Product of Semi Simplicial Sets

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Is it true that given semi-simplicial sets $X,Y$, then $|X|×|Y|$ has a natural semi-simplicial structure?

I would guess that it is true, but I cannot find anything about it online. I know that the product in the category of semi-simplicial sets would not be a solution, since, for example, it occurs at every level on its own. And indeed, the realization functor does not preserve limits.

algebraic-topology simplicial-stuff simplicial-complex

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asked Jul 26 at 21:23

Or Kedar

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up vote
3
down vote

favorite

Is it true that given semi-simplicial sets $X,Y$, then $|X|×|Y|$ has a natural semi-simplicial structure?

I would guess that it is true, but I cannot find anything about it online. I know that the product in the category of semi-simplicial sets would not be a solution, since, for example, it occurs at every level on its own. And indeed, the realization functor does not preserve limits.

algebraic-topology simplicial-stuff simplicial-complex

share|cite|improve this question

asked Jul 26 at 21:23

Or Kedar

1777

add a comment | 

up vote
3
down vote

favorite

up vote
3
down vote

favorite

Is it true that given semi-simplicial sets $X,Y$, then $|X|×|Y|$ has a natural semi-simplicial structure?

I would guess that it is true, but I cannot find anything about it online. I know that the product in the category of semi-simplicial sets would not be a solution, since, for example, it occurs at every level on its own. And indeed, the realization functor does not preserve limits.

algebraic-topology simplicial-stuff simplicial-complex

share|cite|improve this question

asked Jul 26 at 21:23

Or Kedar

1777

Is it true that given semi-simplicial sets $X,Y$, then $|X|×|Y|$ has a natural semi-simplicial structure?

I would guess that it is true, but I cannot find anything about it online. I know that the product in the category of semi-simplicial sets would not be a solution, since, for example, it occurs at every level on its own. And indeed, the realization functor does not preserve limits.

algebraic-topology simplicial-stuff simplicial-complex

share|cite|improve this question

asked Jul 26 at 21:23

Or Kedar

1777

share|cite|improve this question

share|cite|improve this question

asked Jul 26 at 21:23

Or Kedar

1777

asked Jul 26 at 21:23

Or Kedar

1777

asked Jul 26 at 21:23

Or Kedar

1777

1777

add a comment | 

add a comment | 

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Yes. Let $X'$ and $Y'$ be the simplicial sets obtained from $X$ and $Y$ by adjoining degeneracies. Then $X'times Y'$ is a simplicial set, which turns out to have the property that every face of a nondegenerate simplex is nondegenerate. So, if you let $Z$ consist of all the nondegenerate simplices of $X'times Y'$, then $Z$ naturally has the structure of a semisimplicial set. We have canonical homeomorphisms $|Z|cong |X'times Y'|cong |X'|times |Y'|cong |X|times |Y|$, so this $Z$ is the semisimplicial set you seek.

Explicitly, a product of two simplices can be triangulated using simplices indexed by shuffles, as described at https://ncatlab.org/nlab/show/product+of+simplices. We apply this decomposition to each pair of simplices from $X$ and $Y$, and then glue them together to get $Z$.

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answered Jul 26 at 21:46

Eric Wofsey

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1 Answer
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1 Answer
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up vote
2
down vote

Yes. Let $X'$ and $Y'$ be the simplicial sets obtained from $X$ and $Y$ by adjoining degeneracies. Then $X'times Y'$ is a simplicial set, which turns out to have the property that every face of a nondegenerate simplex is nondegenerate. So, if you let $Z$ consist of all the nondegenerate simplices of $X'times Y'$, then $Z$ naturally has the structure of a semisimplicial set. We have canonical homeomorphisms $|Z|cong |X'times Y'|cong |X'|times |Y'|cong |X|times |Y|$, so this $Z$ is the semisimplicial set you seek.

Explicitly, a product of two simplices can be triangulated using simplices indexed by shuffles, as described at https://ncatlab.org/nlab/show/product+of+simplices. We apply this decomposition to each pair of simplices from $X$ and $Y$, and then glue them together to get $Z$.

share|cite|improve this answer

answered Jul 26 at 21:46

Eric Wofsey

162k12188299

add a comment | 

up vote
2
down vote

Yes. Let $X'$ and $Y'$ be the simplicial sets obtained from $X$ and $Y$ by adjoining degeneracies. Then $X'times Y'$ is a simplicial set, which turns out to have the property that every face of a nondegenerate simplex is nondegenerate. So, if you let $Z$ consist of all the nondegenerate simplices of $X'times Y'$, then $Z$ naturally has the structure of a semisimplicial set. We have canonical homeomorphisms $|Z|cong |X'times Y'|cong |X'|times |Y'|cong |X|times |Y|$, so this $Z$ is the semisimplicial set you seek.

Explicitly, a product of two simplices can be triangulated using simplices indexed by shuffles, as described at https://ncatlab.org/nlab/show/product+of+simplices. We apply this decomposition to each pair of simplices from $X$ and $Y$, and then glue them together to get $Z$.

share|cite|improve this answer

answered Jul 26 at 21:46

Eric Wofsey

162k12188299

add a comment | 

up vote
2
down vote

up vote
2
down vote

Yes. Let $X'$ and $Y'$ be the simplicial sets obtained from $X$ and $Y$ by adjoining degeneracies. Then $X'times Y'$ is a simplicial set, which turns out to have the property that every face of a nondegenerate simplex is nondegenerate. So, if you let $Z$ consist of all the nondegenerate simplices of $X'times Y'$, then $Z$ naturally has the structure of a semisimplicial set. We have canonical homeomorphisms $|Z|cong |X'times Y'|cong |X'|times |Y'|cong |X|times |Y|$, so this $Z$ is the semisimplicial set you seek.

Explicitly, a product of two simplices can be triangulated using simplices indexed by shuffles, as described at https://ncatlab.org/nlab/show/product+of+simplices. We apply this decomposition to each pair of simplices from $X$ and $Y$, and then glue them together to get $Z$.

share|cite|improve this answer

answered Jul 26 at 21:46

Eric Wofsey

162k12188299

Yes. Let $X'$ and $Y'$ be the simplicial sets obtained from $X$ and $Y$ by adjoining degeneracies. Then $X'times Y'$ is a simplicial set, which turns out to have the property that every face of a nondegenerate simplex is nondegenerate. So, if you let $Z$ consist of all the nondegenerate simplices of $X'times Y'$, then $Z$ naturally has the structure of a semisimplicial set. We have canonical homeomorphisms $|Z|cong |X'times Y'|cong |X'|times |Y'|cong |X|times |Y|$, so this $Z$ is the semisimplicial set you seek.

Explicitly, a product of two simplices can be triangulated using simplices indexed by shuffles, as described at https://ncatlab.org/nlab/show/product+of+simplices. We apply this decomposition to each pair of simplices from $X$ and $Y$, and then glue them together to get $Z$.

share|cite|improve this answer

answered Jul 26 at 21:46

Eric Wofsey

162k12188299

share|cite|improve this answer

share|cite|improve this answer

answered Jul 26 at 21:46

Eric Wofsey

162k12188299

answered Jul 26 at 21:46

Eric Wofsey

162k12188299

answered Jul 26 at 21:46

Eric Wofsey

162k12188299

162k12188299

add a comment | 

add a comment | 

 

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