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 whi…
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 op…
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 …
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 \(X_1,...,X_n \rightarrow Y\) …
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 …
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…
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 \(k\) subsquar…
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 …
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 \(I_1\)…
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 …
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 cond…
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 …
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 m…
Welcome, Johan!
I studied c.s., but have a gap in my learning when it comes to Haskell.
Can you write a little bit about some of the key ways in which constructs from category theory are used in Haskell?
Of course I could find this on the web, bu…
The number of comparisons QUICKSORT makes to sort a list of \(n\) values is a random variable.
Let \(X_n\) be the number of comparisons QUICKSORT makes to sort a list of \(n\) values, and let \(M_n = E[X_n]\).
Let \(Y\) be rank of the number \(x_i…
From Introduction, cont'd.
Analysis of QUICKSORT.
QUICKSORT(\(x_1, ..., x_n\)):
If \(n = 0\) or \(n = 1\), no sorting needing, so return.
Randomly choose \(x_i\).
Divide up the remaining values into two sets \(L\) and \(H\), where \(L\) is the …
Notes from Introduction.
Examples of stochastic systems: CPU with jobs arriving in random fashion; network multiplexor with packets arriving randomly; a store with random demands on its inventory.
Prob. A monkey hits keys on a typewriter randoml…
Hi Maria, I am curious to know about the connections between music and category theory that you have drawn. Would it be possible for you to post a link to a paper or two that you consider to be of interest?
I tried to download a couple from your …
I think of meet as what is common to the two -- the point at which they meet -- so that's the intersection. Joining implies combining them to make something bigger, which is the union.
Let P be a poset.
Then the least upper bound of {} is, naturally, the least of the upper bounds of {}. Every member of P is an upper bound for {}. So the least upper bound for {} is the least member of P, i.e., the minimum element of P -- if suc…