equivalences in/of $(\infty,1)$-categories
$(\infty,1)Cat$ is the (∞,2)-category of all small (∞,1)-categories.
Its full subcategory on ∞-groupoids is ∞Grpd.
One incarnation of (∞,2)-categories is given by quasi-category-enriched categories (see there for details). As such $(\infty,1)Cat$ is the full SSet-enriched subcategory of SSet on those simplicial sets that are quasi-categories. By the fact described at (∞,1)-category of (∞,1)-functors this is indeed a quasi-category-enriched category.
The model category presenting this (∞,2)-category is the Joyal model structure for quasi-categories $sSet_{Joyal}$. Its full sSet-subcategory is the quasi-category enriched category of quasi-categories from above.
Sometimes it is useful to consider inside the full $(\infty,2)$-catgeory of $(\infty,1)$-categories just the maximal $(\infty,1)$-category and discarding all non-invertible 2-morphisms. This is the (∞,1)-category of (∞,1)-categories.
As an SSet-enriched category the (∞,1)-category of (∞,1)-categories is obtained from the quasi-category-enriched version by picking in each hom-object simplicial set of $(\infty,1)Cat$ the maximal Kan complex.
One model category structure presenting this is the model structure on marked simplicial sets. As a plain model category this is Quillen equivalent to $sSet_{Joyal}$, but as an enriched model category it is $sSet_{Quillen}$ enriched, so that its full SSet-subcategory on fibrant-cofibrant objects presents the $(\infty,1)$-category of $(\infty,1)$-categories.
Limits and colimits over a (∞,1)-functor with values in $(\infty,1)Cat$ may be reformulation in terms of the universal fibration of (infinity,1)-categories $Z \to \infty Grpd^{op}$
Then let $X$ be any (∞,1)-category and
an (∞,1)-functor. Recall that the coCartesian fibration $E_F \to X$ classified by $F$ is the pullback of the universal fibration of (∞,1)-categories $Z$ along F:
Let the assumptions be as above. Then:
The colimit of $F$ is equivalent to $E_F$:
The limit of $F$ is equivalent to the (infinity,1)-category of cartesian section of $E_F \to X$
This is HTT, section 3.3.
The full subcategory of the (∞,1)-category of (∞,1)-categories $Func((\infty,1)Cat, (\infty,1)Cat)$ on those (∞,1)-functors that are equivalences is equivalent to $\{Id, op\}$: it contains only the identity functor and the one that sends an $(\infty,1)$-category to its opposite (infinity,1)-category.
This is due to
It appears as HTT, theorem 5.2.9.1 (arxiv v4+ only)
First of all the statement is true for the ordinary category of posets. This is prop. 5.2.9.14.
From this the statement is deduced for $(\infty,1)$ -categories by observing that posets are characterized by the fact that two parallel functors into them that are objectwise equivalent are already equivalent, prop. 5.2.9.11, which means that posets $C$ are characterized by the fact that
is an injection for all $D \in (\infty,1)Cat$.
This is preserved under automorphisms of $(\infty,1)Cat$, hence any such automorphism preserves posets, hence restricts to an automorphism of the category of posets, hence must be either the identity or $(-)^{op}$ there, by the above statement for posets.
Now finally the main point of the proof is to see that the linear posets $\Delta \subset (\infty,1)Cat$ are dense in $(\infty,1)Cat$, i.e. that the identity transformation of the inclusion functor $\Delta \hookrightarrow (\infty,1)Cat$ exhibits $Id_{(\infty,1)Cat}$ as the left Kan extension
The entries of the following table display models, model categories, and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third table) and of $\mathcal{O}$-monoidal (∞,1)-categories (fourth table).
general pattern | ||||
---|---|---|---|---|
strict enrichment | (∞,1)-category/(∞,1)-operad | |||
$\downarrow$ | $\downarrow$ | |||
enriched (∞,1)-category | $\hookrightarrow$ | internal (∞,1)-category | ||
(∞,1)Cat | ||||
SimplicialCategories | $-$homotopy coherent nerve$\to$ | SimplicialSets/quasi-categories | RelativeSimplicialSets | |
$\downarrow$simplicial nerve | $\downarrow$ | |||
SegalCategories | $\hookrightarrow$ | CompleteSegalSpaces | ||
(∞,1)Operad | ||||
SimplicialOperads | $-$homotopy coherent dendroidal nerve$\to$ | DendroidalSets | RelativeDendroidalSets | |
$\downarrow$dendroidal nerve | $\downarrow$ | |||
SegalOperads | $\hookrightarrow$ | DendroidalCompleteSegalSpaces | ||
$\mathcal{O}$Mon(∞,1)Cat | ||||
DendroidalCartesianFibrations |
$(\infty,1)$Cat, (∞,1)Operad