nLab n-image

Contents

Context

$(\infty,1)$-Category theory

(∞,1)-category theory

Contents

Idea

The generalization of the notion of image from category theory to (∞,1)-category theory.

Definition

By epi/mono factorization in an $(\infty,1)$-topos

Let $\mathbf{H}$ be an (∞,1)-topos. Then by the discussion at (n-connected, n-truncated) factorization system there is a orthogonal factorization system

$(Epi(\mathbf{H}), Mono(\mathbf{H})) \subset Mor(\mathbf{H}) \times Mor(\mathbf{H})$

whose left class consists of the (-1)-connected morphisms , the (∞,1)-effective epimorphisms, while the right class consists of the (-1)-truncated morphisms morphisms, hence the (∞,1)-monomorphisms.

So given a morphism $f : X \to Y$ in $\mathbf{H}$ with epi-mono factorization

$f : X \to im_1(f) \hookrightarrow Y \,,$

we may call $im_1(f) \hookrightarrow Y$ the image of $f$.

As the ∞-colimit of the kernel ∞-groupoid

In a sufficiently well-behaved 1-category, the (co)image of a morphism $f \colon X \to Y$ may be defined as the coequalizer of its kernel pair, hence by the fact that

$X \times_Y X \stackrel{\longrightarrow}{\longrightarrow} X \stackrel{}{\longrightarrow} im(f)$

is a colimiting cocone under the parallel morphism diagram.

In an (∞,1)-topos the 1-image is the (∞,1)-colimit not just of these two morphisms, but of the full “$\infty$-kernel”, namely of the full Cech nerve simplicial object

$\cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} X \times_Y X \times_Y X \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} X \times_Y X \stackrel{\longrightarrow}{\longrightarrow} X \stackrel{}{\longrightarrow} im(f) \,.$

(Here all degeneracy maps are notionally suppressed.)

To see that this gives the same notion of image as given by the epi-mono factorization as discussed above, let $f \colon X \stackrel{f}{\longrightarrow} im(f) \hookrightarrow Y$ be such a factorization. Then using (by the discussion at truncated morphism – Recursive characterization) that the (∞,1)-pullback of a monomorphism is its domain, we find a pasting diagram of (∞,1)-pullback squares of the form

$\array{ X \times_{im(f)} X &\to & X &\stackrel{\simeq}{\to} & X \\ \downarrow && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f}} \\ X &\stackrel{f}{\to}& im(f) &\stackrel{\simeq}{\to}& im(f) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow \\ X &\stackrel{f}{\to}& im(f) &\hookrightarrow& Y } \,,$

which shows, by the pasting law, that $X \times_{Y} X \simeq X \times_{im(f)} X$ and hence that the Cech nerve of $X \stackrel{f}{\to} Y$ is equivalently that of the effective epimorphism $X \stackrel{f}{\to} im(f)$. Now, by one of the Giraud-Rezk-Lurie axioms satisfied by (∞,1)-toposes, the (∞,1)-colimit over the Cech nerve of an effective epimorphism is that morphism itself. Therefore the 1-image defined this way coincides with the one defined by the epi/mono factorization.

Tower of $n$-images

More generally for $f \colon X \to Y$ a morphism we have a tower of $n$-images

$X \simeq im_\infty(f) \to \cdots \to im_2(f) \to im_1(f) \to im_0(f) \simeq Y$

factoring $f$, such that for each $n \in \mathbb{N}$ the factorization $X \to im_{n+2}(f) \to Y$ is the (n-connected, n-truncated) factorization system of $f$.

This is the relative Postnikov system of $f$.

Properties

Syntax in homotopy type theory

Let the ambient (∞,1)-category be an (∞,1)-topos $\mathbf{H}$. Then its internal language is homotopy type theory. We discuss the syntax of 1-images in this theory.

Proposition

If

$a\colon A \;\vdash \; f(a) \colon B$

is a term whose categorical semantics is a morphism $A \xrightarrow{f} B$ in $\mathbf{H}$, then the 1-image of that morphism, when regarded as an object of $\mathbf{H}/B$, is the categorical semantics of the dependent type

$(b:B) \; \vdash \; \Big(\Big[ \sum_{a \colon A} (b = f(a)) \Big] : Type\Big).$

Here $=$ denotes the identity type or “path type”, $\sum$ denotes the dependent sum type, and $[-]$ denotes the bracket type (which is constructible in homotopy type theory either as a higher inductive type or using the univalence axiom and a resizing rule to obtain a subobject classifier).

Proof

Let $\mathbf{M}$ be a suitable model category presenting $\mathbf{H}$. Then by the rules for categorical semantics of identity types and substitution, the interpretation of

$(b:B),\, (a:A) \;\vdash\; ((b = f(a)) : Type)$

in $\mathbf{M}$ a is the following pullback $\tilde A$ (see homotopy pullback for more details):

$\array{ \tilde A &\to& B^{I} \\ \downarrow && \downarrow \\ A \times B &\stackrel{(f,id_B)}{\to}& B \times B }.$

Here all objects now denote fibrant object representatives of the given objects in $\mathbf{H}$, and the right-hand morphism is the fibration out of a path space object for $B$. By the factorization lemma the composite $\tilde A \to A \times B \to B$ here is a fibration resolution of the original $A \stackrel{f}{\to} B$ and $\tilde A \to A \times B$ is a fibration resolution of $A \stackrel{(id_A,f)}{\to} A \times B$. Regarded in the slice category $\mathbf{M}/(A \times B)$, this now interprets the syntax $(b = f(a))$ as an $(A \times B)$-dependent type.

Now the interpretation of the sum $\sum_{a:A}$ is simply that we forget the map to $A$ (or equivalently compose with the projection $A\times B\to B$), regarding $\tilde A$ as an object of $\mathbf{M}/B$. Of course, this is just a fibration resolution of $f$ itself.

Finally, the interpretation of the bracket type of this is precisely the (-1)-truncation of this morphism, which by the discussion there is its 1-image $im_1(\tilde A \to B)$, regarded as a dependent type over $B$. Thus, it is precisely the the 1-image of $f$.

By additionally forgetting the remaining map to $B$, we obtain:

Corollary

In the above situation, the 1-image of $f$, regarded as an object of $\mathbf{H}$ itself, is the semantics of the non-dependent type

$\vdash \; \Big(\Big( \sum_{b:B}\Big[\sum_{a:A} (b = f(a))\Big] \Big) : Type\Big).$
Remark

The bracket type of a dependent sum is the propositions as some types version of the existential quantifier, so we can write the dependent type in Prop. as

$(b:B) \; \vdash \; \left(\exists a:A . (b = f(a)) \;:\; hProp\right).$

The dependent sum of an h-proposition is then the propositions-as-some-types version of the comprehension rule, $\{b \in B | \phi(b)\}$, so the non-dependent type in Cor. may be written as

$\left\{ b \in B \,|\, \exists a \in A . (b = f(a)) \right\}$

which is manifestly the naive definition of image.

Compatibility with limits

Proposition

In an (∞,1)-topos $\mathbf{H}$, the $n$-image of a product is the product of $n$-images:

$im_n\left(X_1 \times X_2 \stackrel{(f,g)}{\longrightarrow}) X_2 \times Y_2\right) \simeq \left( im_n(f) \times im_n(g) \longrightarrow X_2 \times Y_2 \right) \,.$
Proof

The morphism $(f,g)$ is the product of $(f,id)$ with $(g,id)$ in the slice (∞,1)-topos over $X_2 \times Y_2$, namely there is a homotopy fiber product in $\mathbf{H}$ of the form

$\array{ && X_1 \times Y_1 \\ & {}^{\mathllap{(id,g)}}\swarrow && \searrow^{\mathrlap{(f,id)}} \\ X_1 \times Y_2 && \swArrow_{\simeq} && X_2 \times Y_1 \\ & {}_{\mathllap{(f,id)}}\searrow && \swarrow_{\mathrlap{(id,g)}} \\ && X_2 \times Y_2 }$

But by this proposition, $n$-truncation in the slice $\infty$-topos preserves products, so that

\begin{aligned} im_n(f,g) & = \tau_n (f,g) \\ & \simeq \tau_n ((f,id)\times_{X_2 \times Y_1} (id,g)) \\ &\simeq (\tau_n(f,id)) \times_{X_2 \times Y_2} (\tau_n(id,g)) \\ & \simeq (im_n f, id)\times_{X_2 \times Y_2} (id, im_n g) \\ & \simeq (im_n f, im_n g) \end{aligned} \,.
Proposition

The $n$-image of the looping of a morphism $f \colon X \to Y$ of pointed objects is the looping of its $(n+1)$-image

$im_n (\Omega(f)) \simeq \Omega(im_{n+1}(f)) \,.$

i.e. we have the following diagram, where the columns are homotopy fiber sequences

$\array{ \Omega f \colon & \Omega X &\longrightarrow& im_n(\Omega f) &\longrightarrow& \Omega Y \\ & \downarrow && \downarrow && \downarrow \\ id_\ast \colon & \ast &\longrightarrow & im_{n+1}(id_\ast) \simeq \ast &\longrightarrow& \ast \\ & \downarrow && \downarrow && \downarrow \\ f\colon & X &\stackrel{}{\longrightarrow}& im_{n+1}(f) &\stackrel{}{\longrightarrow}& Y }$

Examples

Automorphisms

Let $V \in \mathbf{H}$ be a $\kappa$-small object, and let

$* \stackrel{\vdash V}{\to} Obj_\kappa$

the corresponding morphism to the object classifier. Then the 1-image of this is $\mathbf{B}\mathbf{Aut}(V)$, the delooping of the internal automorphism ∞-group of $V$.

Of functors between groupoids

The simplest nontivial case of higher images after the ordinary case of images of functions of sets is images of functors between groupoids hence betwee 1-truncated objects $X, Y \in Grpd \hookrightarrow$ ∞Grpd.

Proposition

For $f \colon X \to Y$ a functor between groupoids, its image factorization is

$f \colon X \to im_2(f) \to im_1(f) \to Y \,,$

where (up to equivalence of groupoids)

• $im_1(f) \to Y$ is the full subgroupoid of $Y$ on those objects $y$ such that there is an object $x \in X$ with $f(x) \simeq y$;

• $im_2(f)$ is the groupoid whose objecs are those of $X$ and whose morphisms are equivalence classes of morphisms in $X$ where $\alpha,\beta \in Mor(X)$ are equivalent if they have the same domain and codomain in $X$ and if $f(\alpha) = f(\beta)$

• $im_2(f)\to im_1(f)$ is the identity on objects and the canonical inclusion on sets of morphisms;

• $X \to im_2(f)$ is the identity on objects and the defining quotient map on sets of morphisms.

Proof

Evidently $im_1(f) \to Y$ is a fibration (isofibration) and so the homotopy fiber over every point is given by the 1-categorical pullback

$\array{ F_y &\to& im_1(y) \\ \downarrow && \downarrow \\ * &\stackrel{y}{\to}& Y } \,.$

If there is no $x \in X$ such that $f(x) \simeq y$ then fiber $F_y$ is empty set. If there is such $x$ then the fiber is the groupoid with that one object and all morphisms in $X$ on that object which are mapped to the identity morphism, which by construction is only the identity morphism itself, hence the fiber is the point. Hence indeed the homotopy fibers of $im_1(f) \to Y$ are (-1)-truncated objects and the map from homotopy fiber of $f$ to those of $im_1(f)$ is their (-1)-truncation.

Next, the homotopy fibers of $im_2(f) \to Y$ over a point $y \in Y$ are the groupoids whose objects are pairs $(x, (f(x) \to y))$ and whose morphisms are pairs

$\left( \array{ x_1 &\stackrel{[\alpha]}{\to}& x_2 \\ f(x_1) &\stackrel{f(\alpha)}{\to}& f(x_2) \\ \downarrow && \downarrow \\ y &=& y } \right) \,.$

Notice first that the above is 0-truncated: an automorphism of $(x,(f(x) \to y))$ is of the form $x \stackrel{[\alpha]}{to} x$ such that $f(\alpha) = id_{f(x)}$ and so there is precisely one such, namely the equivalence class of $id_x$.

Notice second that the homotopy fibers of $f$ itself have the same form, only that $\alpha \colon x_1 \to x_2$ appears itself, not as its equivalence class. Also if two objects in the homotopy fiber of $f$ are connected by a morphism, then by construction so they are in the homotopy fiber of $im_2(f) \to Y$ and hence the latter is indeed the 0-truncation of the former.

Remark

The factroization $f \colon X \to im_2(f) \to im_1(f) \to Y$ of prop. exhibits a ternary factorization system in Grpd.

Remark

This situation can also be considered from the perspective of the formalization of stuff, structure, property. See there at A factorization system.

Discussion of $n$-image factorization in homotopy type theory is in

A construction of $n$-image factorizations in homotopy type theory using only homotopy pushouts and specifically joins (instead of more general higher inductive types) is described in