# Week 5 Notes

• EX 3.37: From EG 3.36:

1. $$F(m_0) = n_0$$, $$F(m_1) = n_0$$, $$F(f_1) = \text{id}_0$$
2. $$F(m_0) = n_0$$, $$F(m_1) = n_1$$, $$F(f_1) = g_1$$
3. $$F(m_0) = n_0$$, $$F(m_1) = n_2$$, $$F(f_1) = g_2 \circ g_1$$

The other three functors:

1. $$F(m_0) = n_1$$, $$F(m_1) = n_1$$, $$F(f_1) = \text{id}_1$$
2. $$F(m_0) = n_1$$, $$F(m_1) = n_2$$, $$F(f_1) = g_2$$
3. $$F(m_0) = n_2$$, $$F(m_1) = n_2$$, $$F(f_1) = \text{id}_2$$
• EX 3.45: For any set $$S$$ in $$\mathbf{Set}$$, define the functor $$F_S : \mathbf{1} \rightarrow \mathbf{Set}$$ as $$F(1) = S$$, $$F(id_1) = id_S$$, where $$id_S$$ is the identity set-function on $$S$$.
• EX 3.48:
1. $$Z := \mathbf{Z} / 2 \mathbf{Z}$$, and $$S := - + _2$$
2. $$A = C := \mathbf{Z} / 2 \mathbf{Z}$$, $$B := \mathbf{Z} / 4 \mathbf{Z}$$, and $$F := id$$, $$G := - \mid 2$$, $$H(b) := (b == 1)$$
• EX 3.55:
1. For functors $$F, G, H$$ and natural transformations $$\alpha : F \rightarrow G$$, $$\beta : G \rightarrow H$$ in $$\mathcal{D}^\mathcal{C}$$, define the composed natural transformation $$\beta \circ \alpha : F \Rightarrow H$$, specifying

1. for each object $$c \in \mathcal{C}$$, a c-component $$(\beta \circ \alpha)_c = \beta_c \circ \alpha_c : F(c) \rightarrow H(c)$$ in $$\mathcal{D}$$

check that the components satisfy the naturality condition, for every morphism $$f : c \rightarrow c'$$ in $$\mathcal{C}$$. Indeed $\begin{equation}\begin{split} & &&\alpha_c' \circ F(f) = G(f) \circ \alpha_c\ \text{and}\ \beta_c' \circ G(f) = H(f) \circ \beta_c \\ &\implies &&\beta_c' \circ (\alpha_c' \circ F(f)) = \beta_c' \circ (G(f) \circ \alpha_c)\ \text{and}\ (\beta_c' \circ G(f)) \circ \alpha_c = (H(f) \circ \beta_c) \circ \alpha_c &&&\text{(multiplication on both sides)} \\ &\implies &&(\beta \circ \alpha)_c' \circ F(f) = H(f) \circ (\beta \circ \alpha)_c &&&\text{(D 3.6 (b) used multiple times)} \\ \end{split}\end{equation}$

2. For any object $$F \in \mathcal{D}^\mathcal{C}$$, the identity natural transformation $$\alpha : F \Rightarrow F$$ specifies the c-component of $$\alpha$$ as the identity morphism, such that the naturality condition $$\text{id}_c' \circ F(f) = G(f) \circ \text{id}_c$$ for every morphism $$f : c \rightarrow c'$$ in $$\mathcal{C}$$. The identity natural transformation is unital, as the naturality condition corresponds exactly to D 3.6 (a).
• EX 3.58: For arbitrary category $$\mathcal{C}$$ and preorder considered as a category $$\mathcal{P}$$
1. For any two functors $$F,G : \mathcal{C} \rightarrow \mathcal{P}$$, for any object $$c \in \mathcal{C}$$, there is at most one c-component $$\alpha_c : F(c) \rightarrow G(c)$$, corresponding to the single morphism between two objects in a preorder. Hence, there is at most one natural transformation $$F \Rightarrow G$$.
2. For some two functors $$F,G : \mathcal{P} \rightarrow \mathcal{C}$$, there can be multple natural transformations $$F \Rightarrow G$$, since for any object $$p \in \mathcal{P}$$, there can be multiple c-components $$\alpha_c : F(p) \rightarrow G(p)$$, each sending the single morphism in $$\mathcal{P}$$ to one of the corresponding morphisms in $$\mathcal{C}$$. Here are two counterexamples
1. Let $$\mathcal{P}$$ be the preorder with one element $$\ast$$, and $$\mathcal{C}$$ be schema 1 of EX 3.48. Then, for functors $\begin{equation}\begin{split} &F(\ast) = z \\ &G(\ast) = z \\ \end{split}\end{equation}$ and there are two possible natural transformations: either $$\alpha_\ast = \text{id}_z$$, or $$\alpha_\ast = s$$.
2. Let $$\mathcal{P} := (\text{Bool}, \leq)$$ and $$\mathcal{C}$$ be schema 2 of EX 3.48. Then, for functors $\begin{equation}\begin{split} &F(\texttt{false}) = b, F(\texttt{true}) = b, F(\texttt{false} \rightarrow \texttt{true}) = id_b \\ &G(\texttt{false}) = c, G(\texttt{true}) = c, G(\texttt{false} \rightarrow \texttt{true}) = id_c \\ \end{split}\end{equation}$ and there are two possible natural transformations: either $$\alpha_\texttt{false} = g$$ and $$\alpha_\texttt{true} = h$$, or $$\alpha_\texttt{false} = h$$ and $$\alpha_\texttt{true} = g$$.
• EX 3.73:
1. Given a morphism $$f : X \rightarrow Y$$, the functor $$(- \times B) : X \times B \rightarrow Y \times B$$ returns the morphism from the product of sets $$X$$ and $$B$$ to the product of sets $$Y$$ and $$B$$.
2. Given a morphism $$f : X \rightarrow Y$$, the functor $$(-)^B : X^B \rightarrow Y^B$$ returns the morphism from the exponential object $$X^B$$ to the exponential object $$Y^B$$.
3. Let $$+ : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$$. Let the adjunction currying $$\mathbb{N}$$ be $$p : \mathbb{N} \rightarrow \mathbb{N}^\mathbb{N}$$. Then $$p(3) = 3 + -$$.
• EX 3.76: $$! : \mathcal{C} \rightarrow \mathbf{1}$$ sends all objects $$c \in \mathcal{C}$$ to the only object in $$\mathbf{1}$$, that is, to $$1$$. $$!$$ sends all morphisms $$f : c \rightarrow c'$$ in $$\mathcal{C}$$ to the only morphism in $$\mathbf{1}$$, that is, to $$\text{id}_1$$.
• EX 3.81:
• $$(\Rightarrow)$$: if $$z$$ is a final object in the category $$\mathcal{P}$$ corresponding to the preorder $$(P, \leq)$$, then for all $$c \in P$$, there exists a unique morphism $$! : c \rightarrow z$$ (D 3.79). By taking the preorder reflection $$(P, \leq)$$ of the category $$\mathcal{P}$$, the morphism $$!$$ becomes the inequality $$c \leq z$$, and so $$c$$ is a top element in $$\mathcal{P}$$.
• $$(\Leftarrow)$$: if $$c$$ is a top element in $$(P, \leq)$$, then we have $$c \leq z$$ for all $$c \in P$$, and there exists a unique morphism $$! : c \rightarrow z$$ in the category $$\mathcal{P}$$ corresponding to $$(P, \leq)$$, and so $$z$$ is a final object in the category (D 3.79).
• EX 3.82: $$\mathbf{1}$$ is a final object in $$\mathbf{Cat}$$, since for all categories $$\mathcal{C} \in \mathbf{Cat}$$, there is a unique morphism $$! : \mathcal{C} \rightarrow \mathbf{1}$$.
• EX 3.83: The discrete category with more than one object does not have a final object.
• EX 3.88: The product of two objects $$x, y$$ in the category $$\mathcal{P}$$ corresponding to the preorder $$(P, \leq)$$ is the object $$x \times y$$ together with morphisms $$p_x : x \times y \rightarrow x$$ and $$p_y : x \times y \rightarrow y$$ such that for all objects $$c \in \mathcal{P}$$ together with morphisms $$f : c \rightarrow x$$ and $$g : c \rightarrow y$$, there exists a unique morphism $$: c \rightarrow x \times y$$, such that the diagram in D 3.86 commutes. So considering the preorder reflection of $$\mathcal{P}$$, we have $$x \times y \leq x$$, $$x \times y \leq y$$, and $$c \leq x \times y$$. Replacing all $$\times$$ with $$\vee$$ gives the definition of a meet (D 1.81).

Created: 2020-01-13 Mon 14:36