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 (P.Baudot and D.B. 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.

翻译：已知的每个人工深心神经网络( DNN) 都对应了古典格罗特芬迪克( Grothendieck) 文中的一个对象; 其学习动态与此图中的形态变化相对应。 层( CNN 或 LSTMs ) 中的不轨结构与 Giraud 的堆叠相对应。 这种差异应该由一般化属性负责, 也就是从受限制的学习数据中推断出来。 纤维代表了先修类( Culoli, Thom), 人为语言被定义, 其内部逻辑、直观、 古典或线性( Girard) 。 网络的语义功能是能够以这种语言表达理论, 解答输入数据中的问题。 语义信息的数量和空间应该与Shannon entrop( P. Baudot 和 D. B. B. 2015) 的同义解释相仿。 它们概括了Carnap 和 Bar- Hillel (1952 ) 所发现的措施。 令人惊讶的是, 上面的语结构结构结构结构结构结构结构由地理纤维化的理论交换 Qredial- dealtaltialtical 。 在封闭式的模型中, 格式上, 它们在模型中, 它们在模型中将它们在模型中组织其运行中, 和正立的轨道内, 和正正立的轨道内, 格式内, 和正的内置。