Kosta DoSen, Zoran Petrić “Proof-Net Categories"
Polimetrica, International Scientific Publisher | 2007-12-11 | ISBN:8876990801 | PDF | 156
pages | 1 Mb
Star-autonomous categories are a brand of symmetric monoidal closed categories of particular
interest for classical linear logic. This work formulates equationally a precise notion of
star-autonomous category without unit objects, which is called proof-net category. A coherence
theorem analogous to the coherence theorem for symmetric monoidal closed categories with
respect to graphs is proved for proof-net categories. It is also proved that the free proof-net
category generated by a set of objects is isomorphic to a full subcategory of the free
star-autonomous category generated by the same set of objects. This yields a very useful
coherence theorem for star-autonomous categories involving the unit objects, exactly analogous
to the coherence theorem for symmetric monoidal closed categories. An analogous coherence
result is proved also for proof-net categories with the mix principle of linear logic. The graphs
involved in these coherence theorems are the relevant portions of proof nets that one needs to
solve the question whether a diagram of arrows commutes. Proofs are inspired by methods of
proof theory. The results of this work are of interest for general proof theory. They show how
generality of proofs provides a criterion for identity of proofs in a fragment of linear logic. They
also make a contribution to the study of coherence in symmetric monoidal closed categories.
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