Continuum mechanics and geometric integration theory. It treats material which will appear simple and familiar to many philosophers, but in an unfamiliar way. If you have students you wish to introduce to the subject, I suggest this delightfully elementary book. Lawvere is one of the greatest visionaries of mathematics in the last half of the twentieth century. He characteristically digs down beneath the foundations of a concept in order to simplify its understanding.
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This truly elementary book on categories introduces retracts, graphs, and adjoints to students and scientists. Conceptual Mathematics introduces this tool for the learning, development, and use of mathematics, to beginning students and also to practising mathematical scientists. This book provides a skeleton key that makes explicit some concepts and procedures that are common to all branches of pure and applied mathematics. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories.
This second edition provides links with more advanced topics of possible study. In the new appendices and annotated bibliography the reader will find concise introductions to adjoint functors and geometrical structures, as well as sketches of relevant historical developments. The Category of Sets: 1. Sets, maps, composition; Part II. The Algebra of Composition: 2. Isomorphisms; Part III. Categories of Structured Sets: 3.
Examples of categories; Part IV. Elementary Universal Mapping Properties: 4. Universal mapping properties; Part V. Higher Universal Mapping Properties: 5. Map objects; 6. The contravariant parts functor; 7. The components functor; Appendix 1. Geometry of figures and algebra of functions; Appendix 2. Adjoint functors; Appendix 3. The emergence of category theory within mathematics; Appendix 4.
Annotated bibliography. Lawvere, Stephen H. Schanuel Schanuel, Stephen H. Stephen H.
Conceptual mathematics - a first introduction to categories
F. William Lawvere