论文信息 - Lifting as a KZ-Doctrine

Lifting as a KZ-Doctrine

In a cartesian closed category with an initial object and a dominance that classifies it, an intensional notion of approximation between maps —the path relation (c.f. link relation)— is defined. It is shown that if such a category admits strict/upper-closed factorisations then it preorderenriches (as a cartesian closed category) with respect to the path relation. By imposing further axioms we can, on the one hand, endow maps and proofs of their approximations (viz. paths) with the 2-dimensional algebraic structure of a sesqui-category and, on the other, characterise lifting as a preorder-enriched lax colimit. As a consequence of the latter the lifting (or partial map classifier) monad becomes a KZ-doctrine.

Marcelo P. Fiore

[1] Eugenio Moggi,et al. Categories of Partial Morphisms and the lambdap - Calculus , 1985, CTCS.

[2] A. Kock. Monads for which Structures are Adjoint to Units , 1995 .

[3] Marcelo P. Fiore. Axiomatic domain theory in categories of partial maps , 1994 .

[4] J. Hyland. First steps in synthetic domain theory , 1991 .

[5] Marcelo P. Fiore. Order-Enrichment for Categories of Partial Maps , 1995, Math. Struct. Comput. Sci..

[6] Paul Taylor,et al. The fixed point property in synthetic domain theory , 1991, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science.

[7] Ronald Brown,et al. Topology, A Geometric Account Of General Topology, Homotopy Types And The Fundamental Groupoid , 1988 .

[8] Gordon D. Plotkin,et al. An axiomatisation of computationally adequate domain theoretic models of FPC , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[9] Anders Kock. Algebras for the partial map classifier monad , 1991 .

[10] A. Kock. Monads on symmetric monoidal closed categories , 1970 .