-homotopy theory.
-groupoids and homotopy types
-injective covers, -coherent rings, and -rings
It is known that a ring is left Noetherian if and only if every left -module has an injective (pre)cover. We show that if is a right -coherent ring, then every right -module has an -injective (pre)cover; if is a ring such that every -injective right -module is -pure extending, and if every right -module has an -injective cover, then is right -coherent. As applications of these results, we give some characterizations of -rings, von Neumann regular rings and semisimple rings....
-purity and orthogonality.
-homotopy theory of schemes
A categorical concept of completion of objects
We introduce the concept of firm classes of morphisms as basis for the axiomatic study of completions of objects in arbitrary categories. Results on objects injective with respect to given morphism classes are included. In a finitely well-complete category, firm classes are precisely the coessential first factors of morphism factorization structures.
A category model proof of the cogluing theorem
A characterization of Ext(G,ℤ) assuming (V = L)
We complete the characterization of Ext(G,ℤ) for any torsion-free abelian group G assuming Gödel’s axiom of constructibility plus there is no weakly compact cardinal. In particular, we prove in (V = L) that, for a singular cardinal ν of uncountable cofinality which is less than the first weakly compact cardinal and for every sequence of cardinals satisfying (where Π is the set of all primes), there is a torsion-free abelian group G of size ν such that equals the p-rank of Ext(G,ℤ) for every...
A Chen model for mapping spaces and the surface product
We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are complexes whose differentials are modeled on an arbitrary simplicial set much in the same way the ordinary Hochschild differential is modeled on the circle. We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold. This is an analogue of the loop product in string topology....
A classification of degree functors, I
A classification of degree functors, II
A classification of small homotopy functors from spectra to spectra
We show that every small homotopy functor from spectra to spectra is weakly equivalent to a filtered colimit of representable functors represented in cofibrant spectra. Moreover, we present this classification as a Quillen equivalence of the category of small functors from spectra to spectra equipped with the homotopy model structure and the opposite of the pro-category of spectra with the strict model structure.
A cohomology theory for colored tangles
We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labeled by irreducible representations of . We show that the corresponding colored invariants of tangles can be assembled into invariants of bigger tangles. For the case of knots and links, the corresponding theory is a categorification of the colored Jones polynomial, and provides a tool for efficient computations of the resulting colored invariant of knots and links. Our theory is...
A comparison theorem for simplicial resolutions.
A construction of simplicial objects
We construct a simplicial locally convex algebra, whose weak dual is the standard cosimplicial topological space. The construction is carried out in a purely categorical way, so that it can be used to construct (co)simplicial objects in a variety of categories --- in particular, the standard cosimplicial topological space can be produced.
A convenient category for directed homotopy.
A counterexample to a conjecture of Barr.
A counterexample to a group completion conjecture of J. C. Moore.
A decomposition theorem for comodules
A duality on simplicial complexes.