  Every known artificial deep neural network (DNN) corresponds to an object in
a canonical Grothendieck's topos; its learning dynamic corresponds to a flow of
morphisms in this topos. Invariance structures in the layers (like CNNs or
LSTMs) correspond to Giraud's stacks. This invariance is supposed to be
responsible of the generalization property, that is extrapolation from learning
data under constraints. The fibers represent pre-semantic categories (Culioli,
Thom), over which artificial languages are defined, with internal logics,
intuitionist, classical or linear (Girard). Semantic functioning of a network
is its ability to express theories in such a language for answering questions
in output about input data. Quantities and spaces of semantic information are
defined by analogy with the homological interpretation of Shannon's entropy of
P.Baudot and D.Bennequin in 2015). They generalize the measures found by Carnap
and Bar-Hillel (1952). Amazingly, the above semantical structures are
classified by geometric fibrant objects in a closed model category of Quillen,
then they give rise to homotopical invariants of DNNs and of their semantic
functioning. Intentional type theories (Martin-Loef) organize these objects and
fibrations between them. Information contents and exchanges are analyzed by
Grothendieck's derivators.
