equivalences in/of $(\infty,1)$-categories
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
Recall the following familiar 1-categorical statement:
The idea of $(\infty,1)$-toposes is to generalize the above situation from $1$ to $(\infty,1)$ (recall the notion of (n,r)-category and see the general discussion at ∞-topos):
A Grothendieck–Rezk–Lurie $(\infty,1)$-topos $\mathbf{H}$ is an accesible left exact reflective sub-(∞,1)-category of an (∞,1)-category of (∞,1)-presheaves
If the localization in is a topological localization then $\mathbf{H}$ is an (∞,1)-category of (∞,1)-sheaves.
Equivalently:
An $(\infty,1)$-topos $\mathbf{H}$ is
an (∞,1)-category that satisfies the $(\infty,1)$-categorical analogs of Giraud's axioms:
This is part of the statement of HTT, theorm 6.1.0.6.
This is derived from the following equivalent one:
An object classifier is a (small) self-reflections of the $\infty$-topos inside itself (type of types, internal universe). See also (WdL, book 2, section 1).
A further equivalent one (essentially by an invocation of the adjoint functor theorem) is:
A morphism between $(\infty,1)$-toposes is an (∞,1)-geometric morphism.
The (∞,1)-category of all $(\infty,1)$-topos is (∞,1)Toposes.
for the moment see
for the moment see
Another main theorem about $(\infty,1)$-toposes is that models for ∞-stack (∞,1)-toposes are given by the model structure on simplicial presheaves. See there for details
Every ∞-stack $(\infty,1)$-topos $\mathbf{H}$ has a canonical (∞,1)-geometric morphism to the terminal $\infty$-stack $(\infty,1)$-topos ∞Grpd: the direct image is the global sections (∞,1)-functor $\Gamma$, the inverse image is the constant ∞-stack functor
Being a locally presentable (∞,1)-category, an $(\infty,1)$-topos $\mathbf{H}$ is powered and copowered over ∞Grpd, as described at (∞,1)-tensoring.
For any $K \in \infty Grpd$ and $X \in \mathbf{H}$ the powering is the (∞,1)-limit over the diagram constant on $X$
and the $(\infty,1)$-copowering is is the (∞,1)-colimit over the diagram constant on $X$
Under Isbell duality the powering operation corresponds to higher order Hochschild cohomology in $X$, as discussed there.
Below we discuss that the powering is equivalently given by the internal hom (mapping stack) out of the constant ∞-stack $LConst K$ on $K$:
Every $(\infty,1)$-topos is a cartesian closed (∞,1)-category.
By the fact that every $(\infty,1)$-topos $\mathbf{H}$ has universal colimits it follows that for every object $X$ the (∞,1)-functor
preserves all (∞,1)-colimits. Since every $(\infty,1)$-topos is a locally presentable (∞,1)-category it follows with the adjoint (∞,1)-functor theorem that there is a right adjoint (∞,1)-functor
For $C$ an (∞,1)-site for $\mathbf{H}$ we have that the internal hom (mapping stack) $[X,-]$ is given on $A \in \mathbf{H}$ by the (∞,1)-sheaf
where $y : C \to \mathbf{H}$ is the (∞,1)-Yoneda embedding and $L : PSh_C \to \mathbf{H}$ denotes ∞-stackification.
The argument is entirely analogous to that of the closed monoidal structure on sheaves.
We use the full and faithful geometric embedding $(L \dashv i) : \mathbf{H} \hookrightarrow PSh_C$ and the (∞,1)-Yoneda lemma to find for all $U \in C$ the value
and then the fact that ∞-stackification $L$ is left adjoint to inclusion to get
Then the defining adjunction $(X \times (-) \dashv [X,-])$ gives
Finite colimits may be taken out of the internal hom: For $I$ a finite $(\infty,1)$-category and $X : I \to \mathbf{H}$ a diagram, we have for all $A \in \mathbf{H}$
By the above proposition we have
By universal colimits in $\mathbf{H}$ this is
Using the fact that the hom-functor sends colimits in the first argument to limits this is
By the internal hom adjunction and Yoneda this is
Since (∞,1)-limits in the (∞,1)-category of (∞,1)-presheaves are computed objectwise, this is
Finally, because $L$ is a left exact (∞,1)-functor this is also the (∞,1)-limit in $\mathbf{H}$.
For $S \in$ ∞Grpd write $LConst S$ for its inverse image under the global section (∞,1)-geometric morphism $(LConst \dashv \Gamma) : \mathbf{H} \to \infty Grpd$: the constant ∞-stack on $S$.
Then the internal hom $[LConst S,A]$ coincides with the (∞,1)-powering of $A$ by $S$:
By the above we have
As the notation indicates, $LConst S$ is precisely $L Const S$: the ∞-stackification of the (∞,1)-presheaf that is literally constant on $S$. Morover $L$ is a left exact (∞,1)-functor and hence commutes with (∞,1)-products, so that
By the defining geometric embedding $(L \dashv i)$ this is
Since limits of (∞,1)-presheaves are taken objectwise, we have in the first argument the tensoring of $y(U)$ over $S$
By the defining property of tensoring and cotensoring (or explicitly writing out $S \cdot y(U) = {\lim_\to}_{S} const y(U)$, taking the colimit out of the hom, thus turning it into a limit and then inserting that back in the second argument) this is
So finally with the (∞,1)-Yoneda lemma we have
For $\mathbf{H}$ an $(\infty,1)$-topos and $X \in \mathbf{H}$ an object, the over-(∞,1)-category $\mathbf{H}_{/X}$ is itself an $(\infty,1)$-topos – an over-(∞,1)-topos. The projection $\pi_! : \mathbf{H}_{/X} \to \mathbf{H}$ part of an essential geometric morphism
This is HTT, prop. 6.3.5.1.
The $(\infty,1)$-topos $\mathbf{H}_{/X}$ could be called the gros topos of $X$. A geometric morphism $\mathbf{K} \to \mathbf{H}$ that factors as $\mathbf{K} \to \mathbf{H}_{/X} \stackrel{\pi}{\to} \mathbf{H}$ is called an etale geometric morphism.
$(\infty,1)$-Toposes provide categorical semantics for homotopy type theory with a univalent Tarskian type of types (which inteprets as the object classifier).
For more on this see at
Most of the standard constructions in topos theory have or should have immediate generalizations to the context of $(\infty,1)$-toposes, since all notions of category theory exist for (∞,1)-categories.
For instance there are evident notions of
Moreover, it turns out that $(\infty,1)$-toposes come with plenty of internal structures, more than canonically present in an ordinary topos. Every $(\infty,1)$-topos comes with its intrinsic notion of
and with an intrinsic notion of
In classical topos theory, cohomology and homotopy of a topos $E$ are defined in terms of simplicial objects in $C$. If $E$ is a sheaf topos with site $C$ and enough points, then this classical construction is secretly really a model for the intrinsic cohomology and homotopy in the above sense of the hypercomplete (∞,1)-topos of ∞-stacks on $C$.
The beginning of a list of all the structures that exist intrinsically in a big $(\infty,1)$-topos is at
But $(\infty,1)$-topos theory in the style of an $\infty$-analog of the Elephant is only barely beginning to be conceived.
There are some indications as to what the
should be.
$(\infty,1)$-topos, elementary (∞,1)-topos
Locally presentable categories: Large categories whose objects arise from small generators under small relations.
(n,r)-categories… | satisfying Giraud's axioms | inclusion of left exact 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 |
In retrospect it turns out that the homotopy categories of (∞,1)-toposes have been known since
And the model category theory models have been known since Andre Joyal proposed the model structure on simplicial sheaves in his letter to Alexander Grothendieck.
This work used 1-categorical sites. The generalization to (∞,1)categorical sites – modeled by model sites – was discussed in
and “model topos”-theory was also developed in
The intrinsic category-theoretic definition of an (∞,1)-topos was given in section 6.1 of
building on ideas by Charles Rezk. There is is also proven that the Brown-Joyal-Jardine-Toën-Vezzosi models indeed precisely model $\infty$-stack $(\infty,1)$-toposes. Details on this relation are at models for ∞-stack (∞,1)-toposes.
An overview is in
A useful collection of facts of simplicial homotopy theory and (infinity,1)-topos theory is in
A quick introduction to the topic is in
A discussion of the $(\infty,1)$-universal colimits in terms of model category presentations is due to
More on this with an eye on associated ∞-bundles is in