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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

1. I corrected a couple of wrong claims and added the link to a counter-example

AG

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 21st 2019

  #Hovey

2. exibits -> exhibits

Ian Coley

3. Added the fact that (weak) localizations are initial and final, together with the fact that the initial vertex map from the category of simplices is a weak localization (and hence initial and final). Added references to Shah and Cisinski about this and noted that this can be used to prove a Bousfield-Kan formula for homotopy (co)limits in infinity categories. Hopefully didn’t screw up the whole page- I’m horrible at html/computers in general…

Dylan

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 3rd 2020
• (edited Jul 3rd 2020)

I have added an Example-section “Example: Cofiber products in coslice categories” (here). Currently it reads as follows:

Consider the inclusion of the walking span-category, into the result of adjoining an initial object $t$:

$\mathllap{ (1) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; } \Big\{ \array{ x &\longleftarrow& b &\longrightarrow& y } \Big\} \;\; \overset{\phantom{AAAA}}{\hookrightarrow} \;\; \left\{ \array{ && t \\ & \swarrow & \downarrow & \searrow \\ x &\longleftarrow& b &\longrightarrow& y } \right\}$

One readily sees that for each object on the right, its comma category over this inclusion has contractible nerve, whence Theorem \ref{Recognition} implies that this inclusion is a final $\infty$-functor.

As an application of the finality of (1), observe that for $\mathcal{C}$ an (∞,1)-category and $T \in \mathcal{C}$ an object, (∞,1)-colimits in the under-(∞,1)-category

$\mathcal{C}^{T/} \overset{\;\;U\;\;}{\longrightarrow} \mathcal{C}$

are given by the $\infty$-colimit in $\mathcal{C}$ itself of the given cone of the original diagram, with tip $X$ (by this Prop.): For

$F \;\colon\; \mathcal{I} \longrightarrow \mathcal{C}^{T/}$

a small diagram, we have

$U \big( \underset{\longrightarrow}{\lim}\, F \big) \;\simeq\; \underset{\longrightarrow}{\lim}\, \big( T/U(F) \big)$

(when either $\infty$-colimit exists).

Now for $\mathcal{I}$ the walking span diagram on the left of (1), this means that homotopy cofiber products in $\mathcal{C}^{T/}$ are computed as $\infty$-colimits in $\mathcal{C}$ of diagrams of the shape on the right of (1). But since the inclusion in (1) is final, these are just homotopy cofiber products in $\mathcal{C}$.

Explicitly: Given

$\array{ T &=& T &=& T \\ {}^{\mathllap{ \phi_X }} \big\downarrow && {}^{\mathllap{ \phi_B }} \big\downarrow && {}^{\mathllap{ \phi_Y }} \big\downarrow \\ X & \underset{ f }{\longleftarrow} & B & \underset{ g }{ \longrightarrow } & Y }$

regarded as a span in $\mathcal{C}^T$, hence with underlying objects

$U\big( (X,\phi_X) \big) \;=\; X \,, \;\;\;\;\;\; U\big( (B,\phi_B) \big) \;=\; B \,, \;\;\;\;\;\; U\big( (Y,\phi_Y) \big) \;=\; Y \,,$

we have:

$U \Big( \; (X,\phi_X) \underset{ (B,\phi_B) }{\coprod} (Y,\phi_Y) \; \Big) \;\;\;\simeq\;\;\; X \underset{B}{\coprod} Y \,.$

In particular, if $(B,\phi_B) \;\coloneqq\; (T,id_T)$ is the initial object in $\mathcal{C}^{T/}$, in which case the cofiber product is just the coproduct

$(X,\phi_X) \coprod (Y,\phi_Y) \;\;=\;\; (X,\phi_X) \underset{ (T,id_T) }{\coprod} (Y,\phi_Y)$

we find that the coproduct in the co-slice category is the co-fiber product under the given tip object in the underlying category

$U \Big( \; (X,\phi_X) \coprod (Y,\phi_Y) \; \Big) \;\;\;\simeq\;\;\; X \underset{T}{\coprod} Y \,.$