Liora and the Adjunction Bridge

(The Fifth Bonus Tale in the Little Relational Ontology Library)

Liora wandered farther than she ever had before. Past the Category Castle, beyond the Functorial Forest, and even deeper than the Yoneda Trick’s secret glade, she found a strange valley where two landscapes faced each other across a wide shimmering gap.

On the left side, the land was full of shapes—triangles, cubes, spirals, and patterns that changed when you touched them. On the right side, the land was full of actions—folding, stretching, sliding, twirling, shrinking.

Liora frowned.

“These worlds look related,” she whispered, “but they don’t know how.”

Just then, a small creature with a bell-shaped hat rolled toward her.

“I’m Adjay,” it chirped, “guardian of the Adjunction Bridge. Only adjoint travellers may cross!”

Liora peered over the valley. “I don’t see a bridge.”

“Oh,” said Adjay, “that’s because it isn’t there until someone makes a good comparison. Two lands… two kinds of things… each needs to be the best possible partner for the other. Otherwise the bridge refuses to show up.”

Liora sat down. “So… I must find what each side does best for the other?”

Adjay nodded vigorously, tiny bells ringing.

“Exactly! An adjunction is like a perfect friendship: each land offers something the other wants, in the most generous way possible. Not too much. Not too little. Just right.”

Liora walked first to the land of Shapes.

“What do you wish the land of Actions could do for you?” she asked.

A triangle sighed.

“We wish they could wrap us or transform us into something useful. Something that makes it easier to travel.”

Liora crossed to the land of Actions.

“And you? What do you wish Shapes would give you?”

A stretching-motion replied,

“We wish Shapes would receive us—give us a place to land, so we can show what we do.”

Liora thought.

Shapes wanted Actions to build useful structures from them.

Actions wanted Shapes to host them.

“Ah,” she smiled. “You’re asking for the most efficient building… and the most accommodating receiving. You’re asking for a pair of functors that are the best possible fit for each other.”

Adjay spun with delight.

“Oh! Oh! You’ve almost found it!”

Liora raised her lantern and spoke clearly across the valley:

“Let Shapes send their possibilities into Actions in the most generous way.

Let Actions send their transformations back to Shapes in the most understanding way.

And let each be the best possible match for the other.”

The valley trembled.

From the mist, a shining Adjunction Bridge appeared—arched, luminous, and perfectly balanced. One end anchored in possibility; the other in transformation.

Shapes crossed the bridge eagerly, becoming useful structured things.

Actions crossed back, finding places that welcomed their movements.

Adjay bowed.

“You’ve built the bridge that only fits when two lands meet each other exactly as they are—each offering what the other needs most. That is an adjunction.”

Liora stepped onto the bridge.

“It’s beautiful,” she said softly.

“It’s… fair.”

Adjay nodded.

“Adjunctions always are. They’re the universe’s way of saying:

Two different worlds can meet—if each becomes the best translator of the other.”

Liora crossed the bridge as it shone beneath her feet.

Somewhere ahead, she knew, another landscape waited—one that only opened when possibility and transformation learned to walk together.

And Liora, as always, walked on.