An “almost” full embedding of the category of graphs into the category of groups

We construct a functor F:Graphs→Groups which is faithful and “almost” full, in the sense that every nontrivial group homomorphism FX→FY is a composition of an inner automorphism of FY and a homomorphism of the form Ff, for a unique map of graphs f:X→Y. When F is composed with the Eilenberg-Mac Lane space construction K(FX,1) we obtain an embedding of the category of graphs into the unpointed homotopy category which is full up to null-homotopic maps.We provide several applications of this construction to localizations (i.e. idempotent functors); we show that the questions:

(1)

Is every orthogonality class reflective?

(2)

Is every orthogonality class a small-orthogonality class?

have the same answers in the category of groups as in the category of graphs. In other words they depend on set theory: (1) is equivalent to weak Vopěnka's principle and (2) to Vopěnka's principle. Additionally, the second question, considered in the homotopy category, is also equivalent to Vopěnka's principle.     

Trunks and classifying spaces

Trunks are objects loosely analogous to categories. Like a category, a trunk has vertices and edges (analogous to objects and morphisms), but instead of composition (which can be regarded as given by preferred triangles of morphisms) it has preferred squares of edges. A trunk has a natural cubical nerve, analogous to the simplicial nerve of a category. The classifying space of the trunk is the realisation of this nerve. Trunks are important in the theory of racks [8]. A rackX gives rise to a trunkT (X) which has a single vertex and the setX as set of edges. Therack space BX ofX is the realisation of the nerveNT (X) ofT(X). The connection between the nerve of a trunk and the usual (cubical) nerve of a category determines in particular a natural mapBX BAs(X) whereBAs(X) is the classifying space of the associated group ofX. There is an extension to give a classifying space for an augmented rack, which has a natural map to the loop space of the Brown-Higgins classifying space of the associated crossed module [8, Section 2] and [3].The theory can be used to define invariants of knots and links since any invariant of the rack space of the fundamental rack of a knot or link is ipso facto an invariant of the knot or link.     

Bernays-Gödel type theory

We study the type-theoretical analogue of Bernays-Gödel set-theory and its models in categories. We introduce the notion of small structure on a category, and if small structure satisfies certain axioms we can think of the underlying category as a category of classes. Our axioms imply the existence of a co-variant powerset monad on the underlying category of classes, which sends a class to the class of its small subclasses. Simple fixed points of this and related monads are shown to be models of intuitionistic Zermelo-Fraenkel set-theory (IZF).     

The global decomposition theorem for Hochschild (co-)homology of singular spaces via the Atiyah-Chern character

We generalize the decomposition theorem of Hochschild, Kostant and Rosenberg for Hochschild (co-)homology to arbitrary morphisms between complex spaces or schemes over a field of characteristic zero. To be precise, we show that for each such morphism X→Y, the Hochschild complex HX/Y, as introduced in [R.-O. Buchweitz, H. Flenner, Global Hochschild (co-)homology of singular spaces, Adv. Math. (2007), doi: 10.1016/j.aim.2007.06.012], decomposes naturally in the derived category D(X) into _⊕p?0Sp(LX/Y[1]), the direct sum of the derived symmetric powers of the shifted cotangent complex, a result due to Quillen in the affine case.Even in the affine case, our proof is new and provides further information. It shows that the decomposition is given explicitly and naturally by the universal Atiyah-Chern character, the exponential of the universal Atiyah class.We further use the decomposition theorem to show that the semiregularity map for perfect complexes factors through Hochschild homology and, in turn, factors the Atiyah-Hochschild character through the characteristic homomorphism from Hochschild cohomology to the graded centre of the derived category.     

Monomorphisms and epimorphisms in pro-categories

A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. We give characterizations of monomorphisms (respectively, epimorphisms) in pro-category pro-C, provided C has direct sums (respectively, pushouts).Let E(C) (respectively, M(C)) be the subcategory of C whose morphisms are epimorphisms (respectively, monomorphisms) of C. We give conditions in some categories C for an object X of pro-C to be isomorphic to an object of pro-E(C) (respectively, pro-M(C)).A related class of objects of pro-C consists of X such that there is an epimorphism X→P∈Ob(C) (respectively, a monomorphism P∈Ob(C)→X). Characterizing those objects involves conditions analogous (respectively, dual) to the Mittag-Leffler property. One should expect that the object belonging to both classes ought to be stable. It is so in the case of pro-groups. The natural environment to discuss those questions are balanced categories with epimorphic images. The last part of the paper deals with that question in pro-homotopy.     

Implications of large-cardinal principles in homotopical localization

The existence of arbitrary cohomological localizations on the homotopy category of spaces has remained unproved since Bousfield settled the same problem for homology theories in the decade of 1970. This is related with another open question, namely whether or not every homotopy idempotent functor on spaces is an f-localization for some map f. We prove that both questions have an affirmative answer assuming the validity of a suitable large-cardinal axiom from set theory (Vopěnka's principle). We also show that it is impossible to prove that all homotopy idempotent functors are f-localizations using the ordinary ZFC axioms of set theory (Zermelo-Fraenkel axioms with the axiom of choice), since a counterexample can be displayed under the assumption that all cardinals are nonmeasurable, which is consistent with ZFC.     

Internal sizes in μ-abstract elementary classes

Working in the context of μ-abstract elementary classes (μ-AECs)—or, equivalently, accessible categories with all morphisms monomorphisms—we examine the two natural notions of size that occur, namely cardinality of underlying sets and internal size. The latter, purely category-theoretic, notion generalizes e.g. density character in complete metric spaces and cardinality of orthogonal bases in Hilbert spaces. We consider the relationship between these notions under mild set-theoretic hypotheses, including weakenings of the singular cardinal hypothesis. We also establish preliminary results on the existence and categoricity spectra of μ-AECs, including specific examples showing dramatic failures of the eventual categoricity conjecture (with categoricity defined using cardinality) in μ-AECs.     

A note on “Extensional PERs”

In the paper “Extensional PERs” by P. Freyd, P. Mulry, G. Rosolini and D. Scott, a category C of “pointed complete extensional PERs” and computable maps is introduced to provide an instance of an algebraically compact category relative to a restricted class of functors. Algebraic compactness is a synthetic condition on a category which ensures solutions of recursive equations involving endofunctors of the category. We extend that result to include all internal functors on C when C is viewed as a full internal category of the effective topos. This is done using two general results: one about internal functors in general, and one about internal functors in the effective topos.     

Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces

Let K(n) be the nth Morava K-theory at a prime p, and let T(n) be the telescope of a v_n-self map of a finite complex of type n. In this paper we study the K(n)_*-homology of Ω^∞X, the 0th space of a spectrum X, and many related matters.We give a sampling of our results.Let PX be the free commutative S-algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map

s_n(X):L_T(n)P(X)→L_T(n)Σ^∞(Ω^∞X)₊     

Quillen-Barr-Beck (co-) homology for restricted Lie algebras

In this paper we use Quillen-Barr-Beck's theory of (co-) homology of algebras in order to define (co-) homology for the category RLie of restricted Lie algebras over a field k of characteristic p≠0. In contrast with the cases of groups, associative algebras and Lie algebras we do not obtain Hochschild (co-) homology shifted by 1.Precisely, we determine for L∈RLie the category of Beck L-modules and the group of Beck derivations of g∈RLie/L to a Beck L-module M. Moreover, we prove a classification theorem which gives a one-to-one correspondence between the one cohomology and the set of equivalent classes of p-extensions. Finally, a universal coefficient theorem is proved, relating the homology to the Hochschild homology via a short exact sequence. This shows that the new homology determines the Hochschild homology.     

Precosheaves of pro-sets and abelian pro-groups are smooth

Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site X, there exists a reflector from the category of precosheaves on X with values in D to the full subcategory of cosheaves. In the case of precosheaves on topological spaces, it is proved that any precosheaf is smooth, i.e. is locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.     

A counter-example to a conjecture of Félix

If X is a simply connected space of finite type, then the rational homotopy groups of the based loop space of X possess the structure of a graded Lie algebra, denoted L_X. The radical of L_X, which is an important rational homotopy invariant of X, is of finite total dimension if the Lusternik-Schnirelmann category of X is finite.Let X be a simply connected space with finite Lusternik-Schnirelmann category. If dimL_X<∞, i.e., if X is elliptic, then L_X is its own radical, and therefore the total dimension of the radical of L_X in odd degrees is less than or equal to its total dimension in even degrees (Friedlander and Halperin (1979) [8]). Félix conjectured that this inequality should hold for all simply connected spaces with finite Lusternik-Schnirelmann category.We prove Félix’s conjecture in some interesting special cases, then provide a counter-example to the general case.     

A folk model structure on omega-cat

The primary aim of this work is an intrinsic homotopy theory of strict ω-categories. We establish a model structure on ωCat, the category of strict ω-categories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generating cofibrations and a class of weak equivalences as basic items. All objects are fibrant while free objects are cofibrant. We further exhibit model structures of this type on n-categories for arbitrary n∈N, as specializations of the ω-categorical one along right adjoints. In particular, known cases for n=1 and n=2 nicely fit into the scheme.     

A remark on K-theory and S-categories

It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (Invent. Math. 150 (2002) 111). The purpose of this note is to show that the K-theory spectrum of a (good) Waldhausen category is completely determined by its Dwyer-Kan simplicial localization, without any additional structure. As the simplicial localization is a refined version of the homotopy category which also determines the triangulated structure, our result is a possible answer to the general question: “To which extent K-theory is not an invariant of triangulated derived categories? ”     

Splitting monoidal stable model categories

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit forms a commutative ring, [S,S]^C. An idempotent e of this ring will split the homotopy category: [X,Y]^C≅e[X,Y]^C⊕(1−e)[X,Y]^C. We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to L_eSC×L_(1−e)SC and [X,Y]^L_eSC≅e[X,Y]^C. This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.     

Lifting homotopy T-algebra maps to strict maps

The settings for homotopical algebra—categories such as simplicial groups, simplicial rings, _A∞$A_{\infty }$ spaces, _E∞$E_{\infty }$ ring spectra, etc.—are often equivalent to categories of algebras over some monad or triple T. In such cases, T is acting on a nice simplicial model category in such a way that T descends to a monad on the homotopy category and defines a category of homotopy T-algebras. In this setting there is a forgetful functor from the homotopy category of T-algebras to the category of homotopy T-algebras.