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This is a subentry of a reading guide to HTT.
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.
Let be a diagram. If are Morphisms we will in general only have an equivalence
and no equality. If for all morphism these equivalences can be chosen in a “coherent” way, is called a coherent diagram.
If is a classical category and is a quasi-category then a homotopy coherent diagram can be defined to be a map of simplicial sets . This encodes the coherence data.
Let , be simplicial sets. A functor from is defined to be a morphism of simplicial sets; i.e. a natural transformation.
We consider only functors from a simplicial set to an -category. By we denote the collection of functors fro, .
Let be a simplicial set.
For every -category , the simplicial set is an -category.
Let be a categorical equivalence of -ctegories. Then the induced map is a categorical equivalence.
Let be an -category. Let be a categorical equivalence of simplicial sets. The the induced map is a categorical equivalence.
The category is a monoidal category where the monoidal structure is induced by the ordinal sum; i.e. the join of simplicial sets is defined by
The empty simplicial set is the monoidal unit. Moreover we have natural isomorphisms
for all .
An important special case of this definition is that of a cone:
Let be a simplicial set. Then is called left cone of and is called right cone of .
If and are quasi-categories, so is .
Compare this notion of cone with the one from classical category theory:
Let and be categories, let , let denote the global element ‘’in ’’, let and be the constant functor in . It is the terminal object in the functor category of its shape.
A natural transformation from the terminal diagram to is called cone for .
These consideration have an application in limits and colimits.
(over-simplicial-set, under-simplicial-set, over-quasi-category, under-quasi-category)
Let , be simplicial sets, let be an arbitrary map. Then there exists a simplicial set satisfying
where the subscript on the right hand side indicates that we only consider those morphisms which restricted to coincide with . We can define by
If is an -category, so is . In this case is called over--category
Dually the under -category is defined analogously by replacing $ with .
If is a classical category, then there is a canonical equivalence
A functor between simplicial sets / simplicially enriched categories / topologically enriched categories is called an essentially surjective functor reps. fully faithful functor if the induced functor between the homotopy categories is.
Let be an -category, let be a subcategory of its homotopy category. Then there is a pullback diagram of simplicial sets
is called a sub--category of spanned by .
(initial object, final object) An object of a simplicial set / a simplicial category / a topological category is called final reps. initial if it is final resp initial in the homotopy category .
(strongly final object) Let be a simplicial set. An object of is called strongly final object if the projection is an acyclic fibration of simplicial sets.
Let be an -category. Let be the full subcategory of spanned by the final vertices of . Then is either empty or a contractible Kan complex.
The following definition says that just as in classical category theory a limit is a terminal cone and a colimit is an initial cocone:
Let be an -category, let be an arbitrary map of simplicial sets.
A colimit for is defined to be an initial object of .
A limit for is defined to be an final object of .
By definition and the formula
a colimit for is equivalently a map extending . We call such a colimit diagram.
An example for a colimit preserving functor is the following: If a functor possessing a colimit factors into another functor followed by a projection out of an over category, then has a colimit and the projection preserves this colimit.
By presentation is meant here (somehow unconcrete) a fibrant replacement of a simplicial set.
If this simplicial set has only finitely many non-degenerate cells this presentation is called finite.
Note that in the Joyal model structure precisely -categories are the fibrant objects and consequently by the axioms of the notion of model category every simplicial set is categorical equivalent to an -category. One such fibrant replacement of a simplicial set is obtained by taking the the nerve of its realization.
For every cardinal we will assume the existence of a strongly inaccessible cardinal . By we denote the collection of sets with cardinality . is a Grothendieck universe.
Let denote the full -category of spanned by the collection of Kan complexes. We regard as a simplicial category. We call the homotopy coherent nerve
the -category of spaces.
Every -category is enriched in .