Covariant functor

The local Hilbert functor associated to the closed subscheme X c Pr is the covariant functor... 

A2.3) Proposition. With the operations defined above Ex (R,l) is an R-moduie whose zero element is [R[l],p]. This construction defines a covariant functor... 

The operation of taking the tangent space is a covariant functor, while the cotangent space is a contravariant functor. 

Let 7 be a small category, and B, a covariant functor from I to the category of (non-commutative) rings. A left R,-module is a collection M = (/ 0)0eMor(/)) such that M is a left Bj-module for each... 

The functors that we have described so far are sometimes called covariant functors. This is done in order to distinguish them from the contravariant functors. A contravariant functor is defined analogously to the covariant one with the only difference that it turns the arrows around, i.e., C... 

Thus we see that 7t is yet another example of a covariant functor Top — Ab. We leave verification of Proposition 6.21 to the reader. 

Proposition 18.5. For any fixed graph T, the Horn construction yields a covariant functor Horn (T, —). 

Let Ki, K2 be A-categories (1.4.2) with translation functors Ti, T2 respectively. A (covariant) A-functor is defined to be a pair (F, 0) consisting of an additive functor F Ki —> K2 together with an isomorphism of functors... 

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