small object argument

Good colimits introduced by J. Lurie generalize transfinite composites and provide an important tool for understanding cofibrant generation in locally presentable categories. We will explore the relation of good colimits to transfinite composites further and show, in particular, how they eliminate the use of large objects in the usual small object argument.     

Direct realism: Proximate causation and the missing object

Direct Realists believe that perception involves direct awareness of an object not dependent for its existence on the perceiver. Howard Robinson rejects this doctrine in favour of a Sense-Datum theory of perception. His argument against Direct Realism invokes the principle ‘same proximate cause, same immediate effect’. Since there are cases in which direct awareness has the same proximate cerebral cause as awareness of a sense datum, the Direct Realist is, he thinks, obliged to deny this causal principle. I suggest that although Direct Realism is in more than one respect implausible, it does not succumb to Robinson’s argument. The causal principle is true only if ‘proximate cause’ means ‘proximate sufficient cause’, and the Direct Realist need not concede that there is a sufficient cerebral cause for direct awareness of independent objects.     

The Isbell monad

In 1966 [7], John Isbell introduced a construction on categories which he termed the “couple category” but which has since come to be known as the Isbell envelope. The Isbell envelope, which combines the ideas of contravariant and covariant presheaves, has found applications in category theory, logic, and differential geometry. We clarify its meaning by exhibiting the assignation sending a locally small category to its Isbell envelope as the action on objects of a pseudomonad on the 2-category of locally small categories; this is the Isbell monad of the title. We characterise the pseudoalgebras of the Isbell monad as categories equipped with a cylinder factorisation system; this notion, which appears to be new, is an extension of Freyd and Kelly's notion of factorisation system [5] from orthogonal classes of arrows to orthogonal classes of cocones and cones.     

Concordant and Monotone Morphisms

Concordant-dissonant and monotone-light factorisation systems on categories, ways to construct them, and conditions for them to coincide, as well as their examples are studied in this article. These factorisation systems are constructed from a reflection induced from a ground adjunction and a specified prefactorisation system. Furthermore, we give additional conditions, under which the monotone-light and the concordant-dissonant factorisations coincide for sub-reflections of the induced reflection. The adjunctions given by right Kan extensions, from the category of presheaves on sets, turn out to be very well-behaved examples, provided they satisfy the cogenerating set condition, which allows to describe the four classes of morphisms in the reflective and concordant-dissonant (= monotone-light) factorisations. It is also noticed that the faithfulness of the composite of the left-adjoint with the Yoneda embedding can be seen as a generalisation of the cogenerating set condition. Using this generalisation it is possible to present a convenient simplified version of the sufficient conditions above for the case of an adjunction from the category of presheaves on sets into a cocomplete category, satisfying the faithfulness of the abovementioned composite. Then, the same is done for induced sub-reflections from categories of models of (limit) sketches; in particular this explains why the monotone-light factorisation for categories via preordered sets is just the restriction of the same factorisation for simplicial sets via ordered simplicial complexes.     

Connected and Disconnected Maps

A new relation between morphisms in a category is introduced—roughly speaking (accurately in the categories Set and Top), f ∥ g iff morphisms w:dom(f)→dom(g) never map subobjects of fibres of f non-constantly to fibres of g. (In the algebraic setting replace fibre with kernel.) This relation and a slight weakening of it are used to define “connectedness” versus “disconnectedness” for morphisms. This parallels and generalises the classical treatment of connectedness versus disconnectedness for objects in a category (in terms of constant morphisms). The central items of study are pairs (F,G)({\mathcal F},{\mathcal G}) of classes of morphisms which are corresponding fixed points of the polarity induced by the ∥-relation. Properties of such pairs are examined and in particular their relation to (pre)factorisation systems is analysed. The main theorems characterise:

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, ?ech 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.     

广义直觉模糊集的范畴

In this paper, category GIFS of generalized intuitionistic fuzzy sets(GIF) is built up. Topoi properties of category GIFS axe studied. Firstly， it is proved that the category GIFS has all topoi properties except that it has no subobject classifiers， Secondly， it is proved that the category GIFS has middle object and consequently GIFS is a weak topos.Thirdly， by the use of theory of weak topos GIFS，the power object of an object in GIFS is studied.     

The Eckmann-Hilton argument and higher operads

The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its Hom-set is a commutative monoid. A similar argument due to A. Joyal and R. Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category.In this paper we begin to investigate how one can extend this argument to arbitrary dimension. We provide a simple categorical scheme which allows us to formalise the Eckmann-Hilton type argument in terms of the calculation of left Kan extensions in an appropriate 2-category. Then we apply this scheme to the case of n-operads in the author's sense and classical symmetric operads. We demonstrate that there exists a functor of symmetrisation Symn from a certain subcategory of n-operads to the category of symmetric operads such that the category of one object, one arrow, … , one (n−1)-arrow algebras of A is isomorphic to the category of algebras of Symn(A). Under some mild conditions, we present an explicit formula for Symn(A) which involves taking the colimit over a remarkable categorical symmetric operad.We will consider some applications of the methods developed to the theory of n-fold loop spaces in the second paper of this series.     

Cohomology of categories and extensions of twisted diagrams of groups

A twisted diagram of groups assigns a group to every object of an indexing category and a homomorphism of groups to every morphism. However, it does not have to be completely functorial — it preserves composition only up to a compatible family of inner automorphisms. A. Haefliger defined a special case: the complex of groups. We prove that there exists a natural bijective correspondence between equivalence classes of epimorphisms of twisted diagrams of groups and elements of the second cohomology group of a certain small category. If this category is defined by a discrete group, then we obtain the well known classification of extensions of groups.     

Quillen's small object argument in the category of firm modules

Firm modules over a nonunital ring are introduced by Quillen to study properties of Morita invariance extending thus the case of unitary rings. In the present paper we introduce new homology groups in the category of firm modules which are preserved under Morita equivalences in very general terms. We see that there are long exact sequences of homology attached to certain morphisms of firm modules. Our arguments are based in Quillen's small object argument.     

Transfinite Hausdorff dimension

Making extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorphisms), in fact like Hausdorff dimension, it does not increase under Lipschitz maps, and it also satisfies the intermediate dimension property (Theorem 2.7). The primary goal of transfinite Hausdorff dimension is to classify metric spaces with infinite Hausdorff dimension. Indeed, if tHD(X)?ω₀, then HD(X)=+∞. We prove that tHD(X)?ω₁ for every separable metric space X, and, as our main theorem, we show that for every ordinal number α<ω₁ there exists a compact metric space Xα (a subspace of the Hilbert space l₂) with tHD(Xα)=α and which is a topological Cantor set, thus of topological dimension 0. In our proof we develop a metric version of Smirnov topological spaces and we establish several properties of transfinite Hausdorff dimension, including its relations with classical Hausdorff dimension.     

An orthogonal approach to algebraic weak factorisation systems

We describe an equivalent formulation of algebraic weak factorisation systems, not involving monads and comonads, but involving double categories of morphisms equipped with a lifting operation satisfying lifting and factorisation axioms.     

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

We show that in a locally -presentable category, every -injectivity class (i.e., the class of all the objects injective with respect to some class of -presentable morphisms) is a weakly reflective subcategory determined by a functorial weak factorization system cofibrantly generated by a class of -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 -filtered colimits play the role of the transfinite compositions in the injectivity case. -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 -presentable categories are shown to be cellularly generated by the set of morphisms between -presentable objects.     

Almost periodic solutions of neutral impulsive systems with periodic time-dependent perturbed delays

A neutral impulsive system with a small delay of the argument of the derivative and another delay which differs from a constant by a periodic perturbation of a small amplitude is considered. If the corresponding system with constant delay has an isolated ω-periodic solution and the period of the delay is not rationally dependent on ω, then under a nondegeneracy assumption it is proved that in any sufficiently small neighbourhood of this orbit the perturbed system has a unique almost periodic solution.     

Fate, freedom and contingency

Argument for fatalism attempts to prove that free choice is a logical or conceptual impossibility. The paper argues that the first two premises of the argument are sound: propositions are either true or false and they have their truth-value eternally. But the claim that from the fatalistic premises with the introduction of some innocent further premise dire consequences follow as regards to the possibility of free choice is false. The introduced premise, which establishes the connection between the first two premises (which are about the nature of propositions) and the concept of free choice is not innocent. It creates the impression that the truth of certain propositions can somehow determine the occurrence of certain events. But no proposition can have such an effect since the counterfactuals “If proposition P were true, event E would happen” does not say anything about determination. The argument for fatalism is, however, not a boring sophism. It does reveal something about the nature of propositional representation. It shows that each proposition represents necessarily the fact what it represents, i.e. it shows that propositions have their truth conditions non-contingently. But from this nothing follows as regards to the contingent nature of the facts represented. On the bases of the first two premises of the argument for fatalism we cannot infer to the impossibility of free choice. The argument for fatalism should not be interpreted as an attempt to prove on purely logical or conceptual grounds that we do not have the ability to influence future events by our choices. But it could be used to show something about the nature of propositional representation.     

Reply to Nathan: How to reconstruct the causal argument

Nicholas Nathan tries to resist the current version of the causal argument for sense-data in two ways. First he suggests that, on what he considers to be the correct re-construction of the argument, it equivocates on the sense of proximate cause. Second he defends a form of disjunctivism, by claiming that there might be an extra mechanism involved in producing veridical hallucination, that is not present in perception. I argue that Nathan’s reconstruction of the argument is not the appropriate one, and that, properly interpreted, the argument does not equivocate on proximate cause. Furthermore, I claim that his postulation of a modified mechanism for hallucinations is implausibly ad hoc.     

The Hilton-Heckmann argument for the anti-commutativity of cup products

We present a simple extension of the classical Hilton-Eckmann argument which proves that the endomorphism monoid of the unit object in a monoidal category is commutative. It allows us to recover in a uniform way well-known results on the graded-commutativity of cup products defined on the cohomology theories attached to various algebraic structures, as well as some more recent results.

     

Intelligibility and intensionality

A common argumentative strategy employed by anti-reductionists involves claiming that one kind of entity cannot be identified with or reduced to a second because what can intelligibly be predicated of one cannot be predicated intelligibly of the other. For instance, it might be argued that mind and brain are not identical because it makes sense to say that minds are rational but it does not make sense to say that brains are rational. The scope and power of this kind of argument — if valid — are obvious; but if it turns out that ‘It makes sense to say that...’ creates an opaque context, such arguments will fail. I analyze a possible counterexample to validity and show that it is not conclusive, as it depends on what syntactical construction is given to the premises. This leads to the general observation that the argument form under consideration works for some constructions but not others, and thus to the conclusion that further analysis of intelligibility is called for before it can be known whether the argumentative strategy is open to the anti-reductionist or not.     

On exact categories and applications to triangulated adjoints and model structures

We show that Quillen?s small object argument works for exact categories under very mild conditions. This has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures. In particular, the interplay of different exact structures on the category of complexes of quasi-coherent sheaves leads to a streamlined and generalized version of recent results obtained by Estrada, Gillespie, Guil Asensio, Hovey, Jørgensen, Neeman, Murfet, Prest, Trlifaj and possibly others.