Mackey Functor
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, particularly in
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
and
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, a Mackey functor is a type of
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
that generalizes various constructions in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
and equivariant homotopy theory. Named after American
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
George Mackey George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry. Career Mackey earned his B.A. at Rice University in 1938 ...
, these functors were first introduced by
German German(s) may refer to: * Germany, the country of the Germans and German things **Germania (Roman era) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizenship in Germany, see also Ge ...
mathematician
Andreas Dress Andreas Dress (26 August 1938 – 23 February 2024) was a German mathematician specialising in geometry, combinatorics and mathematical biology. Biography Dress earned his PhD from the University of Kiel in 1962, under the supervision of Fried ...
in 1971.Dress, A. W. M. (1971). "Notes on the theory of representations of finite groups. Part I: The Burnside ring of a finite group and some AGN-applications". Bielefeld.


Definition


Classical definition

Let G be a
finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
. A Mackey functor M for G consists of: * For each
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
H \leq G, an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
M(H), * For each pair of subgroups H, K \leq G with H \subseteq K: ** A restriction homomorphism R^K_H: M(K) \to M(H), ** A transfer homomorphism I^K_H: M(H) \to M(K). These maps must satisfy the following axioms: :Functoriality: For nested subgroups H \subseteq K \subseteq L, R^L_H = R^K_H \circ R^L_K and I^L_H = I^L_K \circ I^K_H. :Conjugation: For any g \in G and H \leq G, there are
isomorphisms In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them ...
c_g: M(H) \to M(gHg^) compatible with restriction and transfer. :
Double coset In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups, generalizing the notion of a single coset. Definition Let be a group, and let and b ...
formula: For subgroups H, K \leq G, the following identity holds: ::R^G_H I^G_K = \sum_ I^H_ \circ c_x \circ R^K_.


Modern definition

In modern
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, a Mackey functor can be defined more elegantly using the language of spans. Let \mathcal be a disjunctive (\infty, 1)-category and \mathcal be an additive (\infty, 1)-category ((\infty, 1)-categories are also known as quasi-categories). A Mackey functor is a product-preserving functor M: \text(\mathcal) \to \mathcal where \text(\mathcal) is the (\infty, 1)-category of correspondences in \mathcal.Barwick, C. (2017). "Spectral Mackey functors and equivariant algebraic K-theory (I)". ''Advances in Mathematics'', 304:646–727.


Applications


In equivariant homotopy theory

Mackey functors play an important role in
equivariant stable homotopy theory In mathematics, more specifically in topology, the equivariant stable homotopy theory is a subfield of equivariant topology that studies a spectrum with group action In mathematics, a group action of a group G on a set S is a group homomorph ...
. For a genuine G-spectrum E, its equivariant homotopy groups form a Mackey functor given by: :\pi_n(E): G/H \mapsto /H_+ \wedge S^n, XG where ,-G denotes
morphisms In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
in the equivariant stable homotopy category.May, J. P. (1996). "Equivariant homotopy and cohomology theory". ''CBMS Regional Conference Series in Mathematics'', vol. 91.


Cohomology with Mackey functor coefficients

For a pointed G-CW complex X and a Mackey functor A, one can define
equivariant In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
with coefficients in A as: :H^n_G(X,A) := H^n(\text(C_\bullet(X), A)) where C_\bullet(X) is the
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel o ...
of Mackey functors given by stable equivariant
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
s of quotient spaces.Kronholm, W. (2010). "The RO(G)-graded Serre spectral sequence". ''Homology, Homotopy and Applications'', 12(1):75-92.


References


Further reading

*Dieck, T. (1987). ''Transformation Groups''. de Gruyter. {{ISBN, 978-3110858372 *Webb, P. "A Guide to Mackey Functors" *Bouc, S. (1997). "Green Functors and G-sets". ''Lecture Notes in Mathematics'' 1671. Springer-Verlag. Representation theory Algebraic topology Functors Homological algebra