Abstract
The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain numerous exercises, and hopefully will prove useful for self-study by those seeking a first introduction to the subject, with fairly minimal prerequisites. The coverage is by no means comprehensive, but should provide a good basis for further study; a guide to further reading is included.
The main prerequisite is a basic familiarity with the elements of discrete mathematics: sets, relations and functions. An Appendix contains a summary of what we will need, and it may be useful to review this first. In addition, some prior exposure to abstract algebra—vector spaces and linear maps, or groups and group homomorphisms—would be helpful.
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Notes
- 1.
A review of basic ideas about sets, functions and relations, and some of the notation we will be using, is provided in Appendix A.
- 2.
We shall use the notation “:=” for “is defined to be” throughout these notes.
- 3.
This would be a “multigraph” in normal parlance, since multiple edges between a given pair of vertices are allowed.
- 4.
The last clause can be replaced by any of the following:
-
… if, for some y not appearing in \(t\,t'\), \((y x) \bullet t =_\alpha (y\,x') \bullet t'\).
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… if, for all y not appearing free in \(t\,t'\), \((y\,x) \bullet t =_\alpha (y\,x') \bullet t'\).
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… if, for some y not appearing free in \(t\,t'\), \((y\,x) \bullet t =_\alpha (y\,x') \bullet t'\).
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- 5.
In some texts, this is called a coKleisli category.
References
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Goldblatt, R.: Topoi, the Categorial Analysis of Logic. North-Holland, Amsterdam (1984). Reprinted by Dover Books, 2006.
Herrlich, H., Strecker, G.: Category Theory, 3rd edn. Heldermann, Berline (2007)
Lambek, J., Scott, P.J.: Introduction to Higher-Order Categorical Logic. Cambridge University Press, Cambridge (1986)
Lawvere, F., Schanuel, S.: Conceptual Mathematics: A First Introduction to Categories. Cambridge University Press, Cambridge (1997)
Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, New York (1998)
Mac Lane, S., Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer, New York (1994)
Pierce, B.: Basic Category Theory for Computer Scientists. MIT Press, Cambridge (1991)
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Abramsky, S., Tzevelekos, N. (2010). Introduction to Categories and Categorical Logic. In: Coecke, B. (eds) New Structures for Physics. Lecture Notes in Physics, vol 813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12821-9_1
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