Category Theory for the Sciences by David I. Spivak

Author:David I. Spivak
Language: eng
Format: mobi, epub, pdf
Publisher: MIT Press
Published: 0101-01-01T00:00:00+00:00

Section 5.4.2 formally shows how to turn a schema into a category (the category it presents). For now, it seems better not to be so formal, because the idea is fairly straightforward. Suppose given a schema S, which consists of a graph G = (V, A, src, tgt) equipped with a congruence ~ (see Definition 4.5.2.3). It presents a category C defined as follows. The set of objects in C is defined to be the vertices V; the set of morphisms in C is defined to be the quotient Paths(G)/ ~; and the composition formula is given by concatenation of paths. The path equivalences making up ~ become commutative diagrams in C.

Example 5.2.2.2. The following schema Loop has no path equivalence declarations. As a graph it has one vertex and one arrow.

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