Spahn HTT, 1.2, the language of higher category theory (Rev #1, changes)

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1.2.1 the opposite of an ∞\infty-category

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.

Definition

Let SS be a simplicial set. Let JJ be a linear ordered set. Then the face and degeneracy maps on S opS^{op} are given by.

d i:S n op→S n−1 op)=(d n−i:S n→S n−1)d_i:S_n^{op}\to S_{n-1}^{op})=(d_{n-i}:S_n\to S_{n-1})
s i:S n op→S n+1 op)=(s n−i:S n→S n+1)s_i:S_n^{op}\to S_{n+1}^{op})=(s_{n-i}:S_n\to S_{n+1})

1.2.2 mapping spaces in higher category theory

Definition

Let SS be a simplicial set. Let x,y∈Sx,y\in S be vertices. Then the simplicial mapping space is defined by

Map S(x,y):=Map ℭ(x,y)Map_S (x,y):=Map_\mathfrak{C} (x,y)

where ℭ:sSet→sSet\mathfrak{C}:sSet\to sSet denotes the adjoint of the?homotopy coherent nerve?.

1.2.3 the homotopy category

1.2.4 objects morphisms and equivalences

1.2.5 groupoids and classical homotopy theory

1.2.6 homotopy commutativity versus homotopy coherence

1.2.7 functors between higher categories

1.2.8 joins of ∞\infty-categories

1.2.9 overcategories and undercategories

1.2.10 fully faithful and essentially surjective functors

1.2.11 subcategories of ∞\infty-categories

1.2.12 initial and final objects

1.2.13 limits and colimits

1.2.14 presentations of ∞\infty-categories

1.2.15 Set-theoretic technicalties

1.2.16 the ∞\infty-category of spaces

Revision on June 21, 2012 at 23:58:15 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

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