A Dundas-Goodwillie-McCarthy theorem for split square-zero extensions of exact categories

We prove the analog for the $K$ -theory of forms of the $Q=+$ theorem in algebraic $K$ -theory. That is, we show that the $K$ -theory of forms defined in terms of an $S_{\bullet }$ -construction is a group completion of the category of quadratic spaces for form categories in which all admissible exact sequences split. This applies for instance to quadratic and hermitian forms defined with respect to a form parameter.

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Relations for exact categories

Journal of Algebra ◽

10.1016/0021-8693(69)90108-2 ◽

1969 ◽

Vol 13 (4) ◽

pp. 465-480 ◽

Cited By ~ 5

Author(s):

Hans-Berndt Brinkmann

Keyword(s):

Exact Categories

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Simply Connected Limits

Canadian Journal of Mathematics ◽

10.4153/cjm-1990-038-6 ◽

1990 ◽

Vol 42 (4) ◽

pp. 731-746 ◽

Cited By ~ 15

Author(s):

Robert Paré

Keyword(s):

Left Adjoint ◽

Topos Theory ◽

Essential Ingredient ◽

Original Definition ◽

The Subject ◽

Definition Of ◽

Simply Connected ◽

Geometric Morphisms ◽

Exact Categories

The importance of finite limits in completeness conditions has been long recognized. One has only to consider elementary toposes, pretoposes, exact categories, etc., to realize their ubiquity. However, often pullbacks suffice and in a sense are more natural. For example it is pullbacks that are the essential ingredient in composition of spans, partial morphisms and relations. In fact the original definition of elementary topos was based on the notion of partial morphism classifier which involved only pullbacks (see [6]). Many constructions in topos theory, involving left exact functors, such as coalgebras on a cotriple and the gluing construction, also work for pullback preserving functors. And pullback preserving functors occur naturally in the subject, e.g. constant functors and the Σα. These observations led Rosebrugh and Wood to introduce partial geometric morphisms; functors with a pullback preserving left adjoint [9]. Other reasons led Kennison independently to introduce the same concept under the name semi-geometric functors [5].

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