Initial objects on $infty$-categories
$begingroup$
Let $X in mathbf{Set}_{Delta}$ an $infty$-category and $tau_1$ the left adjoint functor to the nerve $mathrm{N} colon mathbf{Cat} to mathbf{Set}_{Delta}$.
Show that if $x$ is an initial object in $X$, then $x$ is an initial object in $tau_1(X)$.
I don't really know where to start here. An object $x in X$ is initial if the map $h colon X_{/x} to X$ is a trivial Kan fibration (by which I guess it means that it is both a weak homotopy equivalence and a Kan fibration), where $h$ is defined by restricting
$$Delta^0 star Delta^n to mathcal{C}$$
to $Delta^n$.
On the other hand, objects in $tau_1(X)$ are the $0$-simplices of $X_0$. But I don't know how to prove the result from that. Thank you in advance.
category-theory homotopy-theory simplicial-stuff higher-category-theory
asked Dec 31 '18 at 19:09
user313212user313212
331520
$endgroup$
add a comment |
$begingroup$
Let $X in mathbf{Set}_{Delta}$ an $infty$-category and $tau_1$ the left adjoint functor to the nerve $mathrm{N} colon mathbf{Cat} to mathbf{Set}_{Delta}$.
Show that if $x$ is an initial object in $X$, then $x$ is an initial object in $tau_1(X)$.
I don't really know where to start here. An object $x in X$ is initial if the map $h colon X_{/x} to X$ is a trivial Kan fibration (by which I guess it means that it is both a weak homotopy equivalence and a Kan fibration), where $h$ is defined by restricting
$$Delta^0 star Delta^n to mathcal{C}$$
to $Delta^n$.
On the other hand, objects in $tau_1(X)$ are the $0$-simplices of $X_0$. But I don't know how to prove the result from that. Thank you in advance.
category-theory homotopy-theory simplicial-stuff higher-category-theory
asked Dec 31 '18 at 19:09
user313212user313212
331520
$endgroup$
add a comment |
$begingroup$
Let $X in mathbf{Set}_{Delta}$ an $infty$-category and $tau_1$ the left adjoint functor to the nerve $mathrm{N} colon mathbf{Cat} to mathbf{Set}_{Delta}$.
Show that if $x$ is an initial object in $X$, then $x$ is an initial object in $tau_1(X)$.
I don't really know where to start here. An object $x in X$ is initial if the map $h colon X_{/x} to X$ is a trivial Kan fibration (by which I guess it means that it is both a weak homotopy equivalence and a Kan fibration), where $h$ is defined by restricting
$$Delta^0 star Delta^n to mathcal{C}$$
to $Delta^n$.
On the other hand, objects in $tau_1(X)$ are the $0$-simplices of $X_0$. But I don't know how to prove the result from that. Thank you in advance.
category-theory homotopy-theory simplicial-stuff higher-category-theory
asked Dec 31 '18 at 19:09
user313212user313212
331520
$endgroup$
Let $X in mathbf{Set}_{Delta}$ an $infty$-category and $tau_1$ the left adjoint functor to the nerve $mathrm{N} colon mathbf{Cat} to mathbf{Set}_{Delta}$.
Show that if $x$ is an initial object in $X$, then $x$ is an initial object in $tau_1(X)$.
I don't really know where to start here. An object $x in X$ is initial if the map $h colon X_{/x} to X$ is a trivial Kan fibration (by which I guess it means that it is both a weak homotopy equivalence and a Kan fibration), where $h$ is defined by restricting
$$Delta^0 star Delta^n to mathcal{C}$$
to $Delta^n$.
On the other hand, objects in $tau_1(X)$ are the $0$-simplices of $X_0$. But I don't know how to prove the result from that. Thank you in advance.
category-theory homotopy-theory simplicial-stuff higher-category-theory
category-theory homotopy-theory simplicial-stuff higher-category-theory
asked Dec 31 '18 at 19:09
user313212user313212
331520
asked Dec 31 '18 at 19:09
user313212user313212
331520
asked Dec 31 '18 at 19:09
user313212user313212
331520
asked Dec 31 '18 at 19:09
user313212user313212
331520
asked Dec 31 '18 at 19:09
user313212user313212
331520
331520
add a comment |
add a comment |
1 Answer
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$begingroup$
Here are some hints. First, you need a much better idea of what a trivial (Kan is often omitted) fibration is. It's a Kan fibration and a weak equivalence, but much more usefully it's a map which permits lifting against all cofibrations. Equivalently, against all generating cofibrations, namely the inclusions $partial Delta^nto Delta^n$. In short, $x$ is initial when fillings of a sphere under $x$ (that is, a map $partialDelta^nto X_{x/}$) in $X$ lift to fillings in $X_{x/}$.
Now, you should consider what this means for small $n$. Note that the boundary of $Delta^0$ is empty; so what do we know about an initial $x$ from the $n=0$ case? Next consider the case $n=1$, especially when the filler $Delta^1to X$ is an identity edge. The cases $n=0,1$ are all that are needed for, and are equivalent to, your result.
answered Dec 31 '18 at 20:46
Kevin CarlsonKevin Carlson
32.7k23271
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Here are some hints. First, you need a much better idea of what a trivial (Kan is often omitted) fibration is. It's a Kan fibration and a weak equivalence, but much more usefully it's a map which permits lifting against all cofibrations. Equivalently, against all generating cofibrations, namely the inclusions $partial Delta^nto Delta^n$. In short, $x$ is initial when fillings of a sphere under $x$ (that is, a map $partialDelta^nto X_{x/}$) in $X$ lift to fillings in $X_{x/}$.
Now, you should consider what this means for small $n$. Note that the boundary of $Delta^0$ is empty; so what do we know about an initial $x$ from the $n=0$ case? Next consider the case $n=1$, especially when the filler $Delta^1to X$ is an identity edge. The cases $n=0,1$ are all that are needed for, and are equivalent to, your result.
answered Dec 31 '18 at 20:46
Kevin CarlsonKevin Carlson
32.7k23271
$endgroup$
add a comment |
$begingroup$
Here are some hints. First, you need a much better idea of what a trivial (Kan is often omitted) fibration is. It's a Kan fibration and a weak equivalence, but much more usefully it's a map which permits lifting against all cofibrations. Equivalently, against all generating cofibrations, namely the inclusions $partial Delta^nto Delta^n$. In short, $x$ is initial when fillings of a sphere under $x$ (that is, a map $partialDelta^nto X_{x/}$) in $X$ lift to fillings in $X_{x/}$.
Now, you should consider what this means for small $n$. Note that the boundary of $Delta^0$ is empty; so what do we know about an initial $x$ from the $n=0$ case? Next consider the case $n=1$, especially when the filler $Delta^1to X$ is an identity edge. The cases $n=0,1$ are all that are needed for, and are equivalent to, your result.
answered Dec 31 '18 at 20:46
Kevin CarlsonKevin Carlson
32.7k23271
$endgroup$
add a comment |
$begingroup$
Here are some hints. First, you need a much better idea of what a trivial (Kan is often omitted) fibration is. It's a Kan fibration and a weak equivalence, but much more usefully it's a map which permits lifting against all cofibrations. Equivalently, against all generating cofibrations, namely the inclusions $partial Delta^nto Delta^n$. In short, $x$ is initial when fillings of a sphere under $x$ (that is, a map $partialDelta^nto X_{x/}$) in $X$ lift to fillings in $X_{x/}$.
Now, you should consider what this means for small $n$. Note that the boundary of $Delta^0$ is empty; so what do we know about an initial $x$ from the $n=0$ case? Next consider the case $n=1$, especially when the filler $Delta^1to X$ is an identity edge. The cases $n=0,1$ are all that are needed for, and are equivalent to, your result.
answered Dec 31 '18 at 20:46
Kevin CarlsonKevin Carlson
32.7k23271
$endgroup$
Here are some hints. First, you need a much better idea of what a trivial (Kan is often omitted) fibration is. It's a Kan fibration and a weak equivalence, but much more usefully it's a map which permits lifting against all cofibrations. Equivalently, against all generating cofibrations, namely the inclusions $partial Delta^nto Delta^n$. In short, $x$ is initial when fillings of a sphere under $x$ (that is, a map $partialDelta^nto X_{x/}$) in $X$ lift to fillings in $X_{x/}$.
Now, you should consider what this means for small $n$. Note that the boundary of $Delta^0$ is empty; so what do we know about an initial $x$ from the $n=0$ case? Next consider the case $n=1$, especially when the filler $Delta^1to X$ is an identity edge. The cases $n=0,1$ are all that are needed for, and are equivalent to, your result.
answered Dec 31 '18 at 20:46
Kevin CarlsonKevin Carlson
32.7k23271
edited Jan 1 at 21:16
answered Dec 31 '18 at 20:46
Kevin CarlsonKevin Carlson
32.7k23271
answered Dec 31 '18 at 20:46
Kevin CarlsonKevin Carlson
32.7k23271
answered Dec 31 '18 at 20:46
Kevin CarlsonKevin Carlson
32.7k23271
32.7k23271
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