equivalences in/of $(\infty,1)$-categories
The notion of accessible $(\infty,1)$-category is the generalization of the notion of accessible category from category theory to (∞,1)-category theory.
It is a means to handle $(\infty,1)$-categories that are not essentially small in terms of small data.
An accessible $(\infty,1)$-category is one which may be large, but can entirely be accessed as an $(\infty,1)$-category of “conglomerates of objects” in a small $(\infty,1)$-category – precisely: that it is a category of $\kappa$-small ind-objects in some small $(\infty,1)$-category $C$.
An accessible $(\infty,1)$-category which is also locally presentable is called a compactly generated (∞,1)-category.
Let $\kappa$ be a regular cardinal. spring
A (∞,1)-category $\mathcal{C}$ is $\kappa$-accessible if it satisfies the following equivalent conditions:
There is a small (∞,1)-category $\mathcal{C}^0$ and an equivalence of (∞,1)-categories
of $\mathcal{C}$ with the (∞,1)-category of ind-objects, relative $\kappa$, in $\mathcal{C}^0$.
The $(\infty,1)$-category $\mathcal{C}$
has all $\kappa$-filtered colimits
the full sub-(∞,1)-category $\mathcal{C}^\kappa \hookrightarrow \mathcal{C}$ of $\kappa$-compact objects is an essentially small (∞,1)-category;
$\mathcal{C}^\kappa \hookrightarrow \mathcal{C}$ generates $\mathcal{C}$ under $\kappa$-filtered (∞,1)-colimits.
The $(\infty,1)$-category $\mathcal{C}$
has all $\kappa$-filtered colimits
there is some essentially smallsub-(∞,1)-category $\mathcal{C}' \hookrightarrow \mathcal{C}$ of $\kappa$-compact objects which generates $\mathcal{C}$ under $\kappa$-filtered (∞,1)-colimits.
$\mathcal{C}$ is an idempotent-complete (∞,1)-category.
Generally, $\mathcal{C}$ is called an accessible $(\infty,1)$-category if it is $\kappa$-accessible for some regular cardinal $\kappa$.
These conditions are indeed equivalent.
For the first few this is HTT, prop. 5.4.2.2. The last one is in HTT, section 5.4.3.
An (∞,1)-functor between accessible $(\infty,1)$-categories that preserves $\kappa$-filtered colimits is called an accessible (∞,1)-functor .
Write $(\infty,1)AccCat \subset (\infty,1)Cat$ for the 2-sub-(∞,1)-category of (∞,1)Cat on
those objects that are accessible $(\infty,1)$-categories;
those morphisms for which there is a $\kappa$ such that the (∞,1)-functor is $\kappa$-continuous and preserves $\kappa$-compact objects.
So morphisms are the accessible (∞,1)-functors that also preserves compact objects. (?)
This is HTT, def. 5.4.2.16.
If $C$ is an accessible $(\infty,1)$-category then so are
for $K$ a small simplicial set the (∞,1)-category of (∞,1)-functors $Func(K,C)$;
for $p : K \to C$ a small diagram, the over quasi-category $C_{/p}$ and under-quasi-category $C_{p/}$.
This is HTT section 5.4.4, 5.4.5 and 5.4.6.
The (∞,1)-pullback of accessible $(\infty,1)$-categories in (∞,1)Cat is again accessible.
This is HTT, section 5.4.6.
Generally:
The $(\infty,1)$-category $(\infty,1)AccCat$ has all small (∞,1)-limits and the inclusion
preserves these.
This is HTT, proposition 5.4.7.3.
Locally presentable categories: Large categories whose objects arise from small generators under small relations.
(n,r)-categories… | satisfying Giraud's axioms | inclusion of left exaxt localizations | generated under colimits from small objects | localization of free cocompletion | generated under filtered colimits from small objects | ||
---|---|---|---|---|---|---|---|
(0,1)-category theory | (0,1)-toposes | $\hookrightarrow$ | algebraic lattices | $\simeq$ Porst’s theorem | subobject lattices in accessible reflective subcategories of presheaf categories | ||
category theory | toposes | $\hookrightarrow$ | locally presentable categories | $\simeq$ Adámek-Rosický’s theorem | accessible reflective subcategories of presheaf categories | $\hookrightarrow$ | accessible categories |
model category theory | model toposes | $\hookrightarrow$ | combinatorial model categories | $\simeq$ Dugger’s theorem | left Bousfield localization of global model structures on simplicial presheaves | ||
(∞,1)-topos theory | (∞,1)-toposes | $\hookrightarrow$ | locally presentable (∞,1)-categories | $\simeq$ Simpson’s theorem | accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories | $\hookrightarrow$ | accessible (∞,1)-categories |
The theory of accessible 1-categories is described in
The theory of accessible $(\infty,1)$-categories is the topic of section 5.4 of