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For topological and simplicial categories the definition of the opposite category is the same as the notion from classical category theory.
For a simplicial set we obtain the opposite simplicial set by component- wise reversing the order of the ordinal.
Let be a simplicial set. Let be a linear ordered set. Then the face and degeneracy maps on are given by.
Let be a simplicial set. Let be vertices. Then the simplicial mapping space is defined by
where denotes the adjoint of the homotopy coherent nerve: the homotopy coherent realization?. We have
where denotes the Yoneda embedding and denotes the cosimplicial-thickening functor. We think of as assigning to an ordinal (considered as a category) a simplicially-enriched category which is thickened.
Let be an -category. Two parallel edges of are called homotopic if there is a -simplex joining them.
Homotopy is an equivalence relation on .
Let be a classical category. Then
exhibits as a full reflective subcategory of . Here denotes the (classical) nerve functor an assigns to a simplicial set its homotopy category. Joyal calls the fundamental category of since if is a Kan complex is the fundamental groupoid of .
Moreover can be written as a composition
where denotes the simplicial nerve functor and denotes inclusion.
is a reflective subcategory.
(presentation of the homotopy category by generators and relations) Let be a simplicial set.
We have
For each , there is a morphism .
For each , we have
For each vertex of , the morphism is the identity .
Let be a simplicial set.
Vertices of are called objects of .
Edges are called morphisms of .
A morphism in is called an equivalence if it is an isomorphism in the homotopy category .
Two parallel edges of are called equivalent if there is a -simplex between them which is an equivalence.
Let be a simplicial set. The the following conditions are equivalent:
is an -category and is a groupoid.
satisfies the horn-filling condition.
satisfies the horn-filling condition for all horns except the left outer horn.
satisfies the horn-filling condition for all horns except the right outer horn.