MATH.16:11 - SoTA-Echoing

For choosing an object through the maps it must support, the adopted line is universal construction in category theory. Riehl’s Category Theory in Context, §§2.3, 3.1 and 3.2, gives the general account and concrete set constructions. The present method uses that line to move from a working requirement to a construction, then to its derived maps and equality arguments.

Fong and Spivak’s Seven Sketches in Compositionality, especially the chapter on databases and categories, develops the use of these constructions across applications. Its contribution here is the attention to what transformations and queries the constructed object supports. Example 3.72 supplies the function-as-object construction through currying used in :5.4.

At comparable effort, a direct pair or tagged union is often enough when the operation is already settled. Use the universal account when the ambiguity or later reasoning makes it useful. A more elaborate categorical description adds no benefit to a calculation whose relevant conditions and result are already clear.

The elementary examples use ordinary equality and functions. If the work changes the permitted maps or the meaning of equality, reformulate the comparison and its equations in that setting. A requirement to compute the result can also reopen the choice of construction.