Category Theory: "Think bigger thoughts"

Category_Theory.markdown

http://www.cs.ox.ac.uk/people/bob.coecke/AbrNikos.pdf

Why study categories—what are they good for? We can offer a range of answers for readers coming from different backgrounds:

• For mathematicians: category theory organises your previous mathematical experience in a new and powerful way, revealing new connections and structure, and allows you to “think bigger thoughts”.
• For computer scientists: category theory gives a precise handle on important notions such as compositionality, abstraction, representationindependence, genericity and more. Otherwise put, it provides the fundamental mathematical structures underpinning many key programming concepts.
• For logicians: category theory gives a syntax-independent view of the fundamental structures of logic, and opens up new kinds of models and interpretations.
• For philosophers: category theory opens up a fresh approach to structuralist foundations of mathematics and science; and an alternative to the traditional focus on set theory
• For physicists: category theory offers new ways of formulating physical theories in a structural form. There have inter alia been some striking recent applications to quantum information and computation.

https://www.cs.ox.ac.uk/files/3395/PRG72.pdf

• Formulating definitions and theories. In fields that are not yet very well developed like computing science, it often seems that formulating basic concepts is the most difficult part of research. The five guidelines given below provide relatively explicit measures of elegance and coherence that can be helpful in this regard.
• Carrying out proofs. Once basic concepts have been correctly formulated in a categorical language, it often seems that proofs "just happen": at each step, there is a "natural" thing to try, and it works. Diagram chasing provides nice many examples of this. It could almost be said that the purpose of category theory is to reduce all proofs to such simple calculations.
• Discollering and exploiting relations with other fields. Finding similar structures in different areas suggests trying to find further similarities. For example, an analogy between Petri nets and the lambda-calculus might suggest looking for a closed category structure on nets.
• Formulating conjectures and research directions. For example, if you have found an interesting functor, you might be well advised to investigate its adjoints.
• Dealing with abstraction and representation independence. In computing science, abstract viewpoints are often better, because of the need to achieve independence from the often overwhelmingly complex details of how things are represented or implemented. A corollary of the first guideline is that two objects are "abstractly the same" iff they are isomorphic. Moreover, universal constructions (i.e., adjoints) define their results uniquely up to isomorphism, i.e., abstractly in just this sense.

http://emorehouse.web.wesleyan.edu/research/notes/intro_categorical_semantics.pdf

Remark 1.1.0.5 (unbiased presentation) There is an equivalent presentation of
categories in terms of unbiased composition. There, instead of a single binary
composition operation acting on a compatible pair of arrows, we have a lengthindexed
composition operation for paths of arrows (still with unit and associative
laws). In this presentation, an identity morphism is a nullary composition, a
morphism itself is a unary composition, and in general, any length n path of
arrows has a unique composite. Although more cumbersome to axiomatize, an
unbiased presentation of categories makes it easier to appreciate the idea at the
heart of the definition: every composable configuration of things should have a
unique composite.

http://www.cs.man.ac.uk/~hsimmons/zCATS.pdf

These simple examples tend to give the impression that in any category
an object is a structured set and an arrow is a function of a certain
kind. This is a false impression, and in Section 1.3 we look at some
examples to illustrate this. In particular, these examples show that an
arrow need not be a function (of the kind you first thought of).
An important messages of category theory is that the more important
part of a category is not its objects but the way these are compared, its
arrows. Given this we might expect that a category is named after its
arrows. For historical reasons this often doesn’t happen

https://books.google.co.uk/books?id=6B9MDgAAQBAJ&pg=PA62&lpg=PA62&dq=%22definite+article%22+%22category+theory%22&source=bl&ots=AuNUdpyn3Z&sig=cIkBBe7o4hD4TtX0EaT5Hq758ZE&hl=en&sa=X&ved=0ahUKEwidwb-R-4vXAhVEaVAKHSLZCnUQ6AEIJjAA#v=onepage&q=%22definite%20article%22%20%22category%20theory%22&f=false

There may be many isomorphisms between the objects x and y appearing in the proof of ..., but there is a unique natural isomorphism commuting with the chosen representation. On account of this, one typically refers to /the/ representing object of a representable functor. Category theorists often use the definite article "the" in contexts where the object in question is well-defined up to canonical isomorphism.

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https://ncatlab.org/nlab/show/generalized+the

https://byorgey.wordpress.com/2014/05/13/unique-isomorphism-and-generalized-the/