Abstract
Mathematical categories provide an abstract and general framework for logic and mathematics. As such, they could be used by philosophers in all the basic fields of the discipline: semantics, epistemology and ontology. In this paper, we present the basic definitions and notions and suggest some of the ways categories are starting to infiltrate formal philosophy.
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Notes
- 1.
We should emphasize that this is but one definition and that there are others (equivalent, of course). It very much depends on the background one wants to assume to start with and the goals one has in mind when using categories. If we were to assume that all mathematical entities have to be sets, we would give a slightly different definition. On the other hand, if we were to assume a purely formal set up, a fully specified formal framework, we would give a different definition still. For alternative definitions, see for instance [2, 19, 24].
- 2.
Notice that the latter concept can be defined directly in terms of arrows with no reference to the concept of product. Thus a category can have a terminal object without having all products.
- 3.
The preceding remark concerning the possibility of defining the concept of terminal object directly applies mutatis mutandis to the concept of initial object.
- 4.
This is shocking only if someone sticks firmly to the axiom of extensionality. From a categorical point of view, the axiom of extensionality is not the adequate criterion of identity for abstract sets.
- 5.
Since this object is also defined up to a unique isomorphism, we immediately introduce the standard notation.
- 6.
As far as I know, Bill Lawvere was the first to propose this general definition. For more on its properties, see [18].
- 7.
Notice that we do not have to take the closure of X since in the case of topological spaces, the operations define a coHeyting algebra, i.e. the algebra of closed sets.
- 8.
There are obvious foundational issues arising at this point, but we will simply brush them under the carpet and ignore them altogether.
- 9.
Notice that the functors go from the opposite category C op. This is related to theoretical aspects of category theory that we cannot explain in such a short paper.
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Marquis, JP. (2018). Categories. In: Hansson, S., Hendricks, V. (eds) Introduction to Formal Philosophy. Springer Undergraduate Texts in Philosophy. Springer, Cham. https://doi.org/10.1007/978-3-319-77434-3_11
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