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| | | |
| ==CAT. Category Theory • Discussion== | | ==CAT. Category Theory • Discussion== |
| + | |
| + | ===CAT. Discussion Note 1=== |
| | | |
| <pre> | | <pre> |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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− |
| |
− | CAT. Discussion Note 1
| |
− |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| | | |
| I have tried to get through the IFF proposal several times now, | | I have tried to get through the IFF proposal several times now, |
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| to this basic groundwork whenever it seems appropriate. | | to this basic groundwork whenever it seems appropriate. |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | CAT. Discussion Note 2 | + | ===CAT. Discussion Note 2=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| Links to the first three installments from Mac Lane are given below. | | Links to the first three installments from Mac Lane are given below. |
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| Higher Order Categorical Logic, 01-30. | | Higher Order Categorical Logic, 01-30. |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | CAT. Discussion Note 3 | + | ===CAT. Discussion Note 3=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| JS = John Sowa | | JS = John Sowa |
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| necessarily the commonsense classification, naive ontology modules. | | necessarily the commonsense classification, naive ontology modules. |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | CAT. Discussion Note 4 | + | ===CAT. Discussion Note 4=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| Yesterday's discussion of the "lattice of theories" brings back | | Yesterday's discussion of the "lattice of theories" brings back |
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| Figure 1. Lattice of Objects Inducing a Diversity of Sign Partitions | | Figure 1. Lattice of Objects Inducing a Diversity of Sign Partitions |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | CAT. Discussion Note 5 | + | ===CAT. Discussion Note 5=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| In connection with the ontological use of category theory -- | | In connection with the ontological use of category theory -- |
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| http://www.univ-perp.fr/see/rch/lts/marty/semantic-ns/reference1.htm | | http://www.univ-perp.fr/see/rch/lts/marty/semantic-ns/reference1.htm |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
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− | CAT. Discussion Note 6 | + | ===CAT. Discussion Note 6=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| JS = John Sowa | | JS = John Sowa |
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| > > the IFF system in order to (1) use it or (2) evaluate it. | | > > the IFF system in order to (1) use it or (2) evaluate it. |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | CAT. Discussion Note 7 | + | ===CAT. Discussion Note 7=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| One way to get the motivation for category theory | | One way to get the motivation for category theory |
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| Anyway, that is how I see it getting started. | | Anyway, that is how I see it getting started. |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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− |
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− | CAT. Discussion Note 8
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− |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| </pre> | | </pre> |
| | | |
− | CAT. Category Theory • Work Area | + | ==CAT. Category Theory • Work Area== |
| | | |
| <pre> | | <pre> |
| + | |
| "John kicks the cart." | | "John kicks the cart." |
| | | |
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| ==HOC. Higher Order Categorical Logic== | | ==HOC. Higher Order Categorical Logic== |
| + | |
| + | ===HOC. Note 1=== |
| | | |
| <pre> | | <pre> |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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− |
| |
− | HOC. Note 1
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− |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| | | |
| | Part 0. Introduction to Category Theory | | | Part 0. Introduction to Category Theory |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
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− | HOC. Note 2 | + | ===HOC. Note 2=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
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| | We shall now progress from concrete categories | | | We shall now progress from concrete categories |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
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− | HOC. Note 3 | + | ===HOC. Note 3=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | Example C1'. Any set can be viewed as a category: a small 'discrete' | | | Example C1'. Any set can be viewed as a category: a small 'discrete' |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
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− | HOC. Note 4 | + | ===HOC. Note 4=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | It follows from slogans 1 and 2 that small categories | | | It follows from slogans 1 and 2 that small categories |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 5 | + | ===HOC. Note 5=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | One we admit that functors $A$ -> $B$ are interesting objects to study, | | | One we admit that functors $A$ -> $B$ are interesting objects to study, |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 6 | + | ===HOC. Note 6=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | Definition 1.5. An arrow f : A -> B in a category is called an 'isomorphism' | | | Definition 1.5. An arrow f : A -> B in a category is called an 'isomorphism' |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 7 | + | ===HOC. Note 7=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | 2. Natural Transformations | | | 2. Natural Transformations |
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| should be interpreted as equivalent to "1_(F(A))". | | should be interpreted as equivalent to "1_(F(A))". |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 8 | + | ===HOC. Note 8=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | To appreciate the usefulness of natural transformations, | | | To appreciate the usefulness of natural transformations, |
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| o--------------o o--------------o | | o--------------o o--------------o |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 9 | + | ===HOC. Note 9=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
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| | This may be as good a place as any to mention that | | | This may be as good a place as any to mention that |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 10 | + | ===HOC. Note 10=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | If we compare Slogans 1 and 3, we are led to ask: | | | If we compare Slogans 1 and 3, we are led to ask: |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 11 | + | ===HOC. Note 11=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | Definition 2.6. A 'subcategory' $C$ of a category $B$ is any category whose | | | Definition 2.6. A 'subcategory' $C$ of a category $B$ is any category whose |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 12 | + | ===HOC. Note 12=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | Definition 2.8. A functor H : $A$ -> $B$ is | | | Definition 2.8. A functor H : $A$ -> $B$ is |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 13 | + | ===HOC. Note 13=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| I am skipping ahead to Part 1 of L&S -- we have been reading | | I am skipping ahead to Part 1 of L&S -- we have been reading |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 14 | + | ===HOC. Note 14=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | Historical Perspective on Part 1 | | | Historical Perspective on Part 1 |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 15 | + | ===HOC. Note 15=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| Notation. The Greek letters phi and lambda | | Notation. The Greek letters phi and lambda |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 16 | + | ===HOC. Note 16=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | Both Curry and Church realized the importance of introducing types into | | | Both Curry and Church realized the importance of introducing types into |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 17 | + | ===HOC. Note 17=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | In connection with cartesian closed categories, | | | In connection with cartesian closed categories, |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 18 | + | ===HOC. Note 18=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
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| | Up to this point we have avoided discussing natural numbers. | | | Up to this point we have avoided discussing natural numbers. |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
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− | HOC. Note 19 | + | ===HOC. Note 19=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | 1. Propositional Calculus as a Deductive System | | | 1. Propositional Calculus as a Deductive System |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
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− | HOC. Note 20 | + | ===HOC. Note 20=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | A 'conjunction calculus' is a deductive system dealing with truth and | | | A 'conjunction calculus' is a deductive system dealing with truth and |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
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− | HOC. Note 21 | + | ===HOC. Note 21=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
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| | A 'positive intuitionistic propositional calculus' is a conjunction calculus | | | A 'positive intuitionistic propositional calculus' is a conjunction calculus |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
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− | HOC. Note 22 | + | ===HOC. Note 22=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
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| | An 'intuitionistic propositional calculus' is more than a | | | An 'intuitionistic propositional calculus' is more than a |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
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− | HOC. Note 23 | + | ===HOC. Note 23=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | 2. The Deduction Theorem | | | 2. The Deduction Theorem |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
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− | HOC. Note 24 | + | ===HOC. Note 24=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | 3. Cartesian Closed Categories Equationally Presented | | | 3. Cartesian Closed Categories Equationally Presented |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 25 | + | ===HOC. Note 25=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | A 'cartesian closed category' is a cartesian category $A$ with | | | A 'cartesian closed category' is a cartesian category $A$ with |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
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− | HOC. Note 26 | + | ===HOC. Note 26=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | We shall state another useful equation, | | | We shall state another useful equation, |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 27 | + | ===HOC. Note 27=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| I return to Part 0 to pick up one or two bits | | I return to Part 0 to pick up one or two bits |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 28 | + | ===HOC. Note 28=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | Before carrying out the promised generalization, let us | | | Before carrying out the promised generalization, let us |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 29 | + | ===HOC. Note 29=== |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | We shall now generalize the notion of adjoint functor from | | | We shall now generalize the notion of adjoint functor from |
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| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 30 | + | ===HOC. Note 30=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | Proposition 3.3. Given two categories $A$ and $B$, there is a | | | Proposition 3.3. Given two categories $A$ and $B$, there is a |
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Line 4,236: |
| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Note 31 | + | ===HOC. Note 31=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| | 4. Equivalence of Categories | | | 4. Equivalence of Categories |
| | | | | |
| + | | ... |
| | | | | |
− |
| |
| | L&S, pages 16-17. | | | L&S, pages 16-17. |
| | | | | |
Line 4,264: |
Line 4,254: |
| | http://uk.cambridge.org/mathematics/catalogue/0521356539/ | | | http://uk.cambridge.org/mathematics/catalogue/0521356539/ |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| </pre> | | </pre> |
| | | |
| ==HOC. Higher Order Categorical Logic • Discussion== | | ==HOC. Higher Order Categorical Logic • Discussion== |
| + | |
| + | ===HOC. Discussion Note 1=== |
| | | |
| <pre> | | <pre> |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
− |
| |
− | HOC. Discussion Note 1
| |
− |
| |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| | | |
| MA = Murray Altheim | | MA = Murray Altheim |
Line 4,347: |
Line 4,333: |
| [HOC. Higher Order Categorical Logic. Notes 01-07] | | [HOC. Higher Order Categorical Logic. Notes 01-07] |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Discussion Note 2 | + | ===HOC. Discussion Note 2=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| JA = Jon Awbrey | | JA = Jon Awbrey |
Line 4,515: |
Line 4,501: |
| okay, tomorrow ... | | okay, tomorrow ... |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Discussion Note 3 | + | ===HOC. Discussion Note 3=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| JP = Jack Park | | JP = Jack Park |
Line 4,550: |
Line 4,536: |
| hysterical amnesia for my time on the complexity list. | | hysterical amnesia for my time on the complexity list. |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Discussion Note 4 | + | ===HOC. Discussion Note 4=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| JP = Jack Park | | JP = Jack Park |
Line 4,565: |
Line 4,551: |
| what is the composition? | | what is the composition? |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Discussion Note 5 | + | ===HOC. Discussion Note 5=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| JA = Jon Awbrey | | JA = Jon Awbrey |
Line 4,828: |
Line 4,814: |
| Have to break here. Will pick up at "preserve". | | Have to break here. Will pick up at "preserve". |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Discussion Note 6 | + | ===HOC. Discussion Note 6=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| JA: The next examples are sets plus "structure", | | JA: The next examples are sets plus "structure", |
Line 4,942: |
Line 4,928: |
| help with this mess. | | help with this mess. |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Discussion Note 7 | + | ===HOC. Discussion Note 7=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| Here is a simple example of a morphism f : (X, L) -> (Y, M). | | Here is a simple example of a morphism f : (X, L) -> (Y, M). |
Line 5,001: |
Line 4,987: |
| Parity On, Dude! | | Parity On, Dude! |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Discussion Note 8 | + | ===HOC. Discussion Note 8=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| MW = Matthew West | | MW = Matthew West |
Line 5,061: |
Line 5,047: |
| and Y is a finite set, so f cannot possibly be an isomorphism. | | and Y is a finite set, so f cannot possibly be an isomorphism. |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Discussion Note 9 | + | ===HOC. Discussion Note 9=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| JA = Jon Awbrey | | JA = Jon Awbrey |
Line 5,150: |
Line 5,136: |
| that everybody will want to buy sooner or later, anyway. | | that everybody will want to buy sooner or later, anyway. |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Discussion Note 10 | + | ===HOC. Discussion Note 10=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| While I recover my strength for the imminent trek through Emyn Muil, | | While I recover my strength for the imminent trek through Emyn Muil, |
Line 5,175: |
Line 5,161: |
| CAT. Category Theory. Links 01-23. | | CAT. Category Theory. Links 01-23. |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Discussion Note 11 | + | ===HOC. Discussion Note 11=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| I will now introduce a number of different ways of | | I will now introduce a number of different ways of |
Line 5,243: |
Line 5,229: |
| This is the same information as the equation, f(G(u, v)) = H(f(u), f(v)). | | This is the same information as the equation, f(G(u, v)) = H(f(u), f(v)). |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| | | |
− | HOC. Discussion Note 12 | + | ===HOC. Discussion Note 12=== |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | | |
| JA = Jon Awbrey | | JA = Jon Awbrey |
Line 5,301: |
Line 5,287: |
| will get back to this ... | | will get back to this ... |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
− |
| |
− | HOC. Discussion Note 13
| |
− |
| |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| </pre> | | </pre> |
| | | |
| ==HOC. Higher Order Categorical Logic • Work Area== | | ==HOC. Higher Order Categorical Logic • Work Area== |
| + | |
| + | ===HOC. Discussion Work 1=== |
| | | |
| <pre> | | <pre> |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
− |
| |
− | HOC. Discussion Work 1
| |
− |
| |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| | | |
| Let's go back and take another look at what is most likely every | | Let's go back and take another look at what is most likely every |
Line 5,361: |
Line 5,339: |
| | f(G(u, v)) = H(fu, fv) | | | f(G(u, v)) = H(fu, fv) |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | </pre> |
| + | |
| + | ===HOC. Discussion Work 2=== |
| | | |
− | LOR. Note 57
| + | <pre> |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | LOR. Note 57 |
| | | |
| I'm going to elaborate a little further on the subject | | I'm going to elaborate a little further on the subject |
Line 5,439: |
Line 5,419: |
| distributive multiplication of a sum (u + v) by a factor J. | | distributive multiplication of a sum (u + v) by a factor J. |
| Hence another popular name for a morphism: a "linear" map. | | Hence another popular name for a morphism: a "linear" map. |
| + | |
| </pre> | | </pre> |
| + | |
| + | ==HOC. Higher Order Categorical Logic • Document History== |
| + | |
| + | '''Ontology List (Oct 2001)''' |
| + | |
| + | * http://web.archive.org/web/20130304170501/http://suo.ieee.org/ontology/thrd42.html#03373 |
| + | # http://web.archive.org/web/20081204200346/http://suo.ieee.org/ontology/msg03373.html |
| + | # http://web.archive.org/web/20081204200607/http://suo.ieee.org/ontology/msg03375.html |
| + | # http://web.archive.org/web/20081204200947/http://suo.ieee.org/ontology/msg03376.html |
| + | # http://web.archive.org/web/20081121223306/http://suo.ieee.org/ontology/msg03377.html |
| + | # http://web.archive.org/web/20070302103422/http://suo.ieee.org/ontology/msg03378.html |
| + | # http://web.archive.org/web/20070302103431/http://suo.ieee.org/ontology/msg03381.html |
| + | # http://web.archive.org/web/20070302103442/http://suo.ieee.org/ontology/msg03383.html |
| + | # http://web.archive.org/web/20070302103500/http://suo.ieee.org/ontology/msg03384.html |
| + | # http://web.archive.org/web/20070302175225/http://suo.ieee.org/ontology/msg03392.html |
| + | # http://web.archive.org/web/20070302103242/http://suo.ieee.org/ontology/msg03393.html |
| + | # http://web.archive.org/web/20070302103505/http://suo.ieee.org/ontology/msg03394.html |
| + | # http://web.archive.org/web/20070302103230/http://suo.ieee.org/ontology/msg03395.html |
| + | # http://web.archive.org/web/20070302103517/http://suo.ieee.org/ontology/msg03396.html |
| + | # http://web.archive.org/web/20070302103303/http://suo.ieee.org/ontology/msg03398.html |
| + | # http://web.archive.org/web/20081121223647/http://suo.ieee.org/ontology/msg03399.html |
| + | # http://web.archive.org/web/20070302103539/http://suo.ieee.org/ontology/msg03400.html |
| + | # http://web.archive.org/web/20070302103550/http://suo.ieee.org/ontology/msg03401.html |
| + | # http://web.archive.org/web/20070302103559/http://suo.ieee.org/ontology/msg03402.html |
| + | # http://web.archive.org/web/20070302103609/http://suo.ieee.org/ontology/msg03403.html |
| + | # http://web.archive.org/web/20070302103619/http://suo.ieee.org/ontology/msg03404.html |
| + | # http://web.archive.org/web/20070302103630/http://suo.ieee.org/ontology/msg03405.html |
| + | # http://web.archive.org/web/20070302103639/http://suo.ieee.org/ontology/msg03406.html |
| + | # http://web.archive.org/web/20070302103650/http://suo.ieee.org/ontology/msg03409.html |
| + | # http://web.archive.org/web/20081121223918/http://suo.ieee.org/ontology/msg03410.html |
| + | # http://web.archive.org/web/20070302103712/http://suo.ieee.org/ontology/msg03411.html |
| + | # http://web.archive.org/web/20070302103722/http://suo.ieee.org/ontology/msg03412.html |
| + | # http://web.archive.org/web/20080906120100/http://suo.ieee.org/ontology/msg03415.html |
| + | # http://web.archive.org/web/20070302103742/http://suo.ieee.org/ontology/msg03416.html |
| + | # http://web.archive.org/web/20070302103752/http://suo.ieee.org/ontology/msg03417.html |
| + | # http://web.archive.org/web/20070302103802/http://suo.ieee.org/ontology/msg03418.html |
| | | |
| ==HOC. Higher Order Categorical Logic • Discussion History== | | ==HOC. Higher Order Categorical Logic • Discussion History== |
| | | |
− | Ontology List
| + | '''Inquiry List (Jan 2004)''' |
| | | |
− | # http://suo.ieee.org/ontology/msg05262.html | + | * http://web.archive.org/web/20120428203453/http://stderr.org/pipermail/inquiry/2004-January/thread.html#1037 |
− | # http://suo.ieee.org/ontology/msg05263.html | + | # http://web.archive.org/web/20061013235830/http://stderr.org/pipermail/inquiry/2004-January/001037.html |
− | # http://suo.ieee.org/ontology/msg05264.html | + | # http://web.archive.org/web/20061013235743/http://stderr.org/pipermail/inquiry/2004-January/001038.html |
− | # http://suo.ieee.org/ontology/msg05265.html | + | # http://web.archive.org/web/20040331105109/http://stderr.org/pipermail/inquiry/2004-January/001039.html |
− | # http://suo.ieee.org/ontology/msg05266.html | + | # http://web.archive.org/web/20061014000144/http://stderr.org/pipermail/inquiry/2004-January/001040.html |
− | # http://suo.ieee.org/ontology/msg05267.html | + | # http://web.archive.org/web/20061013235834/http://stderr.org/pipermail/inquiry/2004-January/001041.html |
− | # http://suo.ieee.org/ontology/msg05268.html | + | # http://web.archive.org/web/20040331104547/http://stderr.org/pipermail/inquiry/2004-January/001042.html |
− | # http://suo.ieee.org/ontology/msg05270.html | + | # http://web.archive.org/web/20061013235923/http://stderr.org/pipermail/inquiry/2004-January/001043.html |
− | # http://suo.ieee.org/ontology/msg05271.html | + | # http://web.archive.org/web/20040331104447/http://stderr.org/pipermail/inquiry/2004-January/001044.html |
− | # http://suo.ieee.org/ontology/msg05273.html | + | # http://web.archive.org/web/20061013235859/http://stderr.org/pipermail/inquiry/2004-January/001045.html |
− | # http://suo.ieee.org/ontology/msg05277.html | + | # http://web.archive.org/web/20061013235531/http://stderr.org/pipermail/inquiry/2004-January/001047.html |
− | # http://suo.ieee.org/ontology/msg05285.html | + | # http://web.archive.org/web/20061013235946/http://stderr.org/pipermail/inquiry/2004-January/001050.html |
| + | # http://web.archive.org/web/20061013235926/http://stderr.org/pipermail/inquiry/2004-January/001052.html |
| | | |
− | Inquiry List
| + | '''Ontology List (Jan 2004)''' |
| | | |
− | * http://web.archive.org/web/20120428203453/http://stderr.org/pipermail/inquiry/2004-January/thread.html#1037 | + | * http://web.archive.org/web/20070304201532/http://suo.ieee.org/ontology/thrd42.html#05262 |
− | # http://stderr.org/pipermail/inquiry/2004-January/001037.html | + | # http://web.archive.org/web/20070302153905/http://suo.ieee.org/ontology/msg05262.html |
− | # http://stderr.org/pipermail/inquiry/2004-January/001038.html | + | # http://web.archive.org/web/20070302153917/http://suo.ieee.org/ontology/msg05263.html |
− | # http://stderr.org/pipermail/inquiry/2004-January/001039.html | + | # http://web.archive.org/web/20070302133411/http://suo.ieee.org/ontology/msg05264.html |
− | # http://stderr.org/pipermail/inquiry/2004-January/001040.html | + | # http://web.archive.org/web/20070302164529/http://suo.ieee.org/ontology/msg05265.html |
− | # http://stderr.org/pipermail/inquiry/2004-January/001041.html | + | # http://web.archive.org/web/20070302164538/http://suo.ieee.org/ontology/msg05266.html |
− | # http://stderr.org/pipermail/inquiry/2004-January/001042.html | + | # http://web.archive.org/web/20070302164548/http://suo.ieee.org/ontology/msg05267.html |
− | # http://stderr.org/pipermail/inquiry/2004-January/001043.html | + | # http://web.archive.org/web/20070302164558/http://suo.ieee.org/ontology/msg05268.html |
− | # http://stderr.org/pipermail/inquiry/2004-January/001044.html | + | # http://web.archive.org/web/20070302164608/http://suo.ieee.org/ontology/msg05270.html |
− | # http://stderr.org/pipermail/inquiry/2004-January/001045.html | + | # http://web.archive.org/web/20070302164618/http://suo.ieee.org/ontology/msg05271.html |
− | # http://stderr.org/pipermail/inquiry/2004-January/001047.html | + | # http://web.archive.org/web/20070302164638/http://suo.ieee.org/ontology/msg05273.html |
− | # http://stderr.org/pipermail/inquiry/2004-January/001050.html | + | # http://web.archive.org/web/20070302164659/http://suo.ieee.org/ontology/msg05277.html |
− | # http://stderr.org/pipermail/inquiry/2004-January/001052.html | + | # http://web.archive.org/web/20070302154121/http://suo.ieee.org/ontology/msg05285.html |
| | | |
| ==INF. Information Flow== | | ==INF. Information Flow== |
Line 8,840: |
Line 8,858: |
| * Hoel, P.G., Port, S.C., and Stone, C.J. (1971), ''Introduction to Probability Theory'', Houghton Mifflin, Boston, MA. | | * Hoel, P.G., Port, S.C., and Stone, C.J. (1971), ''Introduction to Probability Theory'', Houghton Mifflin, Boston, MA. |
| | | |
− | ===PAS. Probability And Statistics • Inquiry List • History===
| + | '''Inquiry List''' |
| | | |
| * http://web.archive.org/web/20141205223045/http://stderr.org/pipermail/inquiry/2003-June/thread.html#588 | | * http://web.archive.org/web/20141205223045/http://stderr.org/pipermail/inquiry/2003-June/thread.html#588 |
Line 8,848: |
Line 8,866: |
| # http://web.archive.org/web/20070311001647/http://stderr.org/pipermail/inquiry/2003-June/000592.html | | # http://web.archive.org/web/20070311001647/http://stderr.org/pipermail/inquiry/2003-June/000592.html |
| | | |
− | ===PAS. Probability And Statistics • Ontology List • History===
| + | '''Ontology List''' |
| | | |
| * http://web.archive.org/web/20070301065551/http://suo.ieee.org/ontology/thrd13.html#04885 | | * http://web.archive.org/web/20070301065551/http://suo.ieee.org/ontology/thrd13.html#04885 |