Seven Sketches in Compositionality: An Invitation to Applied Category Theory
OK, moderately well-read in software engineering but not in math; experienced programmer; start reading the intro, sounds promising; but early on I find some presentation problems which I think are going to impede you reaching the audience you describe in the first few paragraphs. Just a bit of a commentary here; if I persist I may send a more detailed review -- although I note that there is no invitation to do so, or address for comments, in the Introduction?<p>Top of p.iv, &quot;almost no introductory material available on monoidal categories.&quot; Wait, it was all about categories, now it is suddenly monoidal ones -- what is monoidal, is that a subset of the book subject, or a more precise term? Badly need a definition when first introducing a new term.<p>(Actually, that just highlights the point that the term &quot;category&quot; has been assumed and not defined to this point -- despite the audience not being expected to know what it is.)<p>Lemon pie diagram: of _course_ I can grasp the basic idea, but in what sense does the diagram illustrate category theory? Where are is the category (categories?), is it the pie, the lines, the nodes? It&#x27;s probably blindingly obvious _after you understand the topic_ but at the opening of the book, the point needs to be explained even if the explanation seems redundant to the author.<p>Top paragraph on p.v flirts with a definition but doesn&#x27;t actually define anything. Is a category &quot;structures and coherence&quot;? One &quot;involves&quot; (vague) &quot;a collection of objects, a collection of morphisms relating objects, and a formula for combining any chain of morphisms into a morphism&quot; and OK, here is a hint of familiar ground: this sounds a lot like category theory might be a generalization of classes, instances and methods, and functional composition, of software engineering. But the sentence about chains of chains ending in a happy &quot;That&#x27;s it!&quot; doesn&#x27;t really seem to follow. Onward.<p>Bottom of page 1, we finally get to some substance, although I really don&#x27;t like the &quot;Y is a meter&quot; explanation because that whole paragraph kind of bodges the distinction between italic-capital-letter X and Y, which I am kinda sure you intend to represent classes or domains, with the word &quot;object&quot; which my prior training really wants to confine to singular instances of a domain. So what comes out of the map isn&#x27;t a &quot;meter Y&quot; but &quot;the reading displayed by the meter Y&quot; but that doesn&#x27;t work either.<p>Anyway, very shortly you drop in another new notation, &quot;functions f from big-R to big-R&quot; What does the big-R mean here? I&#x27;m casting about -- real-world-objects? set of Real numbers? Probably real numbers because the following features are about arithmetic comparisons and operations. But say it, dammit.<p>Order-preserving, ok, but -- why &quot;metric-preserving&quot;? How is absolute-difference a &quot;metric&quot;? But... this is an exercise and a challenging one, and I&#x27;m intrigued. Later.