Spahn HTT, 1.2, the language of higher category theory (Rev #2, changes)

Showing changes from revision #1 to #2: Added | ~~Removed~~ | ~~Chan~~ged

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

(d_{i} : S_{n}^{op} \to S_{n - 1}^{op}) = (d_{n - i} : S_{n} \to S_{n - 1})

~~d_i:S_n^{op}\to~~ (d_i:S_n^{op}\to S_{n-1}^{op})=(d_{n-i}:S_n\to S_{n-1})

(s_{i} : S_{n}^{op} \to S_{n + 1}^{op}) = (s_{n - i} : S_{n} \to S_{n + 1})

~~s_i:S_n^{op}\to~~ (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 $S$ be a simplicial set. Let $x,y\in S$ be vertices. Then the simplicial mapping space is defined by

{Map}_{S} (x, y) : = {Map}_{ℭ | S |} (x, y)

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

where $ℭ | - | : sSet Cat \to sSet s Set$ ~~\mathfrak{C}:sSet\to~~ |-|:sSet ~~sSet~~ Cat\to s Set denotes the adjoint of the~~adjoint of the?~~homotopy coherent nerve~~homotopy coherent nerve?~~: the .homotopy coherent realization?. We have

|-|=Lan_y \mathfrak{C}

where $y:\Delta\hookrightarrow [\Delta^{op},Set]$ denotes the Yoneda embedding and $\mathfrak{C}: \Delta\to sSet Cat$ denotes the cosimplicial-thickening functor. We think of $\mathfrak{C}$ as assigning to an ordinal $[n]$ (considered as a category) a simplicially-enriched category which is thickened.

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 22, 2012 at 01:06:12 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

Spahn HTT, 1.2, the language of higher category theory (Rev #2, changes)

1.2.1 the opposite of an ∞\infty-category

Definition

1.2.2 mapping spaces in higher category theory

Definition

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

1.2.1 the opposite of an $\infty$ -category

1.2.8 joins of $\infty$ -categories

1.2.11 subcategories of $\infty$ -categories

1.2.14 presentations of $\infty$ -categories

1.2.16 the $\infty$ -category of spaces