Prompting as Projection: A Categorical Framework for AI Image Generation

J. Rogers, SE Ohio

Abstract

This paper proposes a categorical interpretation of AI image generation, reframing prompting not as an act of theft or imitation (as long as the images created are  used morally), but as the structured exploration of a vast conceptual-image space. We model the relationship between conceptual descriptions and generated images using a fibration, where prompts act as morphisms in the conceptual category and image outputs arise through Cartesian liftings. This framework provides a rigorous account of prompting as a creative act of navigation within a mathematically structured space of artistic possibility.

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1. Introduction

Debates surrounding AI art often hinge on whether machine-generated images are derivative or original. By treating AI models as stochastic navigators of conceptual space, we can formalize prompting as an act of projection within a categorical structure. This perspective moves beyond questions of ownership and towards a deeper understanding of creativity as structured exploration. Just exploring and using it to generate ideas by itself is not wrong.  Falsely claiming that this is your work is when it would rise to the level of immoral action. 

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2. The Conceptual Category of Art

Let C denote the base category of conceptual descriptions:

• Objects: semantic structures (e.g., "cyberpunk crow," "Renaissance portrait," "cosmic landscape").

• Morphisms: conceptual transformations (e.g., "make it darker," "in watercolor," "add glitch aesthetics").

This category encodes the manifold ways in which ideas about art can be transformed and combined.

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3. Images as Fibers

Let E denote the total category of concrete images. A projection functor
$\pi : E \to C$
maps each image to its conceptual description. For each object $c \in C$, the fiber $\pi^{-1}(c)$ is the set of images that instantiate that concept.

Thus, images exist not as isolated artifacts but as fibers suspended above the conceptual base space.

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4. Prompting as Cartesian Lifting

A prompt corresponds to a morphism in C. To apply a prompt is to perform a Cartesian lifting:
$f : c \to c' \quad \mapsto \quad \hat{f} : e \to e'$
where $e$ is an image over concept $c$, and $e'$ is a generated image over the transformed concept $c'$.

Prompting therefore has a functorial structure: conceptual changes consistently lift to image-space transformations.

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5. The Generative Model as a Functor

The AI generative model can be described as a functor:
$G : C \to \mathbf{Set}$
assigning to each concept $c$ a set of possible images $G(c)$. A prompt morphism $f : c \to c'$ is mapped by $G$ into a structured family of possible outputs, producing variability within a coherent conceptual frame.

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6. Creativity as Navigation

In this framework, creativity emerges not from ex nihilo production but from structured navigation:

• Exploring morphisms in C (concept space).

• Sampling fibers in E (image space).

• Identifying low-density or previously unexplored regions of the fibration.

Originality is thus reinterpreted as the discovery of underexplored paths in the categorical structure of art.  This could then be used as the seed to study and reproduce through more traditional methods of artists.  It would just be a tool to mock up art work that could then be used to create art using the new combinations of elements that have never existed before.

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7. Conclusion

AI image generation can be rigorously understood through categorical theory as a fibration of images over concepts. Prompts serve as morphisms whose Cartesian liftings instantiate concrete images. This reframing allows us to move beyond debates of authorship and theft, toward a recognition of prompting as an exploratory act of creativity within a mathematically structured conceptual-image space. 

The images created in this way should just be used as a seed for human creativity.  The fact that you can combine arbitrary elements in brand new ways is the human creativity.