Given an algebra group G = 1 + g and a map [dagger] as above, we define a Springer

morphism f: G [?

the composite of

morphisms f and g is defined as the composite of two linear maps between two linear spaces.

A relation on

morphisms and pairs of objects, called typing of the

morphisms.

where the groupoid

morphisms ([alpha],[beta]) onto objects, are called the range and source maps, respectively.

Morphisms between A-weighted sets are weight preserving bijections.

X] is coherently filtered if locally there is for every x [member of] X, an open U containing x, such that we have an exact sequence of sheaf

morphismsa family of

morphisms {fi: di [right arrow] x i [member of] I} such that for each arrow g: di [right arrow] dj in D, we have fj o g = fi, as shown in Figure 1(a).

functions and maps, operators and

morphisms bind on type of

Let C be a category and let M be a class of

morphisms in C.

They play a role in Geometric Complexity Theory, which seeks to separate complexity classes such as P and NP by associating group-theoretic varieties to them, and then proving the non-existence of injective

morphisms from one to the other by displaying representation theoretic obstructions [13, 14].

In an extended theoretical study in [8] we develop a formal notion of schema

morphisms, show that the corresponding category of schemata with these

morphisms is finitely complete and co-complete, and also show that the algebra in this paper is well-defined and complete in the sense that all operators give rise to canonical

morphisms, and all finite limits and co-limits can be expressed by the algebra.

i]/i [member of] I}, there exist a unique

morphisms f: [P.