Abstract
Roughly speaking a category is a collection of objects and relations between these objects. These relations are required to satisfy certain properties which make the set of all such relations ‘coherent’. Given a category, it is not the case that every two objects have a relation between them, some do and others don’t. For the ones that do, the number of relations can vary depending on which category we are considering.
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Notes
- 1.
By abstract characterisation here we mean a notion that does not depend on the sets or objects between which the arrow is defined.
- 2.
The symbol ∀ means “ for all”.
- 3.
A singleton is a set with only 1 element.
- 4.
A category \(\mathcal{C}\) is called small if \(\mathit{Ob}(\mathcal{C})\) is a Set.
- 5.
Note that one can also have the empty diagram or the diagram with only the identity arrow
- 6.
The index op stands for opposite and it is synonimus to dual.
- 7.
A function between two sets f:A→B is:
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Injective iff f(x)=f(y) implies that x=y for any two elements x,y∈A.
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Surjective iff ∀y∈B there exists an x∈A such that y=f(x).
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Bijective iff f is both injective and surjective.
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- 8.
Note that the choice of terminal object is irrelevant since they are all isomorphic to each other.
- 9.
The proof of existence is obtained by constructing an object and verifying that it satisfies the requirements of being (in this case) a product.
- 10.
Note that an arrow drawn as
indicates uniqueness of that arrow.
- 11.
In this context pointwise order is defined as follows: (A,B)≤(C,D) iff A≤C and B≤D.
- 12.
The definition of an equivalence relation is as follows: given a set A a (binary) equivalence relation on A is a subset R⊆A×A defined as R={(a,b)|a∼ R b}. Thus R represents the set of all pairs which are related by the relation ∼ R . The relation ∼ R has the following properties:
-
1.
Reflexive: for all a∈A, a∼ R a.
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2.
Transitive: if a∼ R b and b∼ R c then a∼ R c.
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3.
Symmetric: if a∼ R b, then b∼ R a.
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1.
- 13.
A set S is said to be a pre-ordered set if it is equipped with a binary relation ≤ which satisfies the following properties:
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Reflexivity: for all a∈S then a≤a.
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Transitive: if a≤b and b≤c then a≤c.
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References
C.J. Isham, J. Butterfield, A topos perspective on the Kochen-Specker theorem: I. Quantum states as generalized valuations (1998). arXiv:quant-ph/9803055
J. Butterfield, C.J. Isham, A topos perspective on the Kochen-Specker theorem: II. Conceptual aspects, and classical analogues (1998). arXiv:quant-ph/9808067
S. MacLane, Categories for the Working Mathematician (Springer, London, 1997)
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Flori, C. (2013). Introducing Category Theory. In: A First Course in Topos Quantum Theory. Lecture Notes in Physics, vol 868. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35713-8_4
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DOI: https://doi.org/10.1007/978-3-642-35713-8_4
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