Ext groups in Homotopy Type Theory

by   J. Daniel Christensen, et al.

We develop the theory of Yoneda Ext groups over a ring in homotopy type theory (HoTT) and describe their interpretation into an ∞-topos. This is an abstract approach to Ext groups which does not require projective or injective resolutions. While it produces group objects that are a priori large, we show that the Ext^1 groups are equivalent to small groups, leaving open the question of whether the higher Ext groups are essentially small as well. We also show that the Ext^1 groups take on the usual form as a product of cyclic groups whenever the input modules are finitely presented and the ring is a PID (in the constructive sense). When interpreted into an ∞-topos of sheaves on a 1-category, our Ext groups recover (and give a resolution-free approach to) sheaf Ext groups, which arise in algebraic geometry. (These are also called "local" Ext groups.) We may therefore interpret results about Ext from HoTT and apply them to sheaf Ext. To show this, we prove that injectivity of modules in HoTT interprets to internal injectivity in these models. It follows, for example, that sheaf Ext can be computed using resolutions which are projective or injective in the sense of HoTT, when they exist, and we give an example of this in the projective case. We also discuss the relation between internal ℤ G-modules (for a 0-truncated group object G) and abelian groups in the slice over BG, and study the interpretation of our Ext groups in both settings.


page 1

page 2

page 3

page 4


Formalising Yoneda Ext in Univalent Foundations

Ext groups are fundamental objects from homological algebra which underl...

The isomorphism problem for plain groups is in Σ_3^𝖯

Testing isomorphism of infinite groups is a classical topic, but from th...

Higher Groups in Homotopy Type Theory

We present a development of the theory of higher groups, including infin...

On the Nielsen-Schreier Theorem in Homotopy Type Theory

We give a formulation of the Nielsen-Schreier theorem (subgroups of free...

Subgroup and Coset Intersection in abelian-by-cyclic groups

We consider two decision problems in infinite groups. The first problem ...

Internal sums for synthetic fibered (∞,1)-categories

We give structural results about bifibrations of (internal) (∞,1)-catego...

Domain Theory in Constructive and Predicative Univalent Foundations

We develop domain theory in constructive univalent foundations without V...

Please sign up or login with your details

Forgot password? Click here to reset