Operad theory

Operad theory

Operad theory is a field of abstract algebra concerned with prototypical algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity. Operads generalize the various associativity properties already observed in algebras and coalgebras such as Lie algebras or Poisson algebras by modeling computational trees within the algebra. Algebras are to operads as group representations are to groups. Originating from work in algebraic topology by Boardman and Vogt, and J. Peter May (to whom their name is due), it has more recently found many applications, drawing for example on work by Maxim Kontsevich on graph homology.

An operad can be seen as a set of operations, each one having a fixed finite number of inputs (arguments) and one output, which can be composed one with others; it is a category-theoretic analog of universal algebra.

Contents

Definition

In category theory, an operad without permutations (sometimes called a non-symmetric, non-Σ or plain operad) is a multicategory with one object. More explicitly, such an operad consists of the following:

  • a sequence (P(n))_{n\in\mathbb{N}} of sets, whose elements are called n-ary operations,
  • for each positive integers n, k1, ..., kn a function

\begin{matrix}
P(n)\times P(k_1)\times\cdots\times P(k_n)&\to&P(k_1+\cdots+k_n)\\
(\theta,\theta_1,\ldots,\theta_n)&\mapsto&\theta\circ(\theta_1,\ldots,\theta_n)
\end{matrix}
called composition,
  • an element 1 in P(1) called the identity,

satisfying the following coherence properties:

  • associativity:

\theta\circ(\theta_1\circ(\theta_{1,1},\ldots,\theta_{1,k_1}),\ldots,\theta_n\circ(\theta_{n,1},\ldots,\theta_{n,k_n}))
=
(\theta\circ(\theta_1,\ldots,\theta_n))\circ(\theta_{1,1},\ldots,\theta_{1,k_1},\ldots,\theta_{n,1},\ldots,\theta_{n,k_n})
  • identity:
\theta\circ(1,\ldots,1)=\theta=1\circ\theta

(where the number of arguments correspond to the arities of the operations).

A morphism of operads f:P\to Q consists of a sequence

(f_n:P(n)\to Q(n))_{n\in\mathbb{N}}

which:

  • preserves composition: for every n-ary operation θ and operations θ1, ..., θn,

f(\theta\circ(\theta_1,\ldots,\theta_n))
=
f(\theta)\circ(f(\theta_1),\ldots,f(\theta_n))
  • preserves identity:
f(1) = 1.

Operads were originally defined topologically, by May, but his full definition requires symmetric group actions on the P(n) that are suitably related to the maps θn. The permutation actions are additional structure that is vital to the original and most later applications.

Axiom of associativity

"Associativity" means that composition of operations is associative (the function \circ is associative), analogous to the axiom in category theory that f \circ (g \circ h) = (f \circ g) \circ h; it does not mean that the operations themselves are associative as operations. Compare with the associative operad, below.

Associativity in operad theory means that one can write expressions involving operations without ambiguity from the omitted compositions, just as associativity for operations allows one to write products without ambiguity from the omitted parentheses.

For instance, suppose that θ is a binary operation, which is written as θ(a,b) or (ab). Note that θ may or may not be associative.

Then what is commonly written ((ab)c) is unambiguously written operadically as \theta \circ (\theta,1): (θ,1) sends (a,b,c) to (ab,c) (apply θ on the first two, and the identity on the third), and then the θ on the left "multiplies" ab by c. This is clearer when depicted as a tree:

Tree before composition

which yields a 3-ary operation:

Tree after composition

However, the expression (((ab)c)d) is a priori ambiguous: it could mean \theta \circ ((\theta,1) \circ ((\theta,1),1)), if the inner compositions are performed first, or it could mean (\theta \circ (\theta,1)) \circ ((\theta,1),1), if the outer compositions are performed first (operations are read from right to left). Writing x = θ,y = (θ,1),z = ((θ,1),1), this is x \circ (y \circ z) versus (x \circ y) \circ z. That is, the tree is missing "vertical parentheses":

Tree before composition

If the top two rows of operations are composed first (puts an upward parenthesis at the (ab)c\ \ d line; does the inner composition first), the following results:

Intermediate tree

which then evaluates unambiguously to yield a 4-ary operation. As an annotated expression:

\theta_{(ab)c\cdot d} \circ ((\theta_{ab \cdot c},1_d) \circ ((\theta_{a\cdot b},1_c),1_d))

Tree after composition

If the bottom two rows of operations are composed first (puts a downward parenthesis at the ab\quad c\ \ d line; does the outer composition first), following results:

Intermediate tree

which then evaluates unambiguously to yield a 4-ary operation:

Tree after composition

The operad axiom of associativity is that these yield the same result, and thus that the expression (((ab)c)d) is unambiguous.

Axiom of identity

The axiom of identity (for a binary operation) can be visualized in a tree as:

The axiom of identity in an operad

meaning that the three operations obtained are equal: pre- or post- composing with the identity makes no difference.

Note that, as for categories, 1 \circ 1 = 1 is a corollary of the axiom of identity.

Examples

Operadic composition in the little 2-discs operad.
Operadic composition in the operad of symmetries.

One class of examples of operads are those capturing the structures of algebraic structures, such as associative algebras, commutative algebras and Lie algebras. Each of these can be exhibited as a finitely presented operad, in each of these three generated by binary operations.

"Little something" operads

A little discs operad or, little balls operad or, more specifically, the little n-discs operad is a topological operad defined in terms of configurations of disjoint n-dimensional discs inside a unit n-disc centered in the origin of Rn. The operadic composition for little 2-discs is illustrated in the figure.[1]

Originally the little n-cubes operad or the little intervals operad (initially called little n-cubes PROPs) was defined by Michael Boardman and Rainer Vogt in a similar way, in terms of configurations of disjoint axis-aligned n-dimensional hypercubes (n-dimensional intervals) inside the unit hypercube.[2] Later it was generalized by May[3] to little convex bodies operad, and "little discs" is a case of "folklore" derived from the "little convex bodies".[4]

Associative operad

Thus, the associative operad is generated by a binary operation ψ, subject to the condition that

\psi\circ(\psi,1)=\psi\circ(1,\psi).

This condition does correspond to associativity of the binary operation ψ; writing ψ(a,b) multiplicatively, the above condition is (ab)c = a(bc). This associativity of the operation should not be confused with associativity of composition; see the axiom of associativity, above.

This operad is terminal in the category of non-symmetric operads, as it has exactly one n-ary operation for each n, corresponding to the unambiguous product of n terms: x_1 \dotsb x_n. For this reason, it is sometimes written as 1 by category theorists (by analogy with the one-point set, which is terminal in the category of sets).

Terminal symmetric operad

The terminal symmetric operad is the operad whose algebras are commutative monoids, which also has one n-ary operation for each n, with each Sn acting trivially; this triviality corresponds to commutativity, and whose n-ary operation is the unambiguous product of n-terms, where order does not matter:

x_1 \dotsb x_n = x_{\sigma(1)} \dotsb x_{\sigma(n)}

for any permutation \sigma \in S_n.

Operads in topology

In many examples the P(n) are not just sets but rather topological spaces. Some names of important examples are the little n-disks, little n-cubes, and linear isometries operads. The idea behind the little n-disks operad comes from homotopy theory, and the idea is that an element of P(n) is an arrangement of n disks within the unit disk. Now, the identity is the unit disk as a subdisk of itself, and composition of arrangements is by scaling the unit disk down into the disk that corresponds to the slot in the composition, and inserting the scaled contents there.

Operads from the symmetric and braid groups

There is an operad for which each P(n) is given by the symmetric group Sn. The composite \sigma \circ (\tau_1, \dots, \tau_n) permutes its inputs in blocks according to σ, and within blocks according to the appropriate τi. Similarly, there is an operad for which each P(n) is given by the Artin braid group Bn.

Linear algebra

In linear algebra, one can consider vector spaces to be algebras over the operad \mathbf{R}^\infty (the infinite direct sum, so only finitely many terms are non-zero; this corresponds to only taking finite sums), which parametrizes linear combinations: the vector (2,3,-5,0,\dots) for instance corresponds to the linear combination

2v_1 + 3v_2 -5v_3 + 0v_4 + \cdots.

Similarly, one can consider affine combinations, conical combinations, and convex combinations to correspond to the sub-operads where the terms sum to 1, the terms are all non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes what is meant by \mathbf{R}^n being or the standard simplex being model spaces, and such observations as that every bounded convex polytope is the image of a simplex. Here suboperads correspond to more restricted operations and thus more general theories.

This point of view formalizes the notion that linear combinations are the most general sort of operation on a vector space – saying that a vector space is an algebra over the operad of linear combinations is precisely the statement that all possible algebraic operations in a vector space are linear combinations. The basic operations of vector addition and scalar multiplication are a generating set for the operad of all linear combinations, while the linear combinations operad canonically encode all possible operations on a vector space.

Origins of the term

The word "operad" was also created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer). Regarding its creation, he wrote: "The name 'operad' is a word that I coined myself, spending a week thinking of nothing else." (http://www.math.uchicago.edu/~may/PAPERS/mayi.pdf Page 2)

See also

Notes

  1. ^ Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily (2005) Geometric and Algebraic Topological Methods in Quantum Mechanics, ISBN 981-256-129-3, pp. 474,475
  2. ^ Axiomatic, Enriched and Motivic Homotopy Theory by J. P. C. Greenlees (2004) ISBN 1-4020-1834-7, pp. 154-156
  3. ^ J. P. May, "Infinite loop space theory", Bull. Amer. Math. Soc. 83 (1977), 456–494.
  4. ^ Jim Stasheff, "Grafting Boardman's Cherry Trees to Quantum Field Theory", 31 March 1998, arXiv:math/9803156v1

References

  • Boardman, J. M.; Vogt, R. M. (1973). Homotopy Invariant Algebraic Structures on Topological Spaces. Lecture Notes in Mathematics. 347. Springer-Verlag. ISBN 3540064796. .

Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • A∞-operad — In the theory of operads in algebra and algebraic topology, an A∞ operad is a parameter space for a multiplication map that is associative up to all higher homotopies, but not necessarily commutative. (An operad that describes a multiplication… …   Wikipedia

  • E∞-operad — In the theory of operads in algebra and algebraic topology, an E∞ operad is a parameter space for a multiplication map that is associative and commutative up to all higher homotopies. (An operad that describes a multiplication that is associative …   Wikipedia

  • Timeline of category theory and related mathematics — This is a timeline of category theory and related mathematics. By related mathematics is meant first hand * Homological algebra * Homotopical algebra * Topology using categories, especially algebraic topology * Categorical logic * Foundations of… …   Wikipedia

  • Universal algebra — (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ( models ) of algebraic structures.For instance, rather than take particular groups as the object of study, in universal… …   Wikipedia

  • List of mathematics articles (O) — NOTOC O O minimal theory O Nan group O(n) Obelus Oberwolfach Prize Object of the mind Object theory Oblate spheroid Oblate spheroidal coordinates Oblique projection Oblique reflection Observability Observability Gramian Observable subgroup… …   Wikipedia

  • Planar algebra — In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II1 subfactor [http://www.math.berkeley.edu/ vfr/plnalg1.ps] . They also provide an appropriate algebraic framework for many knot… …   Wikipedia

  • Multicategory — In mathematics (especially category theory), a multicategory is a generalization of the concept of category that allows morphisms of multiple arity. If morphisms in a category are viewed as analogous to functions, then morphisms in a… …   Wikipedia

  • List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

  • List of mathematics articles (E) — NOTOC E E₇ E (mathematical constant) E function E₈ lattice E₈ manifold E∞ operad E7½ E8 investigation tool Earley parser Early stopping Earnshaw s theorem Earth mover s distance East Journal on Approximations Eastern Arabic numerals Easton s… …   Wikipedia

  • Associahedron — In mathematics, an associahedron or Stasheff polytope K n is a convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a word of n letters and the edges correspond to single application… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”