categorical grammar is a grammar in the language of categories.

constituents

• \(A\), a set of “expressions”
• \(C\), a set of categories of “syntax”
• \(\varphi: A \to Pow( C)\), assigning each \(a \in A\) to a set of categories \(c \subset C\)
• \(G\): a family of sets of n-place operations where \(n=1, 2, \ldots\) (what does a “3-place” op mean? idk)
• \(R\): a set of rules encoded as tuples: \((f; \{c_1, \dots c_{k}\}; c_{k+1})\), where \(f\) is a \(k\) place operation, and \(c_{j} \in C\)

requirements

The operations of this grammar behaves like so:

• categorical grammar

A category is an abstract collection of objects

constituents

• collection of objects, where if \(X\) is an object of \(C\) we write \(X \in C\)
• for a pair of objects \(X, Y \in C\), a set of morphisms acting upon the objects which we call the homset

additional information

requirements

• there exists the identity morphism; that is, \(\forall X \in C, \exists I_{X}: X\to X\)
• morphisms are always composable: given \(f: X\to Y\), and \(g: Y\to Z\), exists \(gf: X \to Z\)
• the identity morphism can compose in either direction: given \(f: X \to Y\), then \(f I_{x} = f = I_{y} f\)
• morphism composition is associative: \((hg)f=h(gf)\)

An abstract study of mathematics based on categories, functors, and natural transformations.

stock market crash of 1929

At October 24th, 1929, Black Thursday took place, and the stock market crashed. During this time, a record of 13 million shares traded, over $3b of losses. This began a 4 year slide of the global economy.

Crash theories:

• demand-driven theory
• Monetarist theory

bank failures of 1929

Banks became irrelevant. Lots of risky loans given out, farmers are taken out huge loans and the banks can’t deal.