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In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations.Swapping the position of two arguments of an antisymmetric operation yields a result which is the inverse of the result with unswapped arguments.
The above definition can be succinctly phrased by the following equivalent definition: a model category is a category C and three classes of (so-called) weak equivalences W, fibrations F and cofibrations C so that C has all limits and colimits, (,) is a weak factorization system,
In category theory, a branch of mathematics, the opposite category or dual category C op of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself.
There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category C op.Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite ...
When is a small category, the functor category ^ = is cartesian closed.; The poset of subobjects of form a Heyting algebra, whenever is an object of ^ = for small .; For any morphism : of ^, the pullback functor of subobjects : ^ ^ has a right adjoint, denoted , and a left adjoint, .
In mathematics, a property is any characteristic that applies to a given set. [1] Rigorously, a property p defined for all elements of a set X is usually defined as a function p: X → {true, false}, that is true whenever the property holds; or, equivalently, as the subset of X for which p holds; i.e. the set {x | p(x) = true}; p is its indicator function.
In group theory, a branch of mathematics, an opposite group is a way to construct a group from another group that allows one to define right action as a special case of left action. Monoids , groups, rings , and algebras can be viewed as categories with a single object.
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