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David Tanzer

David Tanzer
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• 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…
Comment by David Tanzer March 2
• Thanks!
Comment by David Tanzer March 2
• 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…
Comment by David Tanzer March 2
• 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…
• Try to picture the natural rule for composing a connected tree of arrangements into a composite arrangement. See Spivak's slide for a picture.
• 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$$ For each object, a designated identity morphism A rule for combining connected "trees" of m…
• Welcome, Johan! I studied c.s., but have a gap in my learning when it comes to Haskell. Can you write a little bit about some of the key ways in which constructs from category theory are used in Haskell? Of course I could find this on the web, bu…
• Welcome, Steele! That sounds really intriguing. Could you describe a small example of how you apply category theory to the design of software?
• The number of comparisons QUICKSORT makes to sort a list of $$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…
• Great!! This was very helpful in explaining -- in a mathematically entertaining way -- the pith of some nice topics in applied category theory.
• Good idea. I just created a category called Applied Category Theory Formula Examples. Thanks Fredrick.
Comment by David Tanzer May 2018
• 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 …
Comment by David Tanzer May 2018
• in Matthew Doty's puzzles MD1 - MD3 we learn that the logical operations "and" and "or" can be described as right and left adjoints. Wow!
• 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.
• Great!!
• Great, Fredrick!! Thank you
• 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…