It looks like you're new here. If you want to get involved, click one of these buttons!

- All Categories 2.4K
- Chat 502
- Study Groups 21
- Petri Nets 9
- Epidemiology 4
- Leaf Modeling 2
- Review Sections 9
- MIT 2020: Programming with Categories 51
- MIT 2020: Lectures 20
- MIT 2020: Exercises 25
- Baez ACT 2019: Online Course 339
- Baez ACT 2019: Lectures 79
- Baez ACT 2019: Exercises 149
- Baez ACT 2019: Chat 50
- UCR ACT Seminar 4
- General 73
- Azimuth Code Project 110
- Statistical methods 4
- Drafts 10
- Math Syntax Demos 15
- Wiki - Latest Changes 3
- Strategy 113
- Azimuth Project 1.1K
- - Spam 1
- News and Information 148
- Azimuth Blog 149
- - Conventions and Policies 21
- - Questions 43
- Azimuth Wiki 718

Options

## Comments

Definition 3.23 defines

Setto be the category with sets as objects and functions as morphisms. Definition 3.34 defines functors as a function connecting the sets of objects and a set of functions connecting the sets of morphisms (one function for each pair of objects).Since there is only one object in

1, each functor \(F:\textbf{1}\to\textbf{Set}\) must select a single set out of \(\mathrm{Ob}(\textbf{Set})\) to be the image of this object under \(F\). As there is only one morphism, each functor must select a single function between sets inSet. Since functors must preserve identity morphisms, the image of the identity morphism must be the identity function on the set selected by \(F\). These constitute the entirety of the constraints on functors from1toSet, so there are exactly as many functors from1toSetas there are sets. Thus, one may index these functors by the set they map to: for every set \(S\in\mathrm{Ob}(\textbf{Set})\), there is a (unique) functor \(F_S:\textbf{1}\to\textbf{Set}\) such that \(F_S(1)=S\) and \(F_S(\mathrm{id}_1)=\mathrm{id}_S\) (\(\forall s\in S\,\,\mathrm{id}_S(s)=s\)).`Definition 3.23 defines **Set** to be the category with sets as objects and functions as morphisms. Definition 3.34 defines functors as a function connecting the sets of objects and a set of functions connecting the sets of morphisms (one function for each pair of objects). Since there is only one object in **1**, each functor \\(F:\textbf{1}\to\textbf{Set}\\) must select a single set out of \\(\mathrm{Ob}(\textbf{Set})\\) to be the image of this object under \\(F\\). As there is only one morphism, each functor must select a single function between sets in **Set**. Since functors must preserve identity morphisms, the image of the identity morphism must be the identity function on the set selected by \\(F\\). These constitute the entirety of the constraints on functors from **1** to **Set**, so there are exactly as many functors from **1** to **Set** as there are sets. Thus, one may index these functors by the set they map to: for every set \\(S\in\mathrm{Ob}(\textbf{Set})\\), there is a (unique) functor \\(F_S:\textbf{1}\to\textbf{Set}\\) such that \\(F_S(1)=S\\) and \\(F_S(\mathrm{id}_1)=\mathrm{id}_S\\) (\\(\forall s\in S\,\,\mathrm{id}_S(s)=s\\)).`