The n-Category Café

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2024-07-10 12:00:10

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

String diagrams are ubiquitous in applied category theory. They originate as a graphical notation for representing terms in monoidal categories and since their origins, they have been used not just as a tool for researchers to make reasoning easier but also to formalize and give algebraic semantics to previous graphical formalisms.

On the other hand, it is well known the relationship between simply typed lambda calculus and Cartesian Closed Categories(CCC) throughout Curry-Howard-Lambeck isomorphism. By adding the necessary notation for the extra structure of CCC, we could also represent terms of Cartesian Closed Categories using string diagrams. By mixing these two ideas, it is not crazy to think that if we represent terms of CCC with string diagrams, we should be able to represent computation using string diagrams. This is the goal of this blog, we will use string diagrams to represent simply-typed lambda calculus terms, and computation will be modeled by the idea of a sequence of rewriting steps of string diagrams (i.e. an operational semantics!).

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