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Blog notes - Meet the operads

edited February 28 in Azimuth Blog

The general topic here is the operads, starting from the ground up.

Here are the blogs:

As a point of departure and reference, I am using this talk:

Comments

  • 1.
    edited February 27

    1. Definition of operad

    An operad consists of:

    • A set of 'objects'
    • For objects \(X_1,...,X_n,Y\), a set of 'morphisms' \(\phi: X_1,...,X_n \rightarrow Y\)
    • For each object, a designated identity morphism
    • A rule for combining connected "trees" of morphisms into a unique composite morphism
    Comment Source:**1. Definition of operad** An **operad** consists of: * A set of 'objects' * For objects \\(X_1,...,X_n,Y\\), a set of 'morphisms' \\(\phi: X_1,...,X_n \rightarrow Y\\) * For each object, a designated identity morphism * A rule for combining connected "trees" of morphisms into a unique composite morphism
  • 2.
    edited February 27

    Here is an example of what I mean by a connected tree:

    • \(f: A,B \rightarrow X_1\)
    • \(g: C,D \rightarrow X_2\)
    • \(h: X_1,X_2 \rightarrow Y\)

    It's 'connected' in the sense that the first input to \(h\) is fed by the output of \(f\), and the second input of \(h\) is fed by the output of \(g\).

    The composite morphism will map \(A,B,C,D \rightarrow Y\).

    Comment Source:Here is an example of what I mean by a connected tree: * \\(f: A,B \rightarrow X_1\\) * \\(g: C,D \rightarrow X_2\\) * \\(h: X_1,X_2 \rightarrow Y\\) It's 'connected' in the sense that the first input to \\(h\\) is fed by the output of \\(f\\), and the second input of \\(h\\) is fed by the output of \\(g\\). The composite morphism will map \\(A,B,C,D \rightarrow Y\\).
  • 3.

    In the technical construction of an operad, the composition rule is defined just for "two-level" trees such as the one I just wrote. Then, in order to ensure that it can be uniquely extended to a rule for all connected trees, the associativity condition is imposed as a requirement on the composition rule.

    Equivalently, one could start by defining a composition rule for all connected trees. Then, the following associativity constraint would be imposed on the rule. Suppose you had a tree of morphisms \(T\), with child trees \(T_1,...,T_n\). Let \(T'\) be the tree whose root is the same as \(T\), with children \(U_1,...U_n\), where \(U_i\) is the composite of \(T_i\). Then the composite of \(T'\) must equal the composite of \(T\).

    In other words, the composite of a tree must equal composite obtained by first collapsing (composing) each of the child subtrees.

    Comment Source:In the technical construction of an operad, the composition rule is defined just for "two-level" trees such as the one I just wrote. Then, in order to ensure that it can be uniquely extended to a rule for all connected trees, the associativity condition is imposed as a requirement on the composition rule. Equivalently, one could start by defining a composition rule for all connected trees. Then, the following associativity constraint would be imposed on the rule. Suppose you had a tree of morphisms \\(T\\), with child trees \\(T_1,...,T_n\\). Let \\(T'\\) be the tree whose root is the same as \\(T\\), with children \\(U_1,...U_n\\), where \\(U_i\\) is the composite of \\(T_i\\). Then the composite of \\(T'\\) must equal the composite of \\(T\\). In other words, the composite of a tree must equal composite obtained by first collapsing (composing) each of the child subtrees.
  • 4.
    edited February 27

    The identity morphism \(Id_A: A \rightarrow A\) for object A is defined by the requirement that it is a 'no-op' with respect to composition.

    This can be expressed as follows. Suppose \(T\) is a morphism tree, which contains \(Id_A: A \rightarrow A\) as a subtree. Let \(T'\) be the result of 'splicing' \(Id_A\) out of the tree.

    Then composite(\(T\)) = composite(\(T'\)).

    Comment Source:The identity morphism \\(Id_A: A \rightarrow A\\) for object A is defined by the requirement that it is a 'no-op' with respect to composition. This can be expressed as follows. Suppose \\(T\\) is a morphism tree, which contains \\(Id_A: A \rightarrow A\\) as a subtree. Let \\(T'\\) be the result of 'splicing' \\(Id_A\\) out of the tree. Then composite(\\(T\\)) = composite(\\(T'\\)).
  • 5.

    From this is follows that the identity morphism for \(A\) must be unique. For suppose that there were two identity morphisms \(I_1,I_2\) for \(A\). Then consider the linear tree which chains \(I_1\) into \(I_2\). The result of splicing \(I_1\) out of the chain is \(I_2\), and the result of splicing \(I_2\) out of the chain is \(I_1\). By the identity requirement, these must be equal.

    Comment Source:From this is follows that the identity morphism for \\(A\\) must be _unique_. For suppose that there were two identity morphisms \\(I_1,I_2\\) for \\(A\\). Then consider the linear tree which chains \\(I_1\\) into \\(I_2\\). The result of splicing \\(I_1\\) out of the chain is \\(I_2\\), and the result of splicing \\(I_2\\) out of the chain is \\(I_1\\). By the identity requirement, these must be equal.
  • 6.
    edited February 27

    2. The little n-cubes operad

    • See Spivak slide 8, "The first operad"

    We illustrate for \(n=2\), which gives the 'little squares' operad.

    There is just a single object \(\square\), which serves only as a placeholder.

    The entire content of this operad consists of the morphisms from \(\square_1,...,\square_n \rightarrow \square\).

    Note: the subscript on \(\square_i\) is only for counting purposes. They're all the same object.

    Comment Source:**2. The little n-cubes operad** * See Spivak slide 8, "The first operad" We illustrate for \\(n=2\\), which gives the 'little squares' operad. There is just a single object \\(\square\\), which serves only as a placeholder. The entire content of this operad consists of the morphisms from \\(\square_1,...,\square_n \rightarrow \square\\). Note: the subscript on \\(\square_i\\) is only for counting purposes. They're all the same object.
  • 7.
    edited February 27

    The morphisms are geometric arrangements.

    Let SQ denote some fixed 'outer square.' For concreteness, we may take this to be the unit square.

    Then a morphism \(f_k: \square_1,...,\square_k \rightarrow \square\) is an arrangement of \(k\) subsquares within SQ.

    Comment Source:The morphisms are geometric arrangements. Let SQ denote some fixed 'outer square.' For concreteness, we may take this to be the unit square. Then a morphism \\(f_k: \square_1,...,\square_k \rightarrow \square\\) is an arrangement of \\(k\\) subsquares within SQ.
  • 8.
    edited February 27

    Try to picture the natural rule for composing a connected tree of arrangements into a composite arrangement.

    See Spivak's slide for a picture.

    Comment Source:Try to picture the natural rule for composing a connected tree of arrangements into a composite arrangement. See Spivak's slide for a picture.
  • 9.
    edited February 27

    Here, I will draw the picture using words.

    Let \(f\) be an arrangement of \(k\) subsquares \(s_1,..s_k\) within the unit square.

    Let \(w_i\) be the width of square \(i\).

    Suppose that we have arrangements \(g_1,...,g_k\) which we wish to compose into \(f\).

    The interpretation of this will be to nest each \(g_i\) as a sub-arrangement of \(f\), installing it at the site of \(s_i\).

    \(s_i\) will function as a frame for the installation of a scaled-down copies of the subsquares that comprise \(g_i\).

    In particular, the subsquares of \(g_i\) will get scaled down by the factor \(w_i\) before being installed into the frame \(s_i\).

    So now, on our mental 'workbench', we have the unit square, subsquares \(s_i\), and sub-subsquares, which have been scaled down.

    The final composite is defined by discarding the intermediate subsquares \(s_i\), and just retaining the outer unit square and the scaled down sub-subsquares.

    Comment Source:Here, I will draw the picture using words. Let \\(f\\) be an arrangement of \\(k\\) subsquares \\(s_1,..s_k\\) within the unit square. Let \\(w_i\\) be the width of square \\(i\\). Suppose that we have arrangements \\(g_1,...,g_k\\) which we wish to compose into \\(f\\). The interpretation of this will be to nest each \\(g_i\\) as a sub-arrangement of \\(f\\), installing it at the site of \\(s_i\\). \\(s_i\\) will function as a frame for the installation of a scaled-down copies of the subsquares that comprise \\(g_i\\). In particular, the subsquares of \\(g_i\\) will get scaled down by the factor \\(w_i\\) before being installed into the frame \\(s_i\\). So now, on our mental 'workbench', we have the unit square, subsquares \\(s_i\\), and sub-subsquares, which have been scaled down. The final composite is defined by discarding the intermediate subsquares \\(s_i\\), and just retaining the outer unit square and the scaled down sub-subsquares.
  • 10.
    edited February 27

    For example, let \(f: \square \rightarrow \square_1,\square_2\) be the arrangement consisting of the lower left and upper right quarters of the unit square.

    The consider the morphism tree consisting of \(f\) at the root, with two children that are also \(f\).

    The composite of this tree (diagram) will be the first iteration of recursively nesting \(f\) within itself.

    Specifically, it consists of four subsquares, each of length \(0.25\), arranged along the diagonal from the lower left to the upper right corners of the unit square.

    Comment Source:For example, let \\(f: \square \rightarrow \square_1,\square_2\\) be the arrangement consisting of the lower left and upper right quarters of the unit square. The consider the morphism tree consisting of \\(f\\) at the root, with two children that are also \\(f\\). The composite of this tree (diagram) will be the first iteration of recursively nesting \\(f\\) within itself. Specifically, it consists of four subsquares, each of length \\(0.25\\), arranged along the diagonal from the lower left to the upper right corners of the unit square.
  • 11.
    edited February 27

    3. General spirit of operads

    Little n-cubes is a "primordial operad" that clearly illustrates the general spirit and intent of a wide range of applications of operads.

    Objects conceived as interfaces, and a morphism \(X_1,...,X_n \rightarrow Y\) is conceived as an arrangement of the sub-interfaces \(X_1,...,X_n\) within the enclosing interface \(Y\).

    Composition of morphisms is the operation of nesting arrangements and removing the intermediate interfaces.

    A tree of connected morphisms can represent a decomposition of a system using a hierarchy of abstractions.

    Composition of the morphisms gives the opposite process: concretization, to give an arrangement that explicitly includes all of the most granular components in the system.

    Comment Source:**3. General spirit of operads** Little n-cubes is a "primordial operad" that clearly illustrates the general spirit and intent of a wide range of applications of operads. Objects conceived as _interfaces_, and a morphism \\(X_1,...,X_n \rightarrow Y\\) is conceived as an arrangement of the sub-interfaces \\(X_1,...,X_n\\) within the enclosing interface \\(Y\\). _Composition_ of morphisms is the operation of nesting arrangements and removing the intermediate interfaces. A tree of connected morphisms can represent a decomposition of a system using a hierarchy of abstractions. Composition of the morphisms gives the opposite process: _concretization_, to give an arrangement that explicitly includes all of the most granular components in the system.
  • 12.
    edited February 28

    An abstract design for a complex machine may include components like Fast Fourier Transform, Convolution, etc. Operad morphisms can describe the decomposition of the system into subcomponents. and their detailed interconnections (wiring, software plumbing). Were it to be interpreted in hardware alone, the composition of the hierarchy of morphisms could yield a detailed circuit diagram comprising a large number of logic gates.


    This completes the unit of information for blog #1.

    Comment Source:An abstract design for a complex machine may include components like Fast Fourier Transform, Convolution, etc. Operad morphisms can describe the decomposition of the system into subcomponents. and their detailed interconnections (wiring, software plumbing). Were it to be interpreted in hardware alone, the composition of the hierarchy of morphisms could yield a detailed circuit diagram comprising a large number of logic gates. * * * This completes the unit of information for blog #1.
  • 13.

    Hi -

    Some comments:

    1) In your initial definition you don't say the identity morphism is called $\mathrm{Id}_A : A \to A$. This is presumably a deliberate trick to make things easier to read. If so, fine.

    2) You also don't include a notation for composition, presumably for the same reason. If so, fine.

    3) You don't actually draw the "tree diagram" as a tree. I would find it impossible to truly understand operads without seeing such trees (and drawing them). Here is my attempt to explain operads - it's packed with such pictures:

    By the way, I was only talking about operads with one object, so you don't see objects labelling the "wires" or "edges" of these pictures. Also, I accidentally left out one of the laws governing operads.

    Comment Source:Hi - Some comments: 1) In your initial definition you don't say the identity morphism is called $\mathrm{Id}_A : A \to A$. This is presumably a deliberate trick to make things easier to read. If so, fine. 2) You also don't include a notation for composition, presumably for the same reason. If so, fine. 3) You don't actually draw the "tree diagram" as a tree. I would find it impossible to truly understand operads without seeing such trees (and drawing them). Here is my attempt to explain operads - it's packed with such pictures: * Review of _[Operads in Algebra, Topology and Physics](http://math.ucr.edu/home/baez/operad.pdf)_. By the way, I was only talking about operads with one object, so you don't see objects labelling the "wires" or "edges" of these pictures. Also, I accidentally left out one of the laws governing operads.
  • 14.

    Thanks for the feedback!

    Agreed it could use some pictures.

    Your explanation is very nice.

    I will try to blend some pictures into my text.

    Also I see that I left out the condition that there is an action of the permutation group on the set of operators. And the compatibility condition between composition and permuting inputs.

    Also, I accidentally left out one of the laws governing operads.

    Which law was this?

    Comment Source:Thanks for the feedback! Agreed it could use some pictures. Your explanation is very nice. I will try to blend some pictures into my text. Also I see that I left out the condition that there is an action of the permutation group on the set of operators. And the compatibility condition between composition and permuting inputs. > Also, I accidentally left out one of the laws governing operads. Which law was this?
  • 15.

    If you don't want to talk about the permutation group actions, just talk about "planar" or "nonpermutative" operads, which are probably more often called "multicategories". They're what you get when you leave out everything about these permutation group actions. They may be good enough for what you're doing, or good enough to get started.

    Which law was this?

    The second law relating permutations to composition. It's much less visually interesting than the law I included. Stasheff actually left it out of his book - the book I was reviewing! So, technically, almost all the theorems in his book are wrong.

    Comment Source:If you don't want to talk about the permutation group actions, just talk about "planar" or "nonpermutative" operads, which are probably more often called ["multicategories"](https://ncatlab.org/nlab/show/multicategory). They're what you get when you leave out everything about these permutation group actions. They may be good enough for what you're doing, or good enough to get started. > Which law was this? The second law relating permutations to composition. It's much less visually interesting than the law I included. Stasheff actually left it out of his book - the book I was reviewing! So, technically, almost all the theorems in his book are wrong.
  • 16.
    edited March 2

    Thanks!

    Comment Source:Thanks!
  • 17.
    edited March 2

    Spivak's talk focuses on operads as a mathematical language for modular systems. Along the way, he asks the interesting question:

    Can you think of a modular environment that is not an operad?

    But there is one operad discussed in the talk which is different: Sets.

    it'll be the only operad which doesn't feel modular. ... The mathematical definition goes beyond the motivating intuition.

    What are some other significant "non-modular" operads? Was modularity the original motivation behind operads, or is modularity a theme which operads were later found to be well suited for?

    Comment Source:Spivak's talk focuses on operads as a mathematical language for modular systems. Along the way, he asks the interesting question: > Can you think of a modular environment that is not an operad? But there is one operad discussed in the talk which is different: Sets. > it'll be the only operad which doesn't feel modular. ... The mathematical definition goes beyond the motivating intuition. What are some other significant "non-modular" operads? Was modularity the original motivation behind operads, or is modularity a theme which operads were later found to be well suited for?
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