6.3 Obsolete Constructions

The tags in this section were removed from the main text because they make use of a definition or convention which was modified.

Remark This tag referred to the use of the term “equivalence” in higher category theory, which has now been replaced by “isomorphism”.

Remark This tag was removed because the definition of $2$-category was changed.

Let $\operatorname{\mathcal{C}}$ be a strictly unitary $2$-category. Then the triangle identity $(T)$ of Definition can be stated more simply as the assertion that for every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$, the associativity constraint $\alpha _{g, \operatorname{id}_ Y, f}$ coincides with the identity map from $g \circ (\operatorname{id}_{Y} \circ f) = g \circ f = (g \circ \operatorname{id}_{Y} ) \circ f$ to itself.

Remark By swapping the roles of the monomorphisms $i: A_{} \hookrightarrow B_{}$ and $i': A'_{} \hookrightarrow B'_{}$ in the proof of Proposition, we obtain a proof of Theorem (which is essentially the same as the proof given in §3.1.3).

Construction (The $\infty $-Category of Pointed Spaces). Let $(\operatorname{Set_{\Delta }})_{\ast }$ denote the category whose objects are pointed simplicial sets $(X,x)$ and pointed morphisms between them. We regard $(\operatorname{Set_{\Delta }})_{\ast }$ as a simplicial category, where the simplicial set of morphisms from $(X,x)$ to $(Y,y)$ is given by the fiber product $\operatorname{Fun}(X,Y) \times _{ \operatorname{Fun}( \{ x\} , Y) } \{ y\} $. Let $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ denote the full subcategory of $(\operatorname{Set_{\Delta }})_{\ast }$ spanned by the pointed Kan complexes, so that $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ inherits the structure of a simplicial category. We let $\operatorname{\mathcal{S}}_{\ast }$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( (\operatorname{Set}_{\Delta }^{\circ })_{\ast } )$. Corollary (and Remark guarantee that the simplicial category $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ is locally Kan, so the simplicial set $\operatorname{\mathcal{S}}_{\ast } = \operatorname{N}_{\bullet }^{\operatorname{hc}}( (\operatorname{Set}_{\Delta }^{\circ })_{\ast } )$ is an $\infty $-category (Theorem We will refer to $\operatorname{\mathcal{S}}_{\ast }$ as the $\infty $-category of pointed spaces.

Remark The definition of the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ can be viewed as a special case of Construction, applied to the simplicial category $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ of Construction Invoking Proposition, we see that the category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ of Construction can be identified with the homotopy category of the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ (as suggested by the notation).

Remark In §, we will give a different description of the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ (at least up to equivalence): it can be realized as the $\infty $-category of pointed objects of the $\infty $-category $\operatorname{\mathcal{S}}$ (that is, the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{S}})$ spanned by those diagrams $f: X \rightarrow Y$ where Kan complex $X$ is contractible; see Example ). Beware that the analogous statement does not hold at the level of homotopy categories: there is no formal mechanism to extract the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ from the unpointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ (see Warning