# Category Theory Notation Cheat Sheet

| Symbol                                        | Meaning                       | Example                                         | Notes                                                                                                                                |
|-----------------------------------------------|-------------------------------|-------------------------------------------------|--------------------------------------------------------------------------------------------------------------------------------------|
| `\mathcal{C}, \mathcal{D}, \mathcal{E}`       | **Categories**                | $$\mathcal{A}, \mathcal{B}, \mathcal{C}$$       | Calligraphic uppercase letters for categories. Multi-letter names allowed. In LaTeX, use 'calrsfs' package for better multi-letters. |
| `A, B, X, Y`                                  | **Objects**                   | $$A, B, A_\mathcal{C}, B_\mathcal{B}$$          | Plain uppercase letters. Subscripts indicate which category the object belongs to.                                                   |
| `f, g, h`                                     | **Morphisms / Arrows**        | $$f: A_\mathcal{C} \to B_\mathcal{C}$$          | Lowercase letters for functions/arrows.                                                                                              |
| `\text{id}_A`                                 | Identity morphism             | $$\text{id}_A: A \to A$$                        | Usually written with object as subscript.                                                                                            |
| `\circ`                                       | Composition of morphisms      | $$(g \circ f): A \to C$$                        | Standard composition, read right-to-left.                                                                                            |
| `\mathbf{F}, \matchbf{G}`                     | Functors                      | $$\mathbf{F}: \mathcal{C} \to \mathcal{D}$$     | Uppercase, Bold or Bold-Italic.                                                                                                      |
| `\eta, \epsilon`                              | Natural transformations       | $$\eta: F \Rightarrow G$$                       | Greek letters, lowercase.                                                                                                            |
| `\Rightarrow`                                 | Natural transformation arrow  | $$\eta: F \Rightarrow G$$                       | Distinguishes from morphism arrow `\to`.                                                                                             |
| `\mathbf{Set}, \mathbf{Grp}`                  | Standard categories           | $$\mathbf{Set}$$                                | Bold upright letters for familiar concrete categories.                                                                               |
| `A_\mathcal{C} \xrightarrow{f} B_\mathcal{C}` | Alternative morphism notation | $$A_\mathcal{C} \xrightarrow{f} B_\mathcal{C}$$ | Arrow with label on top.                                                                                                             |

**Example in context:**

```latex
\mathcal{C} \text{ is a category, with objects }
A_\mathcal{C}, B_\mathcal{C} in \mathcal{C},
\text{ and a morphism } f: A_\mathcal{C} \to B_\mathcal{C}.
```

$$
\mathcal{C} \text{ is a category, with objects }
A_\mathcal{C}, B_\mathcal{C} \text{ in } \mathcal{C},
\text{ and a morphism } f: A_\mathcal{C} \to B_\mathcal{C}.
$$