Abstract
We are approaching the end of our journey. In this chapter we will discuss the porting of CBTp to category theory. In the final section we will point out certain developments of the period 1918–1924, in which appear the gestalt and metaphor of commutative diagrams, a basic tool of category theory.
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Notes
- 1.
Some restrictions apply to prevent S o sets from including paradoxical sets.
- 2.
In this category the above mentioned restrictions apply too.
- 3.
TK talk about transformations but from the context it appears they mean monotransformations.
- 4.
TK added a third condition: (3) If (A′ f , A′ g , B′ f , B′ g ) also satisfy (1) and (2), then A′ f ⊆ A f . TK disregarded the third condition in their discussion and so we have omitted it from the definition of a Banach category. (3) seems to be related to the possibility of partitioning the two sets in different ways, remarked in a footnote to Sect. 29.1.
- 5.
Recall that this was not how Banach proved CBT from his Partitioning Theorem. Banach proved an analog of CBT for relations R that fulfill certain properties.
- 6.
Again, TK talk about transformations but the context suggests monotransformation.
- 7.
TK omit mentioning that K is a Brandt category and that f, g are monotransformations. They must have taken these points for granted because of the preliminaries to the lemma. We change slightly TK’s notation in what follows and supplement their presentation with necessary details.
- 8.
[A(m)](Ai(o)) means the morphism A(m) acting on the object Ai(o).
- 9.
The gestalt behind the inductive procedure applied to define the sequences is the gestalt of frames, which appears in Borel-like proofs of CBT.
- 10.
There exists such, otherwise the category is a Brandt category.
- 11.
A, B are built through two applications of the operation of disjoint union on the functor Hom K (A, ) (http://en.wikipedia.org/wiki/Disjoint_union).
- 12.
The first clause that K is non-Brandt, is redundant because of the second clause about μ.
- 13.
We detail and slightly change TK’s proof.
- 14.
- 15.
- 16.
See Banaschewski-Brümmer 1986 and Sect. 35.7. There are many papers that discuss CBT for various structures. Jan Jakubík should be mentioned who, since 1973, published a number of papers on CBT for various types of lattices and related structures (cf. Jakubík 2002). His work gained a following, especially since the late 1990s, which expanded the discussion. Cf., De Simone et al. 2003, Ionascu 2006, Galego 2010, and the bibliography in those publications.
- 17.
See remarks 9, 11, 12 in the latter mentioned site on generalizations of the proofs of CBT. Some of the remarks reflects views that we share.
- 18.
In categories for which CBT fails, is there any meaning in speaking of objects that fulfill the conditions of CBT as having some kind of an equivalence relation? This suggestion may be inline with Tarski’s comment cited here.
- 19.
Tarski gave no proofs of the theorems in his 1928 paper.
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Hinkis, A. (2013). CBT in Category Theory. In: Proofs of the Cantor-Bernstein Theorem. Science Networks. Historical Studies, vol 45. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0224-6_39
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