Join of simplicial sets induces a functor.
$begingroup$
Given a simplicial set $X$, denote by $(mathrm{Set}_{Delta})/_X$ the over category, whose objects are morphisms of simplicial sets with source $X$. I want to show that the joint $X star Y$ of simplicial sets induces a functor
begin{align*}
X star - colon mathrm{Set}_{Delta} & to (mathrm{Set}_{Delta})/_X\
Y & mapsto X hookrightarrow X star Y
end{align*}
There has to be something really stupid that I'm missing, because I can't make sense of what that functor is supposed to do on morphisms. Given a morphism $p colon Y_1 to Y_2$ of simplicial sets, I'm suppose to define a map $$f colon X star Y_1 to X star Y_2$$
such that the composite of the inclusion $X hookrightarrow X star Y$ and $f$ is the same as the inclusion $X hookrightarrow X star Y_2$.
My only guess is that $p colon Y_1 to Y_2$ induces a morphism
$$tilde{p} colon X star Y_1 to X star Y_2$$
via
begin{align*}
tilde{p}_n colon (X star Y_1)_n & to (X star Y_2)_n\
coprod_{[i]oplus [j] = [n]} X_i times (Y_1)_j & mapsto coprod_{[i]oplus [j] = [n]} X_i times p_j(Y_1)_j
end{align*}
But I don't know why this map satisfy the commutativity condition with respect to the inclusions $X hookrightarrow to X star Y_k, k in {1,2}$.
category-theory simplicial-stuff higher-category-theory functors
asked Dec 25 '18 at 17:35
user313212user313212
363520
$endgroup$
add a comment |
$begingroup$
Given a simplicial set $X$, denote by $(mathrm{Set}_{Delta})/_X$ the over category, whose objects are morphisms of simplicial sets with source $X$. I want to show that the joint $X star Y$ of simplicial sets induces a functor
begin{align*}
X star - colon mathrm{Set}_{Delta} & to (mathrm{Set}_{Delta})/_X\
Y & mapsto X hookrightarrow X star Y
end{align*}
There has to be something really stupid that I'm missing, because I can't make sense of what that functor is supposed to do on morphisms. Given a morphism $p colon Y_1 to Y_2$ of simplicial sets, I'm suppose to define a map $$f colon X star Y_1 to X star Y_2$$
such that the composite of the inclusion $X hookrightarrow X star Y$ and $f$ is the same as the inclusion $X hookrightarrow X star Y_2$.
My only guess is that $p colon Y_1 to Y_2$ induces a morphism
$$tilde{p} colon X star Y_1 to X star Y_2$$
via
begin{align*}
tilde{p}_n colon (X star Y_1)_n & to (X star Y_2)_n\
coprod_{[i]oplus [j] = [n]} X_i times (Y_1)_j & mapsto coprod_{[i]oplus [j] = [n]} X_i times p_j(Y_1)_j
end{align*}
But I don't know why this map satisfy the commutativity condition with respect to the inclusions $X hookrightarrow to X star Y_k, k in {1,2}$.
category-theory simplicial-stuff higher-category-theory functors
asked Dec 25 '18 at 17:35
user313212user313212
363520
$endgroup$
1
$begingroup$
Well it is satisfied dimensionwise, so it is satisfied globally
$endgroup$
– Max
Dec 25 '18 at 19:25
add a comment |
$begingroup$
Given a simplicial set $X$, denote by $(mathrm{Set}_{Delta})/_X$ the over category, whose objects are morphisms of simplicial sets with source $X$. I want to show that the joint $X star Y$ of simplicial sets induces a functor
begin{align*}
X star - colon mathrm{Set}_{Delta} & to (mathrm{Set}_{Delta})/_X\
Y & mapsto X hookrightarrow X star Y
end{align*}
There has to be something really stupid that I'm missing, because I can't make sense of what that functor is supposed to do on morphisms. Given a morphism $p colon Y_1 to Y_2$ of simplicial sets, I'm suppose to define a map $$f colon X star Y_1 to X star Y_2$$
such that the composite of the inclusion $X hookrightarrow X star Y$ and $f$ is the same as the inclusion $X hookrightarrow X star Y_2$.
My only guess is that $p colon Y_1 to Y_2$ induces a morphism
$$tilde{p} colon X star Y_1 to X star Y_2$$
via
begin{align*}
tilde{p}_n colon (X star Y_1)_n & to (X star Y_2)_n\
coprod_{[i]oplus [j] = [n]} X_i times (Y_1)_j & mapsto coprod_{[i]oplus [j] = [n]} X_i times p_j(Y_1)_j
end{align*}
But I don't know why this map satisfy the commutativity condition with respect to the inclusions $X hookrightarrow to X star Y_k, k in {1,2}$.
category-theory simplicial-stuff higher-category-theory functors
asked Dec 25 '18 at 17:35
user313212user313212
363520
$endgroup$
Given a simplicial set $X$, denote by $(mathrm{Set}_{Delta})/_X$ the over category, whose objects are morphisms of simplicial sets with source $X$. I want to show that the joint $X star Y$ of simplicial sets induces a functor
begin{align*}
X star - colon mathrm{Set}_{Delta} & to (mathrm{Set}_{Delta})/_X\
Y & mapsto X hookrightarrow X star Y
end{align*}
There has to be something really stupid that I'm missing, because I can't make sense of what that functor is supposed to do on morphisms. Given a morphism $p colon Y_1 to Y_2$ of simplicial sets, I'm suppose to define a map $$f colon X star Y_1 to X star Y_2$$
such that the composite of the inclusion $X hookrightarrow X star Y$ and $f$ is the same as the inclusion $X hookrightarrow X star Y_2$.
My only guess is that $p colon Y_1 to Y_2$ induces a morphism
$$tilde{p} colon X star Y_1 to X star Y_2$$
via
begin{align*}
tilde{p}_n colon (X star Y_1)_n & to (X star Y_2)_n\
coprod_{[i]oplus [j] = [n]} X_i times (Y_1)_j & mapsto coprod_{[i]oplus [j] = [n]} X_i times p_j(Y_1)_j
end{align*}
But I don't know why this map satisfy the commutativity condition with respect to the inclusions $X hookrightarrow to X star Y_k, k in {1,2}$.
category-theory simplicial-stuff higher-category-theory functors
category-theory simplicial-stuff higher-category-theory functors
asked Dec 25 '18 at 17:35
user313212user313212
363520
asked Dec 25 '18 at 17:35
user313212user313212
363520
asked Dec 25 '18 at 17:35
user313212user313212
363520
asked Dec 25 '18 at 17:35
user313212user313212
363520
asked Dec 25 '18 at 17:35
user313212user313212
363520
363520
1
$begingroup$
Well it is satisfied dimensionwise, so it is satisfied globally
$endgroup$
– Max
Dec 25 '18 at 19:25
add a comment |
1
$begingroup$
Well it is satisfied dimensionwise, so it is satisfied globally
$endgroup$
– Max
Dec 25 '18 at 19:25
1
1
$begingroup$
Well it is satisfied dimensionwise, so it is satisfied globally
$endgroup$
– Max
Dec 25 '18 at 19:25
$begingroup$
Well it is satisfied dimensionwise, so it is satisfied globally
$endgroup$
– Max
Dec 25 '18 at 19:25
add a comment |
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$begingroup$
Well it is satisfied dimensionwise, so it is satisfied globally
$endgroup$
– Max
Dec 25 '18 at 19:25