Brown representability for space-valued functors

In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every contravariant functor from spaces to spaces which takes coproducts to products up to homotopy, and takes homotopy pushouts to homotopy pullbacks is naturally weakly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets taking homotopy pushouts to homotopy pullbacks is equivalent to the restriction of a representable functor. This theorem may be considered as a contravariant analog of Goodwillie’s classification of linear functors [14].     

Failure of Brown representability in derived categories

Let be a triangulated category with coproducts, the full subcategory of compact objects in . If is the homotopy category of spectra, Adams (Topology 10 (1971) 185–198), proved the following: All homological functors are the restrictions of representable functors on , and all natural transformations are the restrictions of morphisms in . It has been something of a mystery, to what extent this generalises to other triangulated categories. In Neeman (Topology 36 (1997) 619–645), it was proved that Adams’ theorem remains true as long as is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in . A puzzling open problem remained: Is every homological functor the restriction of a representable functor on ? In a recent paper, Beligiannis (Relative homological and purity in triangulated categories, 1999, preprint) made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories of rings, and homological functors which are not restrictions of representables.     

The Grothendieck duality theorem via Bousfield's techniques and Brown representability

Grothendieck proved that if is a proper morphism of nice schemes, then has a right adjoint, which is given as tensor product with the relative canonical bundle. The original proof was by patching local data. Deligne proved the existence of the adjoint by a global argument, and Verdier showed that this global adjoint may be computed locally. In this article we show that the existence of the adjoint is an immediate consequence of Brown's representability theorem. 1It follows almost as immediately, by ``smashing' arguments, that the adjoint is given by tensor product with a dualising complex. Verdier's base change theorem is an easy consequence.

     

Brown representability for triangulated categories with a linear action by a graded ring

Archiv der Mathematik - In this paper we give necessary and sufficient conditions for a functor to be representable in a strongly generated triangulated category which has a linear action by a...     

Ellipsoidal mixed-integer representability

Representability results for mixed-integer linear systems play a fundamental role in optimization since they give geometric characterizations of the feasible sets that can be formulated by mixed-integer linear programming. We consider a natural extension of mixed-integer linear systems obtained by adding just one ellipsoidal inequality. The set of points that can be described, possibly using additional variables, by these systems are called ellipsoidal mixed-integer representable. In this work, we give geometric conditions that characterize ellipsoidal mixed-integer representable sets.     

Finite representability of operators

We introduce operator local supportability as a new type of operator finite representability that generalizes Bellenot finite representability. We prove that local supportability and local representability are mutually independent. New examples of both types of finite representability are given. For instance, for every operator T, we prove that is locally supportable in . We also prove that, given an operator T with range in , T∗ is locally representable in .     

Obstructions to determinantal representability

There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also sufficient. To this end Helton and Vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices. We disprove this conjecture. By relating the question of finding LMI representations to the problem of determining whether a polymatroid is representable over the complex numbers, we find a real zero polynomial such that no power of it admits a determinantal representation. The proof uses recent results of Wagner and Wei on matroids with the half-plane property, and the polymatroids associated to hyperbolic polynomials introduced by Gurvits.     

Uniform reducibility of representability problems for algebraic structures

Given a countable algebraic structure \(\mathfrak{B}\) with no degree we find sufficient conditions for the existence of a countable structure \(\mathfrak{A}\) with the following properties: (1) for every isomorphic copy of \(\mathfrak{A}\) there is an isomorphic copy of \(\mathfrak{A}\) Turing reducible to the former; (2) there is no uniform effective procedure for generating a copy of \(\mathfrak{A}\) given a copy of \(\mathfrak{B}\) even having been enriched with an arbitrary finite tuple of constants.     

Choquet representability of submodular functions

Mathematical Programming - Let $$\Omega $$ be an arbitrary set, equipped with an algebra $${\mathcal {A}}\subseteq 2^{\Omega }$$ and let $$f: B({\mathcal {A}}) \rightarrow {\mathbb {R}}$$ be a...     

The representability number of a chain

For each pair of linear orderings (L,M), the representability number repr_M(L) of L in M is the least ordinal α such that L can be order-embedded into the lexicographic power . The case is relevant to utility theory. The main results in this paper are as follows. (i) If κ is a regular cardinal that is not order-embeddable in M, then repr_M(κ)=κ; as a consequence, for each κω₁. (ii) If M is an uncountable linear ordering with the property that A×_lex2 is not order-embeddable in M for each uncountable AM, then for any ordinal α; in particular, . (iii) If L is either an Aronszajn line or a Souslin line, then .     

Reflexive representability and stable metrics

It is well-known that a topological group can be represented as a group of isometries of a reflexive Banach space if and only if its topology is induced by weakly almost periodic functions [see Megrelishvili (Operator topologies and reflexive representability. Nuclear groups and Lie groups (Madrid 1999). Res. Exp. Math., vol. 24, pp. 197?C208. Heldermann, Lemgo, 2001; Topological transformation groups: selected topics. In: Pearl E (ed) Open Problems in Topology II. Elsevier, Amsterdam 2007), Shtern (Russian J. Math. Phys. 2(1):131?C132, 1994)]. We show that for a metrisable group this is equivalent to the property that its metric is uniformly equivalent to a stable metric in the sense of Krivine and Maurey (Isr J Math 39(4):273?C295, 1981). This result is used to give a partial negative answer to a problem of Megrelishvili.     

Conditions for the representability of analytic functions by series of rational functions

With a functionf(z), analytic in the unit circle, we associate by a specific rule the series \(\sum\nolimits_{n = 1}^\infty {\frac{{A_n }}{{1 - \lambda _n z}},\left| {\lambda _n } \right|< 1} \) . we derive a (necessary and sufficient) condition for the convergence of the series in the unit circle. We derive further conditions under which the series converges to the functionf(z) itself.     

Undecidability of representability for lattice-ordered semigroups and ordered complemented semigroups

We prove that the problems of representing a finite ordered complemented semigroup or finite lattice-ordered semigroup as an algebra of binary relations over a finite set are undecidable. In the case that complementation is taken with respect to a universal relation, this result can be extended to infinite representations of ordered complemented semigroups.