Brown's representability theorem

Last updated

In mathematics, Brown's representability theorem in homotopy theory [1] gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor.

Contents

More specifically, we are given

F: HotcopSet,

and there are certain obviously necessary conditions for F to be of type Hom(, C), with C a pointed connected CW-complex that can be deduced from category theory alone. The statement of the substantive part of the theorem is that these necessary conditions are then sufficient. For technical reasons, the theorem is often stated for functors to the category of pointed sets; in other words the sets are also given a base point.

Brown representability theorem for CW complexes

The representability theorem for CW complexes, due to Edgar H. Brown, [2] is the following. Suppose that:

  1. The functor F maps coproducts (i.e. wedge sums) in Hotc to products in Set:
  2. The functor F maps homotopy pushouts in Hotc to weak pullbacks. This is often stated as a Mayer–Vietoris axiom: for any CW complex W covered by two subcomplexes U and V, and any elements uF(U), vF(V) such that u and v restrict to the same element of F(UV), there is an element wF(W) restricting to u and v, respectively.

Then F is representable by some CW complex C, that is to say there is an isomorphism

F(Z) ≅ HomHotc(Z, C)

for any CW complex Z, which is natural in Z in that for any morphism from Z to another CW complex Y the induced maps F(Y) → F(Z) and HomHot(Y, C) → HomHot(Z, C) are compatible with these isomorphisms.

The converse statement also holds: any functor represented by a CW complex satisfies the above two properties. This direction is an immediate consequence of basic category theory, so the deeper and more interesting part of the equivalence is the other implication.

The representing object C above can be shown to depend functorially on F: any natural transformation from F to another functor satisfying the conditions of the theorem necessarily induces a map of the representing objects. This is a consequence of Yoneda's lemma.

Taking F(X) to be the singular cohomology group Hi(X,A) with coefficients in a given abelian group A, for fixed i > 0; then the representing space for F is the Eilenberg–MacLane space K(A, i). This gives a means of showing the existence of Eilenberg-MacLane spaces.

Variants

Since the homotopy category of CW-complexes is equivalent to the localization of the category of all topological spaces at the weak homotopy equivalences, the theorem can equivalently be stated for functors on a category defined in this way.

However, the theorem is false without the restriction to connected pointed spaces, and an analogous statement for unpointed spaces is also false. [3]

A similar statement does, however, hold for spectra instead of CW complexes. Brown also proved a general categorical version of the representability theorem, [4] which includes both the version for pointed connected CW complexes and the version for spectra.

A version of the representability theorem in the case of triangulated categories is due to Amnon Neeman. [5] Together with the preceding remark, it gives a criterion for a (covariant) functor F: CD between triangulated categories satisfying certain technical conditions to have a right adjoint functor. Namely, if C and D are triangulated categories with C compactly generated and F a triangulated functor commuting with arbitrary direct sums, then F is a left adjoint. Neeman has applied this to proving the Grothendieck duality theorem in algebraic geometry.

Jacob Lurie has proved a version of the Brown representability theorem [6] for the homotopy category of a pointed quasicategory with a compact set of generators which are cogroup objects in the homotopy category. For instance, this applies to the homotopy category of pointed connected CW complexes, as well as to the unbounded derived category of a Grothendieck abelian category (in view of Lurie's higher-categorical refinement of the derived category).

Related Research Articles

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In algebraic geometry, motives is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety.

In topology, a branch of mathematics, the smash product of two pointed spaces and is the quotient of the product space X × Y under the identifications (xy0) ∼ (x0y) for all x in X and y in Y. The smash product is itself a pointed space, with basepoint being the equivalence class of. The smash product is usually denoted X ∧ Y or X ⨳ Y. The smash product depends on the choice of basepoints.

In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext1 classifies extensions of one module by another.

In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.

In mathematics, the derived categoryD(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described by complicated spectral sequences.

In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology.

In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different categories, as discussed below.

This is a glossary of properties and concepts in category theory in mathematics.

In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes. The concept was introduced by Daniel G. Quillen (1967).

The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.

In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank Adams (1958) which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre.

In mathematics, especially homological algebra, a differential graded category, often shortened to dg-category or DG category, is a category whose morphism sets are endowed with the additional structure of a differential graded -module.

In mathematics, derivators are a proposed new frameworkpg 190-195 for homological algebra giving a framework for non-abelian homological algebra and various generalisations of it. They were introduced to address the deficiencies of derived categories and provide at the same time a language for homotopical algebra.

In algebra, Schlessinger's theorem is a theorem in deformation theory introduced by Schlessinger (1968) that gives conditions for a functor of artinian local rings to be pro-representable, refining an earlier theorem of Grothendieck.

In category theory, a branch of mathematics, a stable ∞-category is an ∞-category such that

This is a glossary of properties and concepts in algebraic topology in mathematics.

In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a topological space. In terms of category theory, the fundamental groupoid is a certain functor from the category of topological spaces to the category of groupoids.

[...] people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation, which thus get lost on the way. In certain situations (such as descent theorems for fundamental groups à la Van Kampen Theorem it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points, [,,,]

In mathematics, compact objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness condition.

In mathematics, homotopy theory is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry and category theory.

References

  1. Switzer, Robert M. (2002), Algebraic topology---homotopy and homology, Classics in Mathematics, Berlin, New York: Springer-Verlag, pp. 152–157, ISBN   978-3-540-42750-6, MR   1886843
  2. Brown, Edgar H. (1962), "Cohomology theories", Annals of Mathematics , Second Series, 75: 467–484, doi:10.2307/1970209, ISSN   0003-486X, JSTOR   1970209, MR   0138104
  3. Freyd, Peter; Heller, Alex (1993), "Splitting homotopy idempotents. II.", Journal of Pure and Applied Algebra , 89 (1–2): 93–106, doi: 10.1016/0022-4049(93)90088-b
  4. Brown, Edgar H. (1965), "Abstract homotopy theory", Transactions of the American Mathematical Society , 119 (1): 79–85, doi: 10.2307/1994231
  5. Neeman, Amnon (1996), "The Grothendieck duality theorem via Bousfield's techniques and Brown representability", Journal of the American Mathematical Society , 9 (1): 205–236, doi: 10.1090/S0894-0347-96-00174-9 , ISSN   0894-0347, MR   1308405
  6. Lurie, Jacob (2011), Higher Algebra (PDF), archived from the original (PDF) on 2011-06-09