  As Large Language Models are deployed within Artificial Intelligence systems,
that are increasingly integrated with human society, it becomes more important
than ever to study their internal structures. Higher level abilities of LLMs
such as GPT-3.5 emerge in large part due to informative language
representations they induce from raw text data during pre-training on trillions
of words. These embeddings exist in vector spaces of several thousand
dimensions, and their processing involves mapping between multiple vector
spaces, with total number of parameters on the order of trillions. Furthermore,
these language representations are induced by gradient optimization, resulting
in a black box system that is hard to interpret. In this paper, we take a look
at the topological structure of neuronal activity in the "brain" of Chat-GPT's
foundation language model, and analyze it with respect to a metric representing
the notion of fairness. We develop a novel approach to visualize GPT's moral
dimensions. We first compute a fairness metric, inspired by social psychology
literature, to identify factors that typically influence fairness assessments
in humans, such as legitimacy, need, and responsibility. Subsequently, we
summarize the manifold's shape using a lower-dimensional simplicial complex,
whose topology is derived from this metric. We color it with a heat map
associated with this fairness metric, producing human-readable visualizations
of the high-dimensional sentence manifold. Our results show that sentence
embeddings based on GPT-3.5 can be decomposed into two submanifolds
corresponding to fair and unfair moral judgments. This indicates that GPT-based
language models develop a moral dimension within their representation spaces
and induce an understanding of fairness during their training process.
