* 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 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 closed category has a forgetful functor to the category of sets, which in particular takes the internal hom to the external hom.

Every abelian category is an exact category if we just use the standard interpretation of " exact ".

Every soldier in the army has, somewhere, relatives who are close to starvation.

Every woman has had the experience of saying no when she meant yes, and saying yes when she meant no.

Every detail in his interpretation has been beautifully thought out, and of these I would especially cite the delicious laendler touch the pianist brings to the fifth variation ( an obvious indication that he is playing with Viennese musicians ), and the gossamer shading throughout.

Every calculation has been made independently by two workers and checked by one of the editors.

Every retiring person has a different situation facing him.

Every family of Riviera Presbyterian Church has been asked to read the Bible and pray together daily during National Christian Family Week and to undertake one project in which all members of the family participate.

Every community, if it is alive has a spirit, and that spirit is the center of its unity and identity.

`` Every woman in the block has tried that ''.

: Every set has a choice function.

Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.

** Every surjective function has a right inverse.

** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.

The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.

** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.

** Antichain principle: Every partially ordered set has a maximal antichain.

** Every vector space has a basis.

** Every field has an algebraic closure.

** Every field extension has a transcendence basis.

** Every Tychonoff space has a Stone – Čech compactification.

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

Every unit of length has a corresponding unit of area, namely the area of a square with the given side length.

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 ATM cell has an 8-or 12-bit Virtual Path Identifier ( VPI ) and 16-bit Virtual Channel Identifier ( VCI ) pair defined in its header.