  This paper concerns the intersection of natural language and the physical
space around us in which we live, that we observe and/or imagine things within.
Many important features of language have spatial connotations, for example,
many prepositions (like in, next to, after, on, etc.) are fundamentally
spatial. Space is also a key factor of the meanings of many
words/phrases/sentences/text, and space is a, if not the key, context for
referencing (e.g. pointing) and embodiment.
  We propose a mechanism for how space and linguistic structure can be made to
interact in a matching compositional fashion. Examples include Cartesian space,
subway stations, chesspieces on a chess-board, and Penrose's staircase. The
starting point for our construction is the DisCoCat model of compositional
natural language meaning, which we relax to accommodate physical space. We
address the issue of having multiple agents/objects in a space, including the
case that each agent has different capabilities with respect to that space,
e.g., the specific moves each chesspiece can make, or the different velocities
one may be able to reach.
  Once our model is in place, we show how inferences drawing from the structure
of physical space can be made. We also how how linguistic model of space can
interact with other such models related to our senses and/or embodiment, such
as the conceptual spaces of colour, taste and smell, resulting in a rich
compositional model of meaning that is close to human experience and embodiment
in the world.
