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There are several generalizations of Mac Lane's coherence theorem. Each of them has the rough form that "every weak structure of some sort is equivalent to a stricter one".
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The adjectives such as "pseudo-" and "lax-" are used to refer to the fact equalities are weakened in coherent ways; e.g.,
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may be replaced by associativity via coherent isomorphisms. For example, via this process, one gets the notion of a
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In some situations, isomorphisms need to be chosen in a coherent way. Often, this can be achieved by choosing
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Replacing coherent isomorphisms by equalities is usually called strictification or rectification.
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In practice, coherent isomorphisms arise by weakening equalities; e.g., strict
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whose combinatorial structure appears implicitly in Mac Lane's proof.
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Cahiers de
Topologie et Géométrie Différentielle Catégoriques
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Standard that diagrams must satisfy up to isomorphism
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may be too technical for most readers to understand
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497:(1994). "Coxeter-associahedra".
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