nLab Hegelian taco

The Hegelian taco is food for thought from William Lawvere's kitchen. More concretely, it is an eight-element graphic monoid that intends to capture diagrammatically the essence of a certain configuration of stacking subcategories of a given category that occurs in Lawvere’s mathematical rendering of Hegel's dialectical logic.

Here every column of arrows, numbered descendingly 2,1,02, 1, 0 from left to right describes an essential localization j n⊣f n⊣i nj_n\dashv f_n\dashv i_n with j n,i nj_n,i_n fully faithful. These yield a pair of idempotent adjoint modalities whose functor parts are j nf n⊣i nf nj_n f_n\dashv i_n f_n. Importantly, all the junctures vanish in the sense f ni n=1f_n i_n=1, since this composition corresponds to the inclusion of a subcategory followed by a projection back to the subcategory.

If we compose the functors appropriately we obtain four such adjunctions on 𝒜\mathcal{A} which correspond to reflective and coreflective embeddings of descending complexity of the chain of subcategories 𝒜⊃ℬ⊃𝒞⊃1\mathcal{A}\supset\mathcal{B}\supset\mathcal{C}\supset 1:

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