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Every closed category has a forgetful functor to the category of sets, which in particular takes the internal hom to the external hom.
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Every and closed
Every time I closed my eyes, I saw Gray Eyes rushing at me with a knife.
Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.
* Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions.
Every character is automatically continuous from A to C, since the kernel of a character is a maximal ideal, which is closed.
* Every continuous map from a compact space to a Hausdorff space is closed and proper ( i. e., the pre-image of a compact set is compact.
* Every closed subgroup of a profinite group is itself profinite ; the topology arising from the profiniteness agrees with the subspace topology.
Every open subgroup H is also closed, since the complement of H is the open set given by the union of open sets gH for g in G
* Every closed nowhere dense set is the boundary of an open set.
Every map that is injective, continuous and either open or closed is an embedding ; however there are also embeddings which are neither open nor closed.
Every base he closed resulted in a new construction project elsewhere to expand another base, relocation of forces projects and other related spending.
Every year the central business district ( with corners at the Municipal Building, Grand Street Fire House and Croton-Harmon High School ) is closed to automobile traffic for music, American food, local fund raisers, traveling, and local artists.
The generalized Poincaré conjecture states that Every simply connected, closed n-manifold is homeomorphic to the n-sphere.
Every closed subspace of a reflexive space is reflexive.
Every closed 3-manifold has a prime decomposition: this means it is the connected sum of prime three-manifolds ( this decomposition is essentially unique except for a small problem in the case of non-orientable manifolds ).
: Every oriented prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.
* Every integrable subbundle of the tangent bundle — that is, one whose sections are closed under the Lie bracket — also defines a Lie algebroid.
* Every irreducible closed subset of P < sup > n </ sup >( k ) of codimension one is a hypersurface ; i. e., the zero set of some homogeneous polynomial.
Every October the high street is closed for the two Saturdays either side of 11 October for the Marlborough Mop Fair.
Every closed point of Hilb ( X ) corresponds to a closed subscheme of a fixed scheme X, and every closed subscheme is represented by such a point.
Every homeomorphism is open, closed, and continuous.
Every closed curve c on X is homologous to for some simple closed curves c < sub > i </ sub >, that is,
Every and category
* Every small category has a skeleton.
* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).
* Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by " reversing all the arrows ".
Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.
Every adjunction 〈 F, G, ε, η 〉 gives rise to an associated monad 〈 T, η, μ 〉 in the category D. The functor
Every morphism in a concrete category whose underlying function is injective is a monomorphism ; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense.
Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories.
Every morphism in a concrete category whose underlying function is surjective is an epimorphism.
Every pre-abelian category is of course an additive category, and many basic properties of these categories are described under that subject.
Every person born into this world comes from one of these categories in order to help fulfill the kind of function that that category of people is supposed to fulfill in order to keep the community together.
* Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval 1.
Every monoidal category is monoidally equivalent to a strict monoidal category.
Every group G can be viewed as a category with a single object ( morphisms in this category are just the elements of G ).
Every ring can be viewed as a preadditive category with a single object.
Every year a set of students are selected from each sports category for ' Inter IIT Sports Meet ', a set of sporting events occur in the winter at any one of the IIT's.
Every object in a Grothendieck category has an injective hull.
Every NNO is an initial object of the category of diagrams of the form
Every category has a host of untranslatable jokes that rely on linguistic puns, wordplay, and Russian's vocabulary of foul language.
Every U. S. and foreign charity that qualifies as tax-exempt under Section 501 ( c )( 3 ) of the Internal Revenue Code is considered a " private foundation " unless it demonstrates to the IRS that it falls into another category.
Every abelian category is an exact category if we just use the standard interpretation of " exact ".
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