Construction of Ext in Abelian Categories
This identification enables us to define Ext1
Ab(A, B) even for abelian categories Ab without reference to projectives and injectives. We simply take Ext1
Ab(A, B) to be the set of equivalence classes of extensions of A by B, forming an abelian group under the Baer sum. Similarly, we can define higher Ext groups Extn
Ab(A, B) as equivalence classes of n-extensions
under the equivalence relation generated by the relation that identifies two extensions
if there are maps Xm → X′m for all m in {1, 2, ..., n} so that every resulting square commutes.
The Baer sum of the two n-extensions above is formed by letting X′′
1 be the pullback of X′′
1 and X′
1 over A, and X′′
n be the pushout of Xn and X′
n under B quotiented by the skew diagonal copy of B. Then we define the Baer sum of the extensions to be
Read more about this topic: Ext Functor
Famous quotes containing the words construction of, construction and/or categories:
“There is, I think, no point in the philosophy of progressive education which is sounder than its emphasis upon the importance of the participation of the learner in the formation of the purposes which direct his activities in the learning process, just as there is no defect in traditional education greater than its failure to secure the active cooperation of the pupil in construction of the purposes involved in his studying.”
—John Dewey (1859–1952)
“There’s no art
To find the mind’s construction in the face.”
—William Shakespeare (1564–1616)
“All cultural change reduces itself to a difference of categories. All revolutions, whether in the sciences or world history, occur merely because spirit has changed its categories in order to understand and examine what belongs to it, in order to possess and grasp itself in a truer, deeper, more intimate and unified manner.”
—Georg Wilhelm Friedrich Hegel (1770–1831)