Abstract
Category theory and, more specifically, topos categories provide a more expressive type of mathematical modeling and, thereby, open the door to social models that are both rigorous and expressive. The present analysis draws upon four views of topos categories to construct a rich model of a logic based on recognition theory. The topos initially provides support for: (1) set theory (and classical logic), (2) topological regions dualism. The resulting structures support: (3) a local intuitionist logic that can vary by situated circumstances and actor types, and (4) the specification of regions that are defined by finely-differentiated classifiers. These tools construct an integrated topos category of social recognition that supports diverse forms of local logic. The latter are then explored regarding their contributions to a mathematical model of social conflict. Categorial analysis of historical patterns, as well as a complementary simulation model, is used to illustrate the advantages of such an approach.
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Notes
- 1.
Toposes and topoi are both used as a plural of topos. In this discussion, the latter is preferred.
- 2.
To show how mathematically expressive definitions of topoi have become, Johnstone (2002: vii–viii) lists the seven descriptions of topoidal categories first assembled by André Joyal): (1) a category of sheaves on a site, (2) a category with finite limits and power objects, (3) an intuitionistic higher-order theory, (4) a first-order (infinitary) geometric theory, (5) a totally cocomplete object in the meta-2-category of Cartesian categories, (6) a generalized space, and (7) a semantics for intuitionistic formal systems. He then notes that six additional definitions have been formulated since the initial list was formulated: (8) a Morita equivalence class of continuous groupoids, (9) the category of maps of a power allegory, (10) a category whose canonical indexing over itself is complete and well-powered, (11) the spatial manifestation of a Giraud frame, (12) a setting for synthetic differential geometry, and (13) a setting for synthetic domain theory. Formally, these definitions overlap each other to some extent, but they also illustrate how diverse are the contributions that topoi can make.
- 3.
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Sallach, D.L. (2015). Topos Modeling of Social Conflict: Theory and Methods. In: Nakai, Y., Koyama, Y., Terano, T. (eds) Agent-Based Approaches in Economic and Social Complex Systems VIII. Agent-Based Social Systems, vol 13. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55236-9_4
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