categorical algebra

0.1 Introduction: An Outline of Categorical Algebra

This topic entry provides an outline of an important mathematical field called categorical algebra; although specific definitions are in use for various applications of categorical algebra to specific algebraic structures, they do not cover the entire field. In the most general sense, categorical algebras– as introduced by Mac Lane in 1965 – can be described as the study of representations of algebraic structures, either concrete or abstract, in terms of categories, functors and natural transformations.

In a narrow sense, a categorical algebra is an associative algebra, defined for any locally finite category and a commutative ring with unity. This notion may be considered as a generalization of both the concept of group algebra and that of an incidence algebra, much as the concept of category generalizes the notions of group and partially ordered set.

0.2 Extensions of categorical algebra

• Thus, ultimately, since categories are interpretations of the axiomatic elementary theory of abstract categories (ETAC), so are categorical algebras.

  The general definition of representation introduced above can be still further extended by introducing supercategorical algebras as interpretations of ETAS, as explained next.

• Mac Lane (1976) wrote in his Bull. AMS review cited here as a verbatim quotation:

  “On some occasions I have been tempted to try to define what algebra is,can, or should be - most recently in concluding a survey [72] on Recentadvances in algebra. But no such formal definitions hold valid for long, sincealgebra and its various subfields steadily change under the influence of ideasand problems coming not just from logic and geometry, but from analysis,other parts of mathematics, and extra mathematical sources. The progress ofmathematics does indeed depend on many interlocking, unexpected andmultiform developments.”

0.3 Basic definitions

An algebraic representation is generally defined as a morphism $\rho$ from an abstract algebraic structure ${\mathrm{A}}_S$ to a concrete algebraic structure $A_c$, a Hilbert space $\mathscr{H}$, or a family of linear operator spaces.

The key notion of representable functor (http://planetmath.org/RepresentableFunctor) was first reported by Alexander Grothendieck in 1960.

Definition 0.1.

Thus, when the latter concept is extended to categorical algebra, one has a representable functor $S$ from an arbitrary category $𝒞$ to the category of sets $Set$ if $S$ admits a functor representation defined as follows. A functor representation of $S$ is defined as a pair, $(R,\varphi )$, which consists of an object $R$ of $𝒞$ and a family $\varphi$ of equivalences $\varphi (C):{\mathrm{Hom}}_{𝒞}(R,C)\cong S(C)$, which is natural in C, with C being any object in $𝒞$. When the functor $S$ has such a representation, it is also said to be represented by the object $R$ of $𝒞$. For each object $R$ of $𝐂$one writes $h_R:𝒞⟶Set$ for the covariant $\mathrm{Hom}$–functor $h_R(C)\cong {\mathrm{Hom}}_{𝒞}(R,C)$. A representation $(R,\varphi )$ of $S$ is therefore a natural equivalence of functors:

$$\varphi :h_R\cong S.$$(0.1)

Remark 0.1.

The equivalence classes of such functor representations (defined as natural equivalences) determine directly an algebraic (groupoid) structure.

0.4 Note:

See also in Expositions the entry about abstract and concrete algebras.

0.5 Application: Quantum Categories