Martin Escardo, started 2014. An unformalized version of part of this development was published in Mathematical Structures in Computer Science, Cambridge University Press, 5th January 2021. https://doi.org/10.1017/S0960129520000225 See the file InjectiveTypes.Article for an explanation of the role of the file InjectiveTypes.Blackboard, and, more importantly, for an explanation of the published paper. \begin{code} {-# OPTIONS --safe --without-K --exact-split #-} module InjectiveTypes.index where import InjectiveTypes.Article import InjectiveTypes.Blackboard \end{code} The following have been done after the above article was published. \begin{code} import InjectiveTypes.CounterExamples import InjectiveTypes.InhabitedTypesTaboo \end{code} Injectivity plays a major role in the construction of "searchable" or "compact" types: \begin{code} import TypeTopology.index import Ordinals.index \end{code} The following has applications to (in)decomposability. Injective types have non-trivial decidable properties if and only if weak excluded middle (the decidability of all negative propositions) holds. \begin{code} import InjectiveTypes.OverSmallMaps import Taboos.Decomposability \end{code} Since the above publication, a number of new examples of injective types have been found, including the following: * The type of ordinals. * The type of iterative multisets. * The type of iterative sets. * The type of pointed types. * The type of ∞-magmas. * The type of pointed ∞-magmas. * The type of monoids. (And the type of iterative ordinals is injective, simply because it is equivalent to that of ordinals.) \begin{code} import Ordinals.Injectivity import Iterative.Multisets-Addendum import Iterative.Sets-Addendum import Iterative.Ordinals import InjectiveTypes.MathematicalStructures import InjectiveTypes.Sigma import InjectiveTypes.MathematicalStructuresMoreGeneral \end{code} With Tom de Jong we prove more things about resizing, and in particular we show that previous results about injectivity are tight with respect to the universe levels, and that no small type with two distinct points can be injective without Ω¬¬-resizing. \begin{code} open import InjectiveTypes.Resizing \end{code}