User:Lord Farin/Long-Term Projects/Awodey

= Processing of 'Category Theory' =


 * : $\S 1.3$: Definition $1.1$


 * : $\S 1.3$: Definition $1.1$

First page it covers: Definition:Composite Mapping.

A very accessible though reasonably rigid introduction to category theory. Much gentler than 'Categories work'.

Errata are available at http://www.andrew.cmu.edu/user/awodey/CT2errata2010.pdf.

Progress thus far
Initial set-up complete. --Lord_Farin 20:10, 7 August 2012 (UTC)

Up to $\S 1.8$. End with Definition:Free Category. Continue with Definition:Small Category. --Lord_Farin 10:28, 17 August 2012 (UTC)

Up to $\S 1.9$. --Lord_Farin 17:57, 17 August 2012 (UTC)

Up to $\S 2$, p.29. End with Inverse Relation Functor is Contravariant Functor. Continue with Definition:Injection. --Lord_Farin 09:20, 18 August 2012 (UTC)

Up to $\S 2.2$, p.33. Last covered: Epimorphism into Projective Object Splits. Continue with Definition:Initial Object. --Lord_Farin (talk) 12:37, 20 August 2012 (UTC)

Skipped rest of $2.11.4$ ended at Definition:Category of Boolean Algebras. Continue at Smallest Element is Initial Object. --Lord_Farin (talk) 15:11, 29 August 2012 (UTC)

Up to $\S 2.3$, will skip stuff on BA homomorphisms corresponding to ultrafilters for now. --Lord_Farin (talk) 17:14, 29 August 2012 (UTC)

Up to $\S 2.5$ which consists of examples of products; p.41. --Lord_Farin (talk) 14:08, 1 September 2012 (UTC)

Up to $\S 2.7$ about hom-sets; p.48. --Lord_Farin (talk) 08:40, 8 September 2012 (UTC)

Ended at Covariant Representable Functor is Functor; continue at Surjection iff Epimorphism in Category of Sets. --Lord_Farin (talk) 21:32, 8 September 2012 (UTC)

Up to Example $3.5$, p.56. --Lord_Farin (talk) 11:20, 2 October 2012 (UTC)

Skipped from Supremum is Coproduct in Poset Category to Definition:Equalizer, $\S 3.3$, p.62. --Lord_Farin (talk) 14:45, 8 October 2012 (UTC)

Skipped to Chapter $5$ (from Quotient Mapping is Coequalizer to Definition:Subobject); p. 89. --Lord_Farin (talk) 17:22, 18 October 2012 (UTC)

Up to $5.10$, Definition:Pullback Functor. Skipped to $\S 5.4$: Category has Products and Equalizers iff Pullbacks and Terminal Object; p.100 (yay). --Lord_Farin (talk) 22:34, 15 November 2012 (UTC)

Up to $\S 5.5$; ended at Category has Finite Limits iff Finite Products and Equalizers, cont. at Definition:Functor Preserving Limits. p.106. --Lord_Farin (talk) 22:15, 22 November 2012 (UTC)

Up to $\S 5.6$, p.108. --Lord_Farin (talk) 11:46, 24 November 2012 (UTC)

Up to $5.31$. Haven't decided how to cover the rest; skip to $\S 6$, p.119. --Lord_Farin (talk) 19:17, 27 November 2012 (UTC)

Up to $\S 6.3$, p.129. Will deviate into properly covering Boolean algebras first. This deviation may be long. --Lord_Farin (talk) 11:18, 3 December 2012 (UTC)

Remainder of $\S 6$ glosses over theories that deserve entire books discussing them. I skip to $\S 7$ (naturality), p.147. --Lord_Farin (talk) 16:11, 5 December 2012 (UTC)

Missing Proofs

 * Theorem $1.6$ Cayley's Theorem (Category Theory)
 * $\S 1.6.4$ Slice Category of Poset Category
 * Example $1.8$ Category of Pointed Sets as Coslice Category
 * Example $2.11.3$ Trivial Group is Initial Object, Trivial Group is Terminal Object
 * Proposition $2.17$ Product (Category Theory) is Unique
 * From section $2.5$ the proofs of examples of products (starting with Group Direct Product is Product in Category of Groups)
 * More proofs of examples in later chapters (e.g. $\S 3.2, \S3.3$)
 * $\S 5.1$ Subobject Class in Category of Sets
 * Corollary $5.9$ Pullback of Commutative Triangle
 * Proposition $5.10$ Pullback Functor is Functor
 * Proposition $5.21$ Category has Finite Limits iff Finite Products and Equalizers
 * Proposition $5.25$ Covariant Representable Functor is Continuous
 * Example $6.3$ Category of Finite Sets is Cartesian Closed
 * Example $6.4$ Category of Posets is Cartesian Closed
 * Proposition $6.7$ Exponentiation Functor is Functor

Skipped thus far (that is, what needs to be done still)

 * $\S 1.4.10$ Category of Proofs (context and reach too vague)
 * $\S 1.4.11$ Category of Programs (idem)
 * Some "forgetful functors" in $\S 1.6$
 * Rest of $\S 1.7$ due to lack of rigour in the graph theory compartment.
 * Further exercises from chapter $1$
 * Example $2.4$ which shows that the defs of monomorphism are compatible.
 * Rest of Example $2.11.4$ because of lacking background
 * Beginning of $\S 2.3$ concerning correspondence of ultrafilters and boolean homomorphisms
 * Example $2.5.5$, category of topological spaces not introduced yet
 * Example $2.5.6$, lengthy discussion of lambda calculus and type theory, about which I don't know anything
 * Def $2.19$ concerning categories with arbitrary products
 * Rest of $\S 2.7$, which is a preliminary, colloquial prelude to natural transformations (which are only introduced in chapter $7$)
 * Skipped the nontrivial exercises from $\S 2$ to avoid tedious preliminary work which will be easier to do after more is complete
 * Rest of $\S 3.3$, lack of formality due to equalizers/coequalizers not being introduced yet
 * Example $3.21$ about rooted posets, can't be bothered to set up yet another category
 * Rest of $\S 3.4$ about presentations of algebras, which is hard to formalise.
 * Exercises of Chapter $3$.
 * The entirety of Chapter $4$, mainly sloppy examples that serve as motivation for later concepts.
 * Rest of $\S 5.3$, idem.
 * $5.22,5.23$ since preliminary notions are still missing
 * Beginning of $\S 5.6$, need preliminaries
 * Examples $6.5,6.6$, on categories of $\omega$-cocomplete posets and graphs
 * Rest of $\S 6$, from $\S 6.3$

Other things

 * Many duality stuff is implicit in Awodey, explicate it.
 * Put mathops on the $/$ of slice and coslice categories as this improves the aesthetics.
 * Put this as source for some stuff on products that is discussed in passing.
 * A category Category:Functors may be beneficial for site-structural purposes.
 * Categories to be created: Category:Representable Functors, Category:Cartesian Closed Categories
 * The part about Definition:Boolean Lattices needs to be covered again due to the recreation of this area.