If G : A [right arrow] B is a
functor such that every G-structured source has a G-initial lift, then the following conditions are equivalent:
Conversely given a groupoid g and a group G with a star injective
functor F : Q [right arrow]* G (viewing G as a small category with one object), we can associate a partial action of G on the set of objects X = [Q.sup.(0]).
Application of the
functor [[GAMMA].sub.xR](-) to the monomorphism [mathematical expression not reproducible] yields a monomorphism [mathematical expression not reproducible].
Hence [G.sub.2]: NSet(H) [right arrow] ISet(H) is a
functor.
Hence, ' is a contravariant
functor that is obviously period two.
(1) The interlayer operations inherit the mixture of intra- and interlayer operations and (2) there is a procedure by which the interlayer operation can be regarded as an adjoint
functor. If the interlayer operation satisfies the conditions (1) and (2), it is called a prefunctor [55].
Definition 2 A species (with restrictions) is a contravariant
functor P: [set.sup.x] [right arrow] Set from the category of finite sets with injections to the category of all sets.
In contrast with previous works, the current study is based on phonological and
functor innovations.
From Theorems 27 and 28 it follows that the procedure assigning to each soft fuzzy proximity [delta] generated soft fuzzy topology [[tau].sub.[delta]] and leaving morphisms unchanged can be interpreted as a
functor.
Finally we observe that, from the continuity of the Cech cohomology
functor, we can say that any [R.sub.[delta]] set is also an acyclic space.