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Abstract:
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Central place in this thesis occupy the coherence results for certain types of closed
categories. Coherence results in category theory usually serve to provide a simple decision
procedure for equality of arrows in some category. The approach to coherence that we follow
here implies the existence of a faithfull functor from a freely generated category A of certain
type to the category B in which an equality of arrows can be easily checked. Category B,
which is of the same type as A, usually represents formalisation of some graphical language.
Besides coherence, the second most important notion we consider in this thesis is the
biproduct. The notion of biproduct in a category incorporates notions of coproduct and
product. The main results in this thesis are coherence theorems for three types of closed
categories with biproducts – symmetric monoidal closed categories with biproducts, com-
pact closed categories with biproducts and dagger compact closed categories with dagger
biproducts.
Further, we present a new proof of the well-known Kelly-Mac Lane coherence theorem
for symmetric monoidal closed categories. The methods we use in that proof are completely
proof-theoretical, and one of the key elements in it is the cut-elimination theorem. In all the
above coherence results, the graphical language is based on the category of one-dimensional
cobordisms.
Finaly, we give certain criteria for existence of biproducts in monoidal categories. In this
regard, we rely on recent research that characterizes certain type of monoidal categories with
finite biproducts by using the existence of right duals of some distinguished objects. Our
criteria are a generalization of this result. |