TikZ diagram lab: category theory lecture notes

A category has objects, morphisms, identities, and associative composition. In a small local picture, the equality $h = g \circ f$ is represented by the commutative triangle:

The same idea in raw TikZ gives more control over placement and labels:

A functor $F : \mathcal{C} \to \mathcal{D}$ sends objects to objects and morphisms to morphisms while preserving identity arrows and composition.

Functoriality says the following two routes in $\mathcal{D}$ agree whenever the upper route agrees in $\mathcal{C}$:

For functors $F,G : \mathcal{C}\to\mathcal{D}$, a natural transformation $\eta : F \Rightarrow G$ assigns a component $\eta_A : F A \to G A$ to each object $A$. Naturality means every morphism $f : A\to B$ gives a commutative square.

Here is the same square with visual emphasis on the two parallel paths:

A pullback of $f : X\to Z$ and $g : Y\to Z$ is an object $P$ equipped with projections such that every compatible cone factors uniquely through $P$.

The outer equations are $f u = g v$, and the dashed arrow is the unique map whose composites with $p_1,p_2$ are $u,v$.

An adjunction $F \dashv G$ can be recognized by a natural bijection

The unit and counit satisfy the triangle identities:

The hom-set bijection is often the most useful diagram for computation:

The Yoneda embedding sends an object $A$ to the representable functor $\mathcal{C}(-,A)$. A morphism $u:A\to B$ induces a natural transformation by postcomposition.

One way to remember the Yoneda lemma is that every natural transformation out of a representable functor is determined by the image of the identity:

These smaller knowls mix prose, LaTeX display math, and TikZ diagrams so the runtime can be checked on nested knowl content:

• Pullback pasting
• Adjunction naturality
• Whiskering and coherence