catc-notaition.md

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:

\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}.

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