nLab Unity and Identity of Opposites in Calculus and Physics


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  • William Lawvere:

    Unity and Identity of Opposites in Calculus and Physics

    Proceedings of ECCT 1994 Tours Conference,

    Applied Categorical Structures 4 167-174 (1996)

    Kluwer Academic Publishers



on the formalization of unity of opposites in calculus and physics.

Section II re-casts the statement of the Hadamard lemma via the three homomorphisms of rings of smooth functions on a smooth manifold XX and its product manifold X×XX \times X (the article generalizes this to fiber products but that’s not necessary to appreciate the idea):

where pr i:X×XXpr_i \,\colon\, X \times X \xrightarrow{\;} X are the two projections out of the Cartesian product and diag:XX×Xdiag \,\colon\, X \xrightarrow{\;} X \times X denotes the diagonal map. So:

  • for fC (X)f \,\in\, C^\infty(X) we have

    pr 1 *(f):(x 1,x 2)f(x 1),pr 2 *(f):(x 1,x 2)f(x 2) pr_1^\ast(f) \,\colon\, (x_1, x_2) \,\mapsto\, f(x_1) ,\, \;\;\;\; pr_2^\ast(f) \,\colon\, (x_1, x_2) \,\mapsto\, f(x_2)
  • for gC (X×X)g \,\in\, C^\infty(X \times X) we have

    diag *(g):xg(x,x). diag^\ast(g) \,\colon\, x \,\mapsto\, g(x,x) \,.

These are the expressions that enter the formulation of the Hadamard lemma (here) and its corollary (here).

Compare this to the discussion at adjoint cylinder and then to objective and subjective logic and maybe also Aufhebung.


A significant fraction of dialectical philosophy can be modeled mathematically through the use of “cylinders” (diagrams of shape Δ 1\Delta_1) in a category, wherein the two identical subobjects (united by the third map in the diagram) are “opposite”. In a bicategory, oppositeness can be very effectively characterized in terms of adjointness, but even in an ordinary category it may sometimes be given a useful definition. For example, an effective basis for teaching calculus is a ringed category satisfying the Hadamard-Marx property. The description in engineering mechanics of continuous bodies that can undergo cracking is clarified by an example involving lattices, raising a new questions about the foundations of topology.

(Δ 1\Delta_1 is the category of order-preserving maps between totally-ordered sets of one and two elements, respectively.)

category: reference

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