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1 | | -0. title page |
2 | | -1. from set theory as "sets, elements and functions", to thinking of elements as functions from * to a set, so we now only have "sets and functions". |
3 | | -2. in programming languages and type theory, types are not collections of elements, but a symbol with constructors (functions into the type). In Nat example, say that Nat is |
| 1 | +* title page |
| 2 | + |
| 3 | +# Material -> Structural |
| 4 | +* from set theory as "sets, elements and functions", to thinking of elements as functions from * to a set, so we now only have "sets and functions". |
| 5 | +* in programming languages and type theory, types are not collections of elements, but a symbol with constructors (functions into the type). In Nat example, say that Nat is |
4 | 6 | freely generated by these constructors. |
5 | | -3. in algebra, we think of matrices as linear maps (functions) between fixed dimensional spaces. If input and output dimensions are the same, then we have the monoid of square matrices. But if input/output dims are different, they don't form a monoid because multiplication can fail. |
6 | | -4. Instead, they form a category, with natural numbers as objects, and the matrices as morphisms. you can only "multiply" or compose them if the dimensions match up |
7 | | -5. What is a category? Talk about $C^{op}$ the opposite category. |
8 | | -6. Category theory is not just about "sets and functions", but any abstract structure has some notion of "objects and morphisms". Give an example of a small category, generated from a graph. |
9 | | -7. Given a small category $C$, we often ask if it can be "represented" by some sets and functions. In fact, we have many "functors" from C^{op} to the category of Sets that preserves the structure of C^{op} (and therefore of $C$). Pick some object a in C, and map all objects $b$ in $C$ to the set of morphisms $C[b,a]$ from b to a. Describe map on morphisms. A functor C^{op} -> Set is also called a presheaf. |
10 | | -7. what is a functor? |
11 | | -8. Similar to Cayley's theorem in group theory, that any finite group can be represented by a matrix group. What is Yoneda embedding? |
| 7 | +* in algebra, we think of matrices as linear maps (functions) between fixed dimensional spaces. |
| 8 | + |
| 9 | +# Monoids -> Categories |
| 10 | +* For linear maps, If input and output dimensions are the same, then we have the monoid of square matrices. But if input/output dims are different, they don't form a monoid because multiplication can fail. |
| 11 | +* Instead, they form a category, with natural numbers as objects, and the matrices as morphisms. you can only "multiply" or compose them if the dimensions match up |
| 12 | +* What is a category? Talk about $C^{op}$ the opposite category. |
| 13 | +* Category theory is not just about "sets and functions", but any abstract structure has some notion of "objects and morphisms". Give an example of a small category, generated from a graph. |
| 14 | + |
| 15 | +# Representations and Semantics |
| 16 | +* Given a small category $C$, we often ask if it can be "represented" by some sets and functions. In fact, we have many "functors" from C^{op} to the category of Sets that preserves the structure of C^{op} (and therefore of $C$). Pick some object a in C, and map all objects $b$ in $C$ to the set of morphisms $C[b,a]$ from b to a. Describe map on morphisms. A functor C^{op} -> Set is also called a presheaf. |
| 17 | +* what is a functor? |
| 18 | +* Similar to Cayley's theorem in group theory, that any finite group can be represented by a matrix group. What is Yoneda embedding? |
12 | 19 |
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