Preface to the second edition (2005) Preface to the first edition (1981) Part I. The Synthetic Ttheory: 1. Basic structure on the geometric line 2. Differential calculus 3. Taylor formulae - one variable 4. Partial derivatives 5. Taylor formulae - several variables 6. Some important infinitesimal objects 7. Tangent vectors and the tangent bundle 8. Vector fields 9. Lie bracket 10. Directional derivatives 11. Functional analysis - Jacobi identity 12. The comprehensive axiom 13. Order and integration 14. Forms and currents 15. Currents - Stokes' theorem 16. Weil algebras 17. Formal manifolds 18. Differential forms in terms of simplices 19. Open covers 20. Differential forms as quantities 21. Pure geometry Part II. Categorical Logic: 1. Generalized elements 2. Satisfaction (1) 3. Extensions and descriptions 4. Semantics of function objects 5. Axiom 1 revisited 6. Comma categories 7. Dense class of generators 8. Satisfaction (2) 9. Geometric theories Part III. Models: 1. Models for axioms 1, 2, and 3 2. Models for epsilon-stable geometric theories 3. Well-adapted models (1) 4. Well-adapted models (2) 5. The algebraic theory of smooth functions 6. Germ-determined T-infinity-algebras 7. The open cover topology 8. Construction of well-adapted models 9. Manifolds with boundary 10. Field property - germ algebras 11. Order and integration in cahiers topos Appendices Bibliography Index.

Properties of dense and relative adjoint functors

Synthetic Differential Geometry

/pdf/homology-and-standard-constructions-3lvn8hje9h.pdf

Homology and standard constructions

Yoneda structures on 2-categories

Algebraic theories, introduced as a concept in the 1960s, have been a fundamental step towards a categorical view of general algebra. Moreover, they have proved very useful in various areas of mathematics and computer science. This carefully developed book gives a systematic introduction to algebra based on algebraic theories that is accessible to both graduate students and researchers. It will facilitate interactions of general algebra, category theory and computer science. A central concept is that of sifted colimits - that is, those commuting with finite products in sets. The authors prove the duality between algebraic categories and algebraic theories and discuss Morita equivalence between algebraic theories. They also pay special attention to one-sorted algebraic theories and the corresponding concrete algebraic categories over sets, and to S-sorted algebraic theories, which are important in program semantics. The final chapter is devoted to finitary localizations of algebraic categories, a recent research area.

Algebraic Theories: A Categorical Introduction to General Algebra

The language of
 ∞-categories provides an insightful new way of expressing many
 results in higher-dimensional mathematics but can be challenging
 for the uninitiated. To explain what exactly an ∞-category is
 requires various technical models, raising the question of how
 they might be compared. To overcome this, a model-independent
 approach is desired, so that theorems proven with any model
 would apply to them all. This text develops the theory of
 ∞-categories from first principles in a model-independent
 fashion using the axiomatic framework of an ∞-cosmos, the
 universe in which ∞-categories live as objects. An ∞-cosmos is a
 fertile setting for the formal category theory of ∞-categories,
 and in this way the foundational proofs in ∞-category theory
 closely resemble the classical foundations of ordinary category
 theory. Equipped with exercises and appendices with background
 material, this first introduction is meant for students and
 researchers who have a strong foundation in classical 1-category
 theory.

Properties of dense and relative adjoint functors

Citations

Synthetic Differential Geometry

Homology and standard constructions

Yoneda structures on 2-categories

Algebraic Theories: A Categorical Introduction to General Algebra

Elements of ∞-Category Theory

References

Representable functors with values in arbitrary categories

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