#38 Homotopy Type Theory Explained: Natural Transformations, Cat. of Functors

1. Formula
   Observation

   The left side of the board lists `F : C -> D`.

2. Audio
   Observation

   The narrator says, "Suppose we have the functors F and G from the category C to the category D."

F

A functor from category C to category D.

Category C

1. Formula
   Observation

   The left side of the board lists `G : C -> D`.

2. Audio
   Observation

   The narrator says, "Suppose we have the functors F and G from the category C to the category D."

G

A functor from category C to category D.

Category C

1. Formula
   Observation

   The diagram shows a vertical line separating categories labeled `C` and `D`.

2. Audio
   Observation

   The narrator says, "from the category C to the category D."

C

The source category containing objects x and y and morphism h.

$N/A$

1. Formula
   Observation

   The diagram shows a vertical line separating categories labeled `C` and `D`.

2. Audio
   Observation

   The narrator says, "from the category C to the category D."

D

The target category containing objects $F(x)$, $G(x)$, $F(y)$, and $G(y)$.

$N/A$

1. Formula
   Observation

   An object labeled `x` is drawn in category C.

2. Audio
   Observation

   The narrator says, "given an object x in C".

x

An arbitrary object in category C.

Category C

1. Formula
   Observation

   An object labeled `y` is drawn in category C below x.

2. Audio
   Observation

   The narrator says, "If we had an object y in C".

y

Another arbitrary object in category C.

Category C

1. Formula
   Observation

   A pink arrow labeled `h` is drawn from x to y in category C.

2. Audio
   Observation

   The narrator says, "given a morphism h from x to y".

h

A morphism in category C from object x to object y.

Category C

1. Formula
   Observation

   The symbol `$\eta$` is written on the left side of the board.

2. Audio
   Observation

   The narrator says, "A natural transformation $\eta$ from F to G".

$\eta$

A natural transformation between functors F and G.

Functors F and G

1. Formula
   Observation

   The text `$\eta_x$ : hom_D($F(x)$, $G(x)$)` is written on the left side of the board.

2. Audio
   Observation

   The narrator says, "assigns to each object x in C a morphism $\eta_x$ from $F(x)$ to $G(x)$".

$\eta_x$

The component of the natural transformation $\eta$ at object x, which is a morphism in D from $F(x)$ to $G(x)$.

Category D

1. Formula
   Observation

   An arrow labeled `$\eta_y$` is drawn from $F(y)$ to $G(y)$ in category D.

2. Audio
   Observation

   The narrator says, "we could similarly speak of a morphism $\eta_y$ from $F(y)$ to $G(y)$".

$\eta_y$

The component of the natural transformation $\eta$ at object y, which is a morphism in D from $F(y)$ to $G(y)$.

Category D

1. Formula
   Observation

   An object labeled `$F(x)$` is drawn in category D.

2. Audio
   Observation

   The narrator says, "there are the objects $F(x)$ and $G(x)$ in D".

$F(x)$

The image of object x under the functor F.

Category D

1. Formula
   Observation

   An object labeled `$G(x)$` is drawn in category D below $F(x)$.

2. Audio
   Observation

   The narrator says, "there are the objects $F(x)$ and $G(x)$ in D".

$G(x)$

The image of object x under the functor G.

Category D

1. Formula
   Observation

   The text `$\eta_x$ : hom_D($F(x)$, $G(x)$)` is written on the left side of the board.

2. Audio
   Observation

   The narrator says, "A natural transformation $\eta$ from F to G would then be a rule that assigns to each object x in C a morphism $\eta_x$ from $F(x)$ to $G(x)$."

A natural transformation $\eta$ between two functors $F$ and $G$ (both mapping from category $C$ to category $D$) is a rule that assigns to every object $x$ in $C$ a morphism $\eta_x$ in $D$, specifically from $F(x)$ to $G(x)$. This morphism $\eta_x$ is called the component of the natural transformation at object $x$.

1. $F$ and $G$ are functors from category $C$ to category $D$.

2. For each object $x$ in $C$, $\eta_x$ is a morphism in $D$ from $F(x)$ to $G(x)$.

1. Formula
   Observation

   The equation `$\eta_y \circ F(h) = G(h) \circ \eta_x$` is written at the bottom of the screen.

2. Audio
   Observation

   The narrator says, "Finally, for $\eta$ to be a natural transformation, $\eta_y$ composed with $F(h)$ needs to be equal to $G(h)$ composed with $\eta_x$."

For a collection of morphisms $\eta_x$ to form a natural transformation, it must satisfy a specific condition for every morphism $h: x \to y$ in the source category $C$. The diagram formed by the objects $F(x)$, $F(y)$, $G(x)$, and $G(y)$ and the morphisms $\eta_x$, $\eta_y$, $F(h)$, and $G(h)$ must commute. This means that composing $F(h)$ followed by $\eta_y$ yields the same result as composing $\eta_x$ followed by $G(h)$.

1. $h$ is a morphism in $C$ from object $x$ to object $y$.

2. $F$ and $G$ are functors from $C$ to $D$.

3. $\eta_x$ and $\eta_y$ are components of the natural transformation $\eta$.

1. Definition of a Natural Transformation

1. Audio
   Observation

   The speaker says, 'two objects in a diagram are equal like this. We say that it commutes.'

2. Formula
   Observation

   The equation $\eta_y \circ F(h) = G(h) \circ \eta_x$ is displayed.

A diagram is said to commute if any two directed paths between the same pair of objects yield equal composite morphisms.

1. Applies to diagrams formed by morphisms in a category.

1. Formula
   Observation

   The text '$\eta$ : hom_C($F(x)$, $G(x)$)' is shown, along with the commutative square.

1. The notation 'hom_C($F(x)$, $G(x)$)' is likely a typo for 'hom_D($F(x)$, $G(x)$)' since $F(x)$ and $G(x)$ are objects in D.

A natural transformation $\eta$ between functors F and G assigns to each object x in C a morphism $\eta_x$ : $F(x) \to G(x)$ in D such that for every morphism h : $x \to y$ in C, the corresponding square commutes.

1. F and G are functors from C to D.

2. x and y are objects in C.

3. h is a morphism from x to y in C.

1. Commutative Diagram

1. Audio
   Observation

   The speaker says, 'Suppose we wanted to define the category D superscript C, that is, the category of functors from C to D.'

2. Formula
   Observation

   The symbol This formula needs review. Please report it using the page feedback control.mathcal{C} is displayed.

The category of functors from C to D, denoted This formula needs review. Please report it using the page feedback control.mathcal{C}, has functors from C to D as its objects and natural transformations between these functors as its morphisms.

1. C and D are categories.

1. Natural Transformation

1. Audio
   Observation

   The speaker says, 'And for composition, we'll say that the x component of gamma composed with eta is defined as gamma x composed with eta x.'

2. Formula
   Observation

   The equation ($\gamma \circ \eta$)_x := $\gamma_x \circ \eta_x$ is displayed.

The composition of two natural transformations $\eta$ : $F \to G$ and $\gamma$ : $G \to H$ is defined componentwise: for each object x in C, the x-component of $\gamma \circ \eta$ is the composite morphism $\gamma_x \circ \eta_x$ in D.

1. $\eta$ is a natural transformation from F to G.

2. $\gamma$ is a natural transformation from G to H.

3. x is an object in C.

1. Category of Functors

1. Animation
   Observation

   The video draws a diagram showing two categories C and D separated by a vertical line, with arrows representing functors F and G going from C to D.

1. Category C

2. Category D

3. Functor F

4. Functor G

1. Arrows for F and G are drawn from C to D.

1. The vertical line separating C and D remains constant.

This visual establishes the setting: two functors mapping between the same pair of categories, which is the prerequisite for defining a natural transformation between them.

1. Animation
   Observation

   Objects x and y are added to C. Their images $F(x)$, $G(x)$, $F(y)$, $G(y)$ are added to D. Arrows $\eta_x$ and $\eta_y$ are drawn connecting $F(x)$ to $G(x)$ and $F(y)$ to $G(y)$ respectively.

1. Object x

2. Object y

3. Object $F(x)$

4. Object $G(x)$

5. Object $F(y)$

6. Object $G(y)$

7. Morphism $\eta_x$

8. Morphism $\eta_y$

1. Objects and their images are sequentially added.

2. Vertical arrows representing the components of the natural transformation are drawn in category D.

1. The overall structure of categories C and D and functors F and G remains.

This animation illustrates the first part of the definition: a natural transformation assigns a specific morphism (component) in the target category D to each object in the source category C, connecting the images of that object under the two functors.

1. Animation
   Observation

   A morphism h is drawn from x to y in C. Corresponding morphisms $F(h)$ and $G(h)$ are drawn between the images in D. The equation $\eta_y \circ F(h) = G(h) \circ \eta_x$ appears.

1. Morphism h

2. Morphism $F(h)$

3. Morphism $G(h)$

4. Equation $\eta_y \circ F(h) = G(h) \circ \eta_x$

1. Horizontal arrows representing the images of morphism h under functors F and G are added.

2. The commutativity equation is displayed at the bottom.

1. The previously drawn objects and vertical component arrows remain.

This completes the visual representation of a natural transformation. It shows that for any morphism h between two objects in C, the square formed by the images of those objects and the images of h must commute in D. This ensures the natural transformation is consistent with the structure of the categories.

1. Diagram
   Observation

   A commutative square is drawn showing objects $F(x)$, $F(y)$, $G(x)$, $G(y)$ and morphisms $\eta_x$, $\eta_y$, $F(h)$, $G(h)$.

1. $F(x)$

2. $F(y)$

3. $G(x)$

4. $G(y)$

5. $\eta_x$

6. $\eta_y$

7. $F(h)$

8. $G(h)$

1. The diagram illustrates the condition for a natural transformation.

1. The equality of the two paths from $F(x)$ to $G(y)$.

The visual representation confirms that applying F to h and then $\eta_y$ yields the same result as applying $\eta_x$ and then G to h.

1. Diagram
   Observation

   A diagram shows functors F, G, H as objects in This formula needs review. Please report it using the page feedback control.mathcal{C}, with natural transformations $\eta$ and $\gamma$ as morphisms between them.

1. This formula needs review. Please report it using the page feedback control.mathcal{C}

2. F

3. G

4. H

5. $\eta$

6. $\gamma$

1. The diagram illustrates the structure of the category of functors.

1. The composition of morphisms $\gamma \circ \eta$.

The visual representation shows how functors serve as objects and natural transformations as morphisms in the functor category This formula needs review. Please report it using the page feedback control.mathcal{C}.

1. Formula
   Observation

   The text '$\eta$ : hom_C($F(x)$, $G(x)$)' is displayed.

1. It is highly probable that 'hom_C' is a typo for 'hom_D', as $F(x)$ and $G(x)$ are objects in the target category D.

The video displays $\eta$ : hom_C($F(x)$, $G(x)$), which incorrectly places the morphisms in the source category C.

Since F and G map from C to D, their images $F(x)$ and $G(x)$ are objects in D. Therefore, the components $\eta_x$ should be morphisms in D, meaning the correct notation is $\eta_x$ : hom_D($F(x)$, $G(x)$).

1. Audio
   Observation

   The narrator states, 'Finally, for $\eta$ to be a natural transformation...' followed by the commutative condition.

The commutative square condition is an essential component of the full definition of a natural transformation; assigning components is necessary but not sufficient without this structural consistency requirement.

1. Audio
   Observation

   The speaker connects natural transformations to the definition of the category of functors.

Natural transformations are used as the morphisms to define the category of functors This formula needs review. Please report it using the page feedback control.mathcal{C}.

1. Audio
   Observation

   The speaker defines composition within the context of the newly introduced category.

The componentwise composition of natural transformations provides the composition operation for the morphisms in the category of functors This formula needs review. Please report it using the page feedback control.mathcal{C}.

1. Audio
   Observation

   The narrator explicitly defines what a natural transformation is.

1. Definition of a Natural Transformation
2. Commutative Square Condition for Natural Transformations

1. Formula
   Observation

   The equation $\eta_y \circ F(h) = G(h) \circ \eta_x$ is shown as the final condition.

1. Commutative Square Condition for Natural Transformations

1. Audio
   Observation

   The speaker defines when a diagram commutes.

1. Commutative Diagram

1. Formula
   Observation

   The commutative square equation is shown.

1. Natural Transformation

1. Audio
   Observation

   The speaker introduces the category $D^C$.

1. Category of Functors

1. Formula
   Observation

   The composition formula is displayed.

1. Composition of Natural Transformations