  Large Language Models are transformer neural networks which are trained to
produce a probability distribution on the possible next words to given texts in
a corpus, in such a way that the most likely word predicted is the actual word
in the training text. In this paper we find what is the mathematical structure
defined by such conditional probability distributions of text extensions.
Changing the view point from probabilities to -log probabilities we observe
that the subtext order is completely encoded in a metric structure defined on
the space of texts $\mathcal{L}$, by -log probabilities. We then construct a
metric polyhedron $P(\mathcal{L})$ and an isometric embedding (called Yoneda
embedding) of $\mathcal{L}$ into $P(\mathcal{L})$ such that texts map to
generators of certain special extremal rays. We explain that $P(\mathcal{L})$
is a $(\min,+)$ (tropical) linear span of these extremal ray generators. The
generators also satisfy a system of $(\min+)$ linear equations. We then show
that $P(\mathcal{L})$ is compatible with adding more text and from this we
derive an approximation of a text vector as a Boltzmann weighted linear
combination of the vectors for words in that text. We then prove a duality
theorem showing that texts extensions and text restrictions give isometric
polyhedra (even though they look a priory very different). Moreover we prove
that $P(\mathcal{L})$ is the lattice closure of (a version of) the so called,
Isbell completion of $\mathcal{L}$ which turns out to be the $(\max,+)$ span of
the text extremal ray generators. All constructions have interpretations in
category theory but we don't use category theory explicitly. The categorical
interpretations are briefly explained in an appendix. In the final appendix we
describe how the syntax to semantics problem could fit in a general well known
mathematical duality.
