Diagram examples

Creating Diagrams in Category Theory

This page provides examples of different diagram types available for category theory notes.

Commutative Diagrams with MathJax

Use the AMScd package for simple commutative diagrams:

Basic Square Diagram

\[\begin{CD} A @>f>> B\\ @VgVV @VVhV\\ C @>>k> D \end{CD}\]

Code:

$$
\begin{CD}
A @>f>> B\\
@VgVV @VVhV\\
C @>>k> D
\end{CD}
$$

Triangle Diagram

\[\begin{CD} A @>f>> B\\ @| @VVgV\\ A @>>h> C \end{CD}\]

Functor Diagram

\[\begin{CD} F(A) @>F(f)>> F(B)\\ @V\eta_AVV @VV\eta_BV\\ G(A) @>>G(f)> G(B) \end{CD}\]

Product Diagram

\[\begin{CD} @. A \times B @.\\ @. @V\pi_1VV @VV\pi_2V @.\\ @. A @. B \end{CD}\]

String Diagrams (SVG)

For string diagrams (used in monoidal categories), use inline SVG:

Identity Wire

Cup (Creation/Unit)

Map $\eta: I \to A \otimes A^*$ - creates a pair from nothing:

Cap (Evaluation/Counit)

Map $\epsilon: A^* \otimes A \to I$ - evaluates a pair to nothing:

Zigzag (Snake Equation)

The yanking identity: $(1_A \otimes \epsilon) \circ (\eta \otimes 1_A) = 1_A$

Morphism Box

Composition

Tensor Product

Braiding

String Diagram Notation

String diagrams use graphical elements to represent categorical structures:

Wires (straight lines): Represent objects flowing through the diagram
Boxes: Represent morphisms/functions transforming objects
Curves: Special morphisms like cups (creation) and caps (evaluation)
Connections: Composition shown by joining wires vertically

Reading String Diagrams

Convention: Diagrams can be read from bottom to top (objects flow upward) or top to bottom (composition flows downward). In these notes, we read top to bottom.

Commutative Diagram Notation (AMScd)

Arrow styles for commutative diagrams:

@>>> : Long right arrow
@>label>> : Right arrow with label above
@VVV : Down arrow
@VlabelVV : Down arrow with label on left
@AAA : Up arrow
@AAlabelA : Up arrow with label on left
@<<< : Left arrow
@= : Equals sign (for identities)
@| : Vertical bar
@. : Empty cell (for spacing)

Tips for Good Diagrams

Keep it simple: Start with AMScd for basic commutative diagrams
String diagrams: Use the string-diagram class for monoidal category diagrams
Complex cases: Use TikZ only when AMScd isn’t sufficient
Labels: Keep labels concise and readable
Spacing: Use proper spacing for readability
Dark mode: All diagram styles are dark-mode compatible

Common Patterns

Adjunction

\[\begin{CD} @. F(A) @.\\ @. @| @.\\ \text{Hom}(F(A), B) @>{\cong}>> \text{Hom}(A, G(B)) \end{CD}\]

Limit Cone

\[\begin{CD} @. L @.\\ @. @VVV @.\\ A @>>f> B @<g<< C \end{CD}\]

Yoneda Lemma

\[\begin{CD} \text{Nat}(\text{Hom}(A, -), F) @>{\cong}>> F(A)\\ @. @VV{\alpha \mapsto \alpha_A(\text{id}_A)}V\\ @. F(A) \end{CD}\]

See: