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I'd think the construction in question is the homotopy coimage of $f$ (it's unfortunately called "coimage" even though it behaves like the image).

First one forms the homotopy Cech nerve

$$C(f) = \left( \cdots X \times_Y X \times_Y X \stackrel{\to}{\stackrel{\to}{\to}}X \times_Y X \stackrel{\to}{\to} X \right)$$

This is the internal groupoid object that encodes the $\infty$-equivalence relation on the elements of $X$ "is equivalent in $Y$"

Forming its homotopy colimit

$$coim(f) := \lim_{\leftarrow} lim_{\to} C(f)$$

produces the homotopy quotient of $X$ by this equivlence relation.

As an example, take $\mathbf{B}G$ the one-object groupoid of a group $G$ and $* \to \mathbf{B}G$ the point inclusion. One wants to see that the homotopy image of the point inclusion $* \to \mathbf{B}G$ is not just the point, but is $*//G$, i.e. $\mathbf{B}G$ itself, because there is $G$ worth of ways for the point to be equivalent to itself after inclusion into $\mathbf{B}G$.

So one computes the homotopy Cech nerve and finds the familiar

$$C(* \to \mathbf{B}G) = \left( \cdots G \times G \times G \stackrel{\to}{\stackrel{\to}{\to}} G\times G \stackrel{\to}{\to} G \right)$$

but now regarded as a simplicial object in $\infty Grpd$. This is the diagram that encodes the action of $G$ on the point. Its homotopy colimit is indeed again $\mathbf{B}G$, so that we find

$$coim(* \to \mathbf{B}G) = \mathbf{B}G$$

That this comes out this way is an example of Giraud's axioms at work, since $\infty Grpd$ happens to be an $\infty$-topos: this implies that every $\infty$-groupoid object (simplicial diagram as above) in $\infty Grpd$ is effective .

2 added 10 characters in body

The

I'd think the construction in question is the homotopy coimage of $f$ (it's unfortunately called "coimage" even though it behaves like the image).

First one forms the homotopy Cech nerve

$$C(f) = \left( \cdots X \times_Y X \times_Y X \stackrel{\to}{\stackrel{\to}{\to}}X \times_Y X \stackrel{\to}{\to} X \right)$$

This is the internal groupoid object that encodes the $\infty$-equivalence relation on the elements of $X$ "is equivalent in $Y$"

Forming its homotopy colimit

$$coim(f) := \lim_{\leftarrow} C(f)$$

produces the homotopy quotient of $X$ by this equivlence relation.

As an example, take $\mathbf{B}G$ the one-object groupoid of a group $G$ and $* \to \mathbf{B}G$ the point inclusion. One wants to see that the homotopy image of the point inclusion $* \to \mathbf{B}G$ is not just the point, but is $*//G$, i.e. $\mathbf{B}G$ itself, because there is $G$ worth of ways for the point to be equivalent to itself after inclusion into $\mathbf{B}G$.

So one computes the homotopy Cech nerve and finds the familiar

$$C(* \to \mathbf{B}G) = \left( \cdots G \times G \times G \stackrel{\to}{\stackrel{\to}{\to}} G\times G \stackrel{\to}{\to} G \right)$$

but now regarded as a simplicial object in $\infty Grpd$. This is the diagram that encodes the action of $G$ on the point. Its homotopy colimit is indeed again $\mathbf{B}G$, so that we find

$$coim(* \to \mathbf{B}G) = \mathbf{B}G$$

That this comes out this way is an example of Giraud's axioms at work, since $\infty Grpd$ happens to be an $\infty$-topos: this implies that every $\infty$-groupoid object (simplicial diagram as above) in $\infty Grpd$ is effective .

1

The construction in question is the homotopy coimage of $f$ (it's unfortunately called "coimage" even though it behaves like the image).

First one forms the homotopy Cech nerve

$$C(f) = \left( \cdots X \times_Y X \times_Y X \stackrel{\to}{\stackrel{\to}{\to}}X \times_Y X \stackrel{\to}{\to} X \right)$$

This is the internal groupoid object that encodes the $\infty$-equivalence relation on the elements of $X$ "is equivalent in $Y$"

Forming its homotopy colimit

$$coim(f) := \lim_{\leftarrow} C(f)$$

produces the homotopy quotient of $X$ by this equivlence relation.

As an example, take $\mathbf{B}G$ the one-object groupoid of a group $G$ and $* \to \mathbf{B}G$ the point inclusion. One wants to see that the homotopy image of the point inclusion $* \to \mathbf{B}G$ is not just the point, but is $*//G$, i.e. $\mathbf{B}G$ itself, because there is $G$ worth of ways for the point to be equivalent to itself after inclusion into $\mathbf{B}G$.

So one computes the homotopy Cech nerve and finds the familiar

$$C(* \to \mathbf{B}G) = \left( \cdots G \times G \times G \stackrel{\to}{\stackrel{\to}{\to}} G\times G \stackrel{\to}{\to} G \right)$$

but now regarded as a simplicial object in $\infty Grpd$. This is the diagram that encodes the action of $G$ on the point. Its homotopy colimit is indeed again $\mathbf{B}G$, so that we find

$$coim(* \to \mathbf{B}G) = \mathbf{B}G$$

That this comes out this way is an example of Giraud's axioms at work, since $\infty Grpd$ happens to be an $\infty$-topos: this implies that every $\infty$-groupoid object (simplicial diagram as above) in $\infty Grpd$ is effective .