nLab category of being




Traditionally in ontology/metaphysics, a category of being is, vaguely, a “kind or way of being” (e.g WP).

In (Lawvere 91) is a proposal for a formalization of this idea. Lawvere starts from the metaphysics as laid out in (Hegel 12) and first observes that the concept of unity of opposites which controls the discussion there is usefully formalized by the concept of adjoint modalities. In a way the simplest example of such is that given by the monad constant on the terminal object

(*). (\emptyset \dashv \ast) \,.

One observes that under the identification of Hegelian unity of opposites with adjoint modalities, this adjoint pair serves well as a formalization of what (Hegel 12) calls the moments of Nichts (nothing) and reines Sein (pure being).

Therefore any other adjoint modality of the form

\Box \dashv \bigcirc

which necessarily includes the former (as an inclusion of modal types)

* \array{ \emptyset &\subset& \Box \\ \bot && \bot \\ \ast &\subset& \bigcirc }

formalizes a more “determinate” being ( Dasein , Fürsichsein in the language of (Hegel 12)). See also at Aufhebung.


Therefore, following (Lawvere 91, p. 7):


A category of being is a category with initial object and terminal object and equipped with an adjoint modality \Box \dashv \bigcirc, hence with an idempotent monad \bigcirc on the category which has a left adjoint \Box.


In geometric language these are categories equipped with a notion of discrete objects and codiscrete objects.


Examples include cohesive toposes, and these are the examples considered in (Lawvere 91).


Last revised on November 26, 2014 at 21:30:44. See the history of this page for a list of all contributions to it.