Robert Figura wrote:

> Page 51 may be a good entry point compare Spivak's slides. However, looking at the diagrams on that page I'm having trouble to understand the difference between Gr(I) and I. Mostly because of the label on the arrow (b1,b1,b2). Because of that it really looks as if the diagram I had all the info Gr(I) has.

You're right, Gr(I) has exactly the same information as I, but they're different kinds of mathematical objects. I is a functor from C to Set, while Gr(I) is its own category. In Spivak's pictures on slide 51, the pictures for C and Gr(I) represent categories, where the arrows are just abstract morphisms, while the picture for I itself consists of sets with arrows labeled by functions: a diagram in Set with shape C. Roughly speaking, Gr(I) is the category you get by starting with C, making I(c) copies of each object c in C, and then connecting up those objects by morphisms according to the functions I(f) for each morphism in f. More precisely, objects of Gr(I) are pairs (c, x) where c is an object of C and x is an element of I(c); the set of morphisms from (c, x) to (d, y) is the set of morphisms f in C(c,d) such that I(f): I(c) -> I(d) sends x to y.

**Puzzle**: Use the precise definition to construct the category Gr(I) for the functor I in Spivak's slide, and show that, up to relabeling, it matches the picture given there for Gr(I).

You can think of the difference between Gr(I) and I as the difference between a subset B of a set A and its characteristic function f from A to {0,1}, which sends an element to 1 if it is in B or to 0 if not. Here, Gr(I) is analogous to the subset B, while I is analogous to the characteristic function f.

**Puzzle**: Show that this is more than an analogy: If you start with a subset B of A, form its characteristic function f : A -> {0,1}, and then think of f as a functor I from the discrete category A to Set sending each element to a 0- or 1-element set, then Gr(I) is a discrete category with object set B.

(I would label my puzzles but I've lost track of whether the personal numbering "OB1, OB2, etc." is supposed to reset in each thread or if I should be keeping track of how many puzzles I've posted total.)

> Page 51 may be a good entry point compare Spivak's slides. However, looking at the diagrams on that page I'm having trouble to understand the difference between Gr(I) and I. Mostly because of the label on the arrow (b1,b1,b2). Because of that it really looks as if the diagram I had all the info Gr(I) has.

You're right, Gr(I) has exactly the same information as I, but they're different kinds of mathematical objects. I is a functor from C to Set, while Gr(I) is its own category. In Spivak's pictures on slide 51, the pictures for C and Gr(I) represent categories, where the arrows are just abstract morphisms, while the picture for I itself consists of sets with arrows labeled by functions: a diagram in Set with shape C. Roughly speaking, Gr(I) is the category you get by starting with C, making I(c) copies of each object c in C, and then connecting up those objects by morphisms according to the functions I(f) for each morphism in f. More precisely, objects of Gr(I) are pairs (c, x) where c is an object of C and x is an element of I(c); the set of morphisms from (c, x) to (d, y) is the set of morphisms f in C(c,d) such that I(f): I(c) -> I(d) sends x to y.

**Puzzle**: Use the precise definition to construct the category Gr(I) for the functor I in Spivak's slide, and show that, up to relabeling, it matches the picture given there for Gr(I).

You can think of the difference between Gr(I) and I as the difference between a subset B of a set A and its characteristic function f from A to {0,1}, which sends an element to 1 if it is in B or to 0 if not. Here, Gr(I) is analogous to the subset B, while I is analogous to the characteristic function f.

**Puzzle**: Show that this is more than an analogy: If you start with a subset B of A, form its characteristic function f : A -> {0,1}, and then think of f as a functor I from the discrete category A to Set sending each element to a 0- or 1-element set, then Gr(I) is a discrete category with object set B.

(I would label my puzzles but I've lost track of whether the personal numbering "OB1, OB2, etc." is supposed to reset in each thread or if I should be keeping track of how many puzzles I've posted total.)