nLab (infinity,1)Topos

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(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos

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1. Idea

By (,1)Topos(\infty,1)Topos is denoted the collection of all (∞,1)-toposes. This is the (∞,1)-category-theory analog of Topos.

2. Definition

(,1)Topos(\infty,1)\,Topos is the (non-full) sub-(∞,1)-category of (∞,1)Cat on (∞,1)-toposes and (∞,1)-geometric morphisms between them.

Morally, this should actually be an (∞,2)-category, just as Topos is a 2-category, but since the technology of (,2)(\infty,2)-categories is not well developed, this point of view is not often taken yet.

3. Properties

In the following, (,1)Cat(\infty,1)Cat will refer to the (superlarge) (,1)(\infty,1) category of large (,1)(\infty, 1)-categories.

Existence of sites of definition

(…)

Existence of limits and colimits

We discuss existence of (∞,1)-limits and (∞,1)-colimits in (,1)Topos(\infty,1)Topos.

Proposition 3.1. The (,1)(\infty,1)-category (,1)Topos(\infty,1)Topos has all small (,1)(\infty,1)-colimits and functor

(,1)Topos op(,1)Cat (\infty,1)Topos^{op} \to (\infty,1)Cat

preserves small limits.

This is HTT, prop. 6.3.2.3. In the notation there, LTopLTop is the (,1)(\infty,1)-category of toposes whose arrows are the inverse image morphisms, and thus opposite to (,1)Topos(\infty,1)Topos.

Proposition 3.2. The (,1)(\infty,1)-category (,1)Topos(\infty,1)Topos has filtered (∞,1)-limits and the inclusion

(,1)Topos(,1)Cat (\infty,1)Topos \to (\infty,1)Cat

preserves these.

This is HTT, prop. 6.3.3.1.

Proposition 3.3. The (,1)(\infty,1)-category (,1)Topos(\infty,1)Topos has all small (∞,1)-limits.

This is HTT, prop. 6.3.4.7.

Remark 3.4. The (,1)(\infty,1)-limits in (,1)Topos(\infty,1)Topos coincide actually with the proper (,2)(\infty,2)-limits.

This is HTT, remark 6.3.4.10.

Proposition 3.5. The terminal object in (∞,1)Topos is ∞Groupoids.

This is HTT, Prop. 6.3.4.1, for details see at terminal geometric morphism.

Computation of limits and colimits

We discuss more or less explicit descriptions of (∞,1)-limits and (∞,1)-colimits in (,1)Topos(\infty,1)Topos.

Proposition 3.6. Let

𝒳 (g *g *) 𝒴 (f *f *) 𝒵 \array{ && \mathcal{X} \\ && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} }

be a diagram of (∞,1)-toposes. Then its (∞,1)-pullback is computed, roughly, by the (∞,1)-pushout of their (∞,1)-sites of definition (see above).

More in detail: there exist (∞,1)-sites 𝒟˜\tilde \mathcal{D}, 𝒟\mathcal{D}, and 𝒞\mathcal{C} with finite (∞,1)-limit and morphisms of sites

𝒟 g 𝒟˜ f 𝒞 \array{ && \mathcal{D} \\ && \uparrow^{\mathrlap{g}} \\ \tilde \mathcal{D} &\stackrel{f}{\leftarrow}& \mathcal{C} }

such that

( 𝒳 (g *g *) 𝒴 (f *f *) 𝒵)( Sh (,1)(𝒟) (Lan g()g) Sh (,1)(𝒟˜) (Lan f()f) Sh (,1)(𝒞)). \left( \array{ && \mathcal{X} \\ && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} } \right) \,\,\, \simeq \,\,\, \left( \array{ && Sh_{(\infty,1)}(\mathcal{D}) \\ && \downarrow^{\mathrlap{(Lan_g \dashv (-)\circ g)}} \\ Sh_{(\infty,1)}(\tilde \mathcal{D}) &\stackrel{(Lan_f \dashv (-)\circ f)}{\to}& Sh_{(\infty,1)}(\mathcal{C}) } \right) \,.

Let then

𝒟˜ 𝒞𝒟 f 𝒟 g g 𝒟˜ f 𝒞(,1)Cat lex \array{ \tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D} &\stackrel{f'}{\leftarrow}& \mathcal{D} \\ {}^{\mathllap{g'}}\uparrow &\swArrow_{\simeq}& \uparrow^{\mathrlap{g}} \\ \tilde \mathcal{D} &\stackrel{f}{\leftarrow}& \mathcal{C} } \,\,\,\,\, \in (\infty,1)Cat^{lex}

be the (∞,1)-pushout of the underlying (∞,1)-categories in the full sub-(∞,1)-category (∞,1)Cat lex(,1)Cat{}^{lex} \subset (\infty,1)Cat of (,1)(\infty,1)-categories with finite (,1)(\infty,1)-limits.

Let moreover

Sh (,1)(𝒟˜ 𝒞𝒟)PSh (,1)(𝒟˜ 𝒞𝒟) Sh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D}) \hookrightarrow PSh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})

be the reflective sub-(∞,1)-category obtained by localization at the class of morphisms generated by the inverse image Lan f()Lan_{f'}(-) of the coverings of 𝒟\mathcal{D} and the inverse image Lan g()Lan_{g'}(-) of the coverings of 𝒟˜\tilde \mathcal{D}.

Then

Sh (,1)(𝒟˜ 𝒞𝒟) 𝒳 (g *g *) 𝒴 (f *f *) 𝒵 \array{ Sh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D}) &\to& \mathcal{X} \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} }

is an (∞,1)-pullback square.

This is HTT, prop. 6.3.4.6.

Colimits of over-toposes

Proposition 3.7. For any (,1)(\infty,1) topos H\mathbf{H}, there is a colimit preserving functor H(,1)Topos\mathbf{H} \to (\infty,1)Topos sending an object XX to its over-topos H /X\mathbf{H}_{/X}, and sending an arrow f:XYf : X \to Y to the essential geometric morphism f *:H /XH /Yf_* : \mathbf{H}_{/X} \to \mathbf{H}_{/Y}.

Proof. HTT, Prop. 6.1.3.9 implies the “terminal vertex” Cartesian fibration H [1]H\mathbf{H}^{[1]} \to \mathbf{H} is classified by a limit preserving functor H opPr L\mathbf{H}^{op} \to Pr^L, the (,1)(\infty,1)-category of locally presentable (,1)(\infty,1)-categories and colimit-preserving functors between them.

This functor factors through the subcategory (,1)Topos opPr L(\infty,1)Topos^{op} \subseteq Pr^L that sends a geometric morphism to its inverse image part. By [HTT, Prop. 6.3.2.3] and [HTT, Prop. 5.5.1.13], it follows that this is also a limit preserving functor.

The opposite category is then formed by taking right adjoints.  ▮

  • Topos

  • (,1)(\infty,1)Topos

5. References

section 6.3 in

category: category

Last revised on October 18, 2021 at 08:30:34. See the history of this page for a list of all contributions to it.