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1.
Categories (Section 11.1).
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2.
Examples of categories (Section 11.2).
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3.
The quadruple adjunction $\pi _{0}\dashv {(-)_{\mathsf{disc}}}\dashv \operatorname {\mathrm{Obj}}\dashv {(-)_{\mathsf{indisc}}}$ between the category of categories and the category of sets (Section 11.3).
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4.
Groupoids, categories in which all morphisms admit inverses (Section 11.4).
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5.
Functors (Section 11.5).
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6.
The conditions one may impose on functors in decreasing order of importance:
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(a)
Section 11.6 introduces the foundationally important conditions one may impose on functors, such as faithfulness, conservativity, essential surjectivity, etc.
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(b)
Section 11.7 introduces more conditions one may impose on functors that are still important but less omni-present than those of Section 11.6, such as being dominant, being a monomorphism, being pseudomonic, etc.
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(c)
Section 11.8 introduces some rather rare or uncommon conditions one may impose on functors that are nevertheless still useful to explicit record in this chapter.
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(a)
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7.
Natural transformations (Section 11.9).
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8.
The various categorical and 2-categorical structures formed by categories, functors, and natural transformations (Section 11.10).
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•
Fix categories having an underlying set of objects by having them have an underlying setoid of objects (not necessarily by definition, as that’ll likely be bothersome; at least Section 11.3 should be fixed and several remarks should be added at several points). Related: Warning 11.3.1.1.2
11 Categories
This chapter contains some elementary material about categories, functors, and natural transformations. Notably, we discuss and explore:
This chapter is under active revision. TODO:
