*Steve Awodey*

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568612
- eISBN:
- 9780191717567
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568612.003.0008
- Subject:
- Mathematics, Algebra

This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the fact that it ...
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This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the fact that it is a straightforward generalization of facts that are fairly easily shown in relation to monoids and posets. The topics discussed include set-valued functor categories, Yoneda embedding, limits in categories of diagrams, colimits in categories of diagrams, exponentials in categories of diagrams, and Topoi. Exercises are provided in the last part of the chapter.Less

This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the fact that it is a straightforward generalization of facts that are fairly easily shown in relation to monoids and posets. The topics discussed include set-valued functor categories, Yoneda embedding, limits in categories of diagrams, colimits in categories of diagrams, exponentials in categories of diagrams, and Topoi. Exercises are provided in the last part of the chapter.

*Lisa Irene Hau*

- Published in print:
- 2014
- Published Online:
- September 2014
- ISBN:
- 9780748680108
- eISBN:
- 9780748697007
- Item type:
- chapter

- Publisher:
- Edinburgh University Press
- DOI:
- 10.3366/edinburgh/9780748680108.003.0012
- Subject:
- Classical Studies, Ancient Greek, Roman, and Early Christian Philosophy

This chapter focuses on our surviving historiographic texts from the fifth to the first century BC : Herodotus, Thucydides, Xenophon, Polybius, and Diodorus of Sicily. Greek historiography dealt with ...
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This chapter focuses on our surviving historiographic texts from the fifth to the first century BC : Herodotus, Thucydides, Xenophon, Polybius, and Diodorus of Sicily. Greek historiography dealt with a limited number of types of situations and within these situations was concerned with a limited number of themes. This observation leads to a preliminary typology of such stock events and their concomitant themes, or topoi. There are four other distinct characteristics of Greek historiography: interest in causation, didactic purpose (practical or moral), even-handedness in narrative treatment of opponents in conflict, and alternation between remote, omniscient narration – feigning non-narration – and involved, polemical first-person argumentation. Although none of these traits is in itself uniquely Greek, the specific combination of them seen in these five historiographers is.Less

This chapter focuses on our surviving historiographic texts from the fifth to the first century BC : Herodotus, Thucydides, Xenophon, Polybius, and Diodorus of Sicily. Greek historiography dealt with a limited number of types of situations and within these situations was concerned with a limited number of themes. This observation leads to a preliminary typology of such stock events and their concomitant themes, or *topoi*. There are four other distinct characteristics of Greek historiography: interest in causation, didactic purpose (practical or moral), even-handedness in narrative treatment of opponents in conflict, and alternation between remote, omniscient narration – feigning non-narration – and involved, polemical first-person argumentation. Although none of these traits is in itself uniquely Greek, the specific combination of them seen in these five historiographers is.

*Rocco Gangle*

- Published in print:
- 2015
- Published Online:
- September 2016
- ISBN:
- 9781474404174
- eISBN:
- 9781474418645
- Item type:
- chapter

- Publisher:
- Edinburgh University Press
- DOI:
- 10.3366/edinburgh/9781474404174.003.0007
- Subject:
- Philosophy, Metaphysics/Epistemology

This chapter examines two advanced constructions in category theory: adjoint functors and topoi. Pairs of adjoint functors, or adjunctions, are formally defined and then illustrated with several ...
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This chapter examines two advanced constructions in category theory: adjoint functors and topoi. Pairs of adjoint functors, or adjunctions, are formally defined and then illustrated with several concrete examples. Topoi are introduced conceptually by focusing on the role of the subobject classifier within a topos, with examples from set theory and graph theory. The difference between the classical logic of Boolean algebras and the non-classical logic of Heyting algebras is explained in terms of their natural mathematical environments of set theory and topos theory respectively.Less

This chapter examines two advanced constructions in category theory: adjoint functors and topoi. Pairs of adjoint functors, or adjunctions, are formally defined and then illustrated with several concrete examples. Topoi are introduced conceptually by focusing on the role of the subobject classifier within a topos, with examples from set theory and graph theory. The difference between the classical logic of Boolean algebras and the non-classical logic of Heyting algebras is explained in terms of their natural mathematical environments of set theory and topos theory respectively.