Homological algebra in topoi

Abstract

Let A be the category of sheaves of universal algebras of some fixed type on a Grothendieck topology. There is defined a cohomology theory H* such that if A is an object in A and M is an A-module then H 1(A, M) is in one-to-one correspondence with the equivalence classes of singular extensions of A by M. When A is the category of sheaves of R-modules for some sheaf of rings R, then Hn(A, M)_ Extn(A, M) for all n > 0. The purpose of this note is to extend the results of [4] and [5] to general topoi, i.e. to sheaves on a Grothendieck topology. The extension has been made possible by the following theorem: Let E be a topos. Then there exists a complete Boolean algebra B and a left exact cotriple G on B (= sheaves on B for the canonical topology) such that E and BG (= the category of G-coalgebras) are equivalent categories. Barr's proof of this theorem can be found in [1]. Lawvere, who conjectured the theorem, paraphrases it by saying that any topos has enough Boolean-valued points. The value of the theorem is that in B subobjects have complements and epimorphisms split, i.e. B is very set-like. One Received by the editors March 1, 1974 and, in revised form, April 3, 1974. AMS (MOS) subject, classifications (1970). Primary 18C15, 18F10; Secondary 18G 15.