category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the categorical trace is a generalization of the

trace Trace may refer to: Arts and entertainment Music * Trace (Son Volt album), ''Trace'' (Son Volt album), 1995 * Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * The Trace (album), ''The ...

of a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

Definition

The trace is defined in the context of a

symmetric monoidal category In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sen ...

''C'', i.e., a

category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...

equipped with a suitable notion of a product

\otimes

. (The notation reflects that the product is, in many cases, a kind of a

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...

.) An

object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...

''X'' in such a category ''C'' is called dualizable if there is another object

X^\vee

playing the role of a dual object of ''X''. In this situation, the trace of a

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

f: X \to X

is defined as the composition of the following morphisms:

\mathrm(f) : 1 \ \stackrel\ X \otimes X^\vee \ \stackrel\ X \otimes X^\vee \ \stackrel\ X^\vee \otimes X \ \stackrel\ 1

where 1 is the monoidal unit and the extremal morphisms are the coevaluation and evaluation, which are part of the definition of dualizable objects. The same definition applies, to great effect, also when ''C'' is a symmetric monoidal ∞-category.

Examples

* If ''C'' is the

category of vector spaces In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring o ...

over a fixed

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''k'', the dualizable objects are precisely the

finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

s, and the trace in the sense above is the morphism ::

k \to k

:which is the multiplication by the trace of the

endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...

''f'' in the usual sense of

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

. * If ''C'' is the ∞-category of

chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...

es of

modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...

(over a fixed

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

''R''), dualizable objects ''V'' in ''C'' are precisely the

perfect complex In algebra, a perfect complex of modules over a commutative ring ''A'' is an object in the derived category of ''A''-modules that is quasi-isomorphic to a bounded complex of finite projective ''A''-modules. A perfect module is a module that is perfe ...

es. The trace in this setting captures, for example, the

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...

, which is the alternating sum of the ranks of its terms: ::

\mathrm(\operatorname_V) = \sum_i (-1)^i \operatorname  V_i.

Further applications

have used categorical trace methods to prove an algebro-geometric version of the Atiyah–Bott fixed point formula, an extension of the

Lefschetz fixed point formula In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named ...

References

* * {{Citation, first1 = Kate , last1 = Ponto , first2 = Michael , last2 = Shulman, title = Traces in symmetric monoidal categories , journal = Expositiones Mathematicae , volume = 32 , issue = 3 , year = 2014 , pages = 248–273 , doi = 10.1016/j.exmath.2013.12.003 , arxiv = 1107.6032, bibcode = 2011arXiv1107.6032P , s2cid = 119129371 Category theory Fixed-point theorems Geometry