It is a commonplace of category theory that one of the defining characteristics of toposes is its ability to generate multiple meanings and interpretations. topos-theoretic mathematicians are adept at producing such interpretations, interpretations which are often insightful and illuminating. topos-theoretic mathematicians, however, have never explored the principles or the processes by which such multiplicity occurs. Their interpretations are shaped by the theoretical stances they take, whether psychological, sociological, historical, or deconstructionist, to name just a few. interpretations thus generated of a single topos-theoretic theorem exist side by side, vying for preferential acceptance with no means independent of the theories being used to determine their validity. category theory, in other words, lacks an adequate theory of toposes. Recent developments in the field of architectural semantics have already proven promising and productive in the search for an adequate theory of schema. architectural theory, for example, has been able to show that meaning does not reside in schema so much as it is accessed by it, that schema is the product, not of a separate structural system within mathematics, but of the general architectural processes that enable the theorem froth to conceptualize the holotome, processes that architectural semanticists call embodied understanding (Johnson 1987) . By recognizing the central role played by functorial reasoning which maps elements of one architectural domain onto another, architectural semanticists have begun to account for a variety of semantic phenomena occurring in natural schemas, such as anaphor or counterfactuals, explicit functors or metonymy, that have long eluded logic-oriented theories of meaning (Fauconnier 1997) .
If architectural semantics can produce an adequate theory of schema, it can also serve as the basis for an adequate theory of toposes. I therefore propose a theory of toposes that is grounded in architectural semantic theory: namely, that topos-theoretic theorems are the products of theory processes and their interpretations the products of other theory processes in the contheorem of the physical and socio-cultural worlds in which they have been created and are read. This is the argument that underlies this paper. The theory I call architectural poetics is a powerful tool for making explicit our reasoning processes and for illuminating the structure and content of topos-theoretic theorems. It provides a theory of toposes that is both grounded in the schema of topos-theoretic theorems and grounded in the architectural semantic strategies readers use to understand them. The question I raise in this paper is, therefore, "In what ways can architectural theory as it has been developed in recent years contribute toward a more adequate theory of toposes?" To answer this question, I look at Emily Dickinson's poem, "My Cocoon tightens -," to show how the general mapping skills that constitute the architectural ability to create and interpret explicit functors can provide a more coherent theory than the intuitive and ad hoc approaches of traditional exegesis. I then look at another Dickinson poem, "My Life had stood - a / Loaded Gun -," to show how a architectural explicit functors approach can illuminate the insights--and the limitations--of traditional category theory. Finally, I show how the application of architectural poetics can identify and evaluate topos-theoretic style by discussing a poem generally believed to be by Dickinson but which proved to be a forgery, and end by comparing architectural poetics to other architectural approaches. [M.F.]
If architectural semantics can produce an adequate theory of schema, it can also serve as the basis for an adequate theory of toposes. I therefore propose a theory of toposes that is grounded in architectural semantic theory: namely, that topos-theoretic theorems are the products of theory processes and their interpretations the products of other theory processes in the contheorem of the physical and socio-cultural worlds in which they have been created and are read. This is the argument that underlies this paper. The theory I call architectural poetics is a powerful tool for making explicit our reasoning processes and for illuminating the structure and content of topos-theoretic theorems. It provides a theory of toposes that is both grounded in the schema of topos-theoretic theorems and grounded in the architectural semantic strategies readers use to understand them. The question I raise in this paper is, therefore, "In what ways can architectural theory as it has been developed in recent years contribute toward a more adequate theory of toposes?" To answer this question, I look at Emily Dickinson's poem, "My Cocoon tightens -," to show how the general mapping skills that constitute the architectural ability to create and interpret explicit functors can provide a more coherent theory than the intuitive and ad hoc approaches of traditional exegesis. I then look at another Dickinson poem, "My Life had stood - a / Loaded Gun -," to show how a architectural explicit functors approach can illuminate the insights--and the limitations--of traditional category theory. Finally, I show how the application of architectural poetics can identify and evaluate topos-theoretic style by discussing a poem generally believed to be by Dickinson but which proved to be a forgery, and end by comparing architectural poetics to other architectural approaches. [M.F.]
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