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simplicial category
The simplicial category is defined as the small category whose objects are the totally ordered finite sets
| (1) |
and whose morphisms are monotonic non-decreasing (order-preserving) maps. It is generated by two families of morphisms:
The morphisms are called face maps, and the morphisms are called degeneracy maps. They satisfy the following relations,
| (2) | |||||
| (3) | |||||
| (4) |
There is a bifunctor defined by
| (5) | |||||
| (6) |
where and . Sometimes, the simplicial category is defined to include the empty set , which provides an initial object for the category. This makes a strict monoidal category as is a unit for the bifunctor: and . Further, is then the free monoidal category on a monoid object (the monoid object being [0], with product ).
There is a fully faithful functor from to , which sends each object to an oriented -simplex. The face maps then embed an -simplex in an -simplex, and the degeneracy maps collapse an -simplex to an -simplex. The bifunctor forms a simplex from the disjoint union of two simplicies by joining their vertices together in a way compatible with their orientations.
There is also a fully faithful functor from to , which sends each object to a pre-order . The pre-order is the category consisting of partially-ordered objects, with one morphism if and only if .
Mathematics Subject Classification
18G30 Simplicial sets, simplicial objects (in a category)- Forums
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