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A strict weak order that is trichotomous is called a strict total order. [14] The total preorder which is the inverse of its complement is in this case a total order . For a strict weak order < {\displaystyle \,<\,} another associated reflexive relation is its reflexive closure , a (non-strict) partial order ≤ . {\displaystyle \,\leq .}
2 of 3: if f and g are maps in C such that gf is defined and any two of these are weak equivalences then so is the third. Lifting : acyclic cofibrations have the left lifting property with respect to fibrations, and cofibrations have the left lifting property with respect to acyclic fibrations.
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so too are its restrictions. However, the transitive closure of a restriction is a subset of the restriction of the ...
The meanings of different arrows are not entirely standardized: the arrows used for monomorphisms, epimorphisms, and isomorphisms are also used for injections, surjections, and bijections, as well as the cofibrations, fibrations, and weak equivalences in a model category.
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for
Order, an academic journal on order theory; Dense order, a total order wherein between any unequal pair of elements there is always an intervening element in the order; Glossary of order theory; Lexicographical order, an ordering method on sequences analogous to alphabetical order on words; List of order topics, list of order theory topics
The absolute order is defined analogously to the weak order, but with generating set/alphabet consisting of all conjugates of the Coxeter generators. For example, the permutation (1 2 3) in S 3 has only one reduced word, (12)(23), so covers (12) and (23) in the Bruhat order but only covers (12) in the weak order.
Weak Kan complexes, or quasi-categories, are simplicial sets satisfying a weak version of the Kan condition. André Joyal showed that they are a good foundation for higher category theory. Recently, in 2009, the theory has been systematized further by Jacob Lurie who simply calls them infinity categories, though the latter term is also a ...
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