VAR, Algebra and local presentability
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Here1, on the page 282, I would like to understand why precisely Examples of $k$-ary operations are all $k-mathrm{lim}$, BUT $k-textrm{colim}$ must be filtered. Where have we used that $k$ is filtered and how taking $k$-colim becomes an operation? And yet, on that page 282 above those examples, what is the difference between $ast$ and $cdot$ in the equation
$$omega_{Gcdot F}=(Gastomega_F)cdot (omega_Gast F)?$$ What squares these $ast$ and $cdot$ amounts to commute?
I more or less understand what comes next in the paper, but for this page.
1 Jiří Adámek and Jiří Rosický: Algebra and local presentability: how algebraic are they? (A survey); Tbilisi Mathematical Journal 10(3) (2017), pp. 279–295; DOI: 10.1515/tmj-2017-0113
category-theory universal-algebra limits-colimits higher-category-theory
edited Nov 22 at 10:51
user302797
19.7k92252
asked Nov 13 at 20:03
user122424
1,0621616
add a comment |
up vote
2
down vote
favorite
Here1, on the page 282, I would like to understand why precisely Examples of $k$-ary operations are all $k-mathrm{lim}$, BUT $k-textrm{colim}$ must be filtered. Where have we used that $k$ is filtered and how taking $k$-colim becomes an operation? And yet, on that page 282 above those examples, what is the difference between $ast$ and $cdot$ in the equation
$$omega_{Gcdot F}=(Gastomega_F)cdot (omega_Gast F)?$$ What squares these $ast$ and $cdot$ amounts to commute?
I more or less understand what comes next in the paper, but for this page.
1 Jiří Adámek and Jiří Rosický: Algebra and local presentability: how algebraic are they? (A survey); Tbilisi Mathematical Journal 10(3) (2017), pp. 279–295; DOI: 10.1515/tmj-2017-0113
category-theory universal-algebra limits-colimits higher-category-theory
edited Nov 22 at 10:51
user302797
19.7k92252
asked Nov 13 at 20:03
user122424
1,0621616
2
There seems to be no reason to create the tag "var." It seems to be much too specific. It doesn't even have a wikipedia page. See here for guidelines on tag creation: meta.stackexchange.com/questions/146949/…
– jgon
Nov 13 at 21:09
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Here1, on the page 282, I would like to understand why precisely Examples of $k$-ary operations are all $k-mathrm{lim}$, BUT $k-textrm{colim}$ must be filtered. Where have we used that $k$ is filtered and how taking $k$-colim becomes an operation? And yet, on that page 282 above those examples, what is the difference between $ast$ and $cdot$ in the equation
$$omega_{Gcdot F}=(Gastomega_F)cdot (omega_Gast F)?$$ What squares these $ast$ and $cdot$ amounts to commute?
I more or less understand what comes next in the paper, but for this page.
1 Jiří Adámek and Jiří Rosický: Algebra and local presentability: how algebraic are they? (A survey); Tbilisi Mathematical Journal 10(3) (2017), pp. 279–295; DOI: 10.1515/tmj-2017-0113
category-theory universal-algebra limits-colimits higher-category-theory
edited Nov 22 at 10:51
user302797
19.7k92252
asked Nov 13 at 20:03
user122424
1,0621616
Here1, on the page 282, I would like to understand why precisely Examples of $k$-ary operations are all $k-mathrm{lim}$, BUT $k-textrm{colim}$ must be filtered. Where have we used that $k$ is filtered and how taking $k$-colim becomes an operation? And yet, on that page 282 above those examples, what is the difference between $ast$ and $cdot$ in the equation
$$omega_{Gcdot F}=(Gastomega_F)cdot (omega_Gast F)?$$ What squares these $ast$ and $cdot$ amounts to commute?
I more or less understand what comes next in the paper, but for this page.
1 Jiří Adámek and Jiří Rosický: Algebra and local presentability: how algebraic are they? (A survey); Tbilisi Mathematical Journal 10(3) (2017), pp. 279–295; DOI: 10.1515/tmj-2017-0113
category-theory universal-algebra limits-colimits higher-category-theory
category-theory universal-algebra limits-colimits higher-category-theory
edited Nov 22 at 10:51
user302797
19.7k92252
asked Nov 13 at 20:03
user122424
1,0621616
edited Nov 22 at 10:51
user302797
19.7k92252
asked Nov 13 at 20:03
user122424
1,0621616
edited Nov 22 at 10:51
user302797
19.7k92252
edited Nov 22 at 10:51
user302797
19.7k92252
edited Nov 22 at 10:51
user302797
19.7k92252
19.7k92252
asked Nov 13 at 20:03
user122424
1,0621616
asked Nov 13 at 20:03
user122424
1,0621616
asked Nov 13 at 20:03
user122424
1,0621616
1,0621616
2
There seems to be no reason to create the tag "var." It seems to be much too specific. It doesn't even have a wikipedia page. See here for guidelines on tag creation: meta.stackexchange.com/questions/146949/…
– jgon
Nov 13 at 21:09
add a comment |
2
There seems to be no reason to create the tag "var." It seems to be much too specific. It doesn't even have a wikipedia page. See here for guidelines on tag creation: meta.stackexchange.com/questions/146949/…
– jgon
Nov 13 at 21:09
2
2
There seems to be no reason to create the tag "var." It seems to be much too specific. It doesn't even have a wikipedia page. See here for guidelines on tag creation: meta.stackexchange.com/questions/146949/…
– jgon
Nov 13 at 21:09
There seems to be no reason to create the tag "var." It seems to be much too specific. It doesn't even have a wikipedia page. See here for guidelines on tag creation: meta.stackexchange.com/questions/146949/…
– jgon
Nov 13 at 21:09
add a comment |
1 Answer
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Morphisms of varieties preserve limits and filtered colimits, but not general colimits. This is essentially the definition of an operation on VAR. The $*$ denotes whiskering a functor with a natural transformation, while $cdot$ denotes (vertical) composition of natural transformations. All the authors are doing here is spelling out the meaning of pseudonatural transformation, which you can find in more verbose form in many places.
answered Nov 13 at 22:53
Kevin Carlson
32.3k23270
Yes, but could you please possibly draw the few commuting squares with labels so that I would easily understand which compositions are meant here? For whiskering as well as for vertical composition?
– user122424
Nov 14 at 14:55
@user122424 I would prefer you just look up the definition of pseudonatural transformation. It's a bit annoying to type diagrams on here.
– Kevin Carlson
Nov 14 at 17:34
OK, and can you please help by giving me a reference at least?
– user122424
Nov 14 at 20:32
Just try Googling the phrase. It's a very standard concept.
– Kevin Carlson
Nov 14 at 22:09
add a comment |
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1 Answer
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up vote
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Morphisms of varieties preserve limits and filtered colimits, but not general colimits. This is essentially the definition of an operation on VAR. The $*$ denotes whiskering a functor with a natural transformation, while $cdot$ denotes (vertical) composition of natural transformations. All the authors are doing here is spelling out the meaning of pseudonatural transformation, which you can find in more verbose form in many places.
answered Nov 13 at 22:53
Kevin Carlson
32.3k23270
Yes, but could you please possibly draw the few commuting squares with labels so that I would easily understand which compositions are meant here? For whiskering as well as for vertical composition?
– user122424
Nov 14 at 14:55
@user122424 I would prefer you just look up the definition of pseudonatural transformation. It's a bit annoying to type diagrams on here.
– Kevin Carlson
Nov 14 at 17:34
OK, and can you please help by giving me a reference at least?
– user122424
Nov 14 at 20:32
Just try Googling the phrase. It's a very standard concept.
– Kevin Carlson
Nov 14 at 22:09
add a comment |
up vote
0
down vote
Morphisms of varieties preserve limits and filtered colimits, but not general colimits. This is essentially the definition of an operation on VAR. The $*$ denotes whiskering a functor with a natural transformation, while $cdot$ denotes (vertical) composition of natural transformations. All the authors are doing here is spelling out the meaning of pseudonatural transformation, which you can find in more verbose form in many places.
answered Nov 13 at 22:53
Kevin Carlson
32.3k23270
Yes, but could you please possibly draw the few commuting squares with labels so that I would easily understand which compositions are meant here? For whiskering as well as for vertical composition?
– user122424
Nov 14 at 14:55
@user122424 I would prefer you just look up the definition of pseudonatural transformation. It's a bit annoying to type diagrams on here.
– Kevin Carlson
Nov 14 at 17:34
OK, and can you please help by giving me a reference at least?
– user122424
Nov 14 at 20:32
Just try Googling the phrase. It's a very standard concept.
– Kevin Carlson
Nov 14 at 22:09
add a comment |
up vote
0
down vote
up vote
0
down vote
Morphisms of varieties preserve limits and filtered colimits, but not general colimits. This is essentially the definition of an operation on VAR. The $*$ denotes whiskering a functor with a natural transformation, while $cdot$ denotes (vertical) composition of natural transformations. All the authors are doing here is spelling out the meaning of pseudonatural transformation, which you can find in more verbose form in many places.
answered Nov 13 at 22:53
Kevin Carlson
32.3k23270
Morphisms of varieties preserve limits and filtered colimits, but not general colimits. This is essentially the definition of an operation on VAR. The $*$ denotes whiskering a functor with a natural transformation, while $cdot$ denotes (vertical) composition of natural transformations. All the authors are doing here is spelling out the meaning of pseudonatural transformation, which you can find in more verbose form in many places.
answered Nov 13 at 22:53
Kevin Carlson
32.3k23270
answered Nov 13 at 22:53
Kevin Carlson
32.3k23270
answered Nov 13 at 22:53
Kevin Carlson
32.3k23270
answered Nov 13 at 22:53
Kevin Carlson
32.3k23270
32.3k23270
Yes, but could you please possibly draw the few commuting squares with labels so that I would easily understand which compositions are meant here? For whiskering as well as for vertical composition?
– user122424
Nov 14 at 14:55
@user122424 I would prefer you just look up the definition of pseudonatural transformation. It's a bit annoying to type diagrams on here.
– Kevin Carlson
Nov 14 at 17:34
OK, and can you please help by giving me a reference at least?
– user122424
Nov 14 at 20:32
Just try Googling the phrase. It's a very standard concept.
– Kevin Carlson
Nov 14 at 22:09
add a comment |
Yes, but could you please possibly draw the few commuting squares with labels so that I would easily understand which compositions are meant here? For whiskering as well as for vertical composition?
– user122424
Nov 14 at 14:55
@user122424 I would prefer you just look up the definition of pseudonatural transformation. It's a bit annoying to type diagrams on here.
– Kevin Carlson
Nov 14 at 17:34
OK, and can you please help by giving me a reference at least?
– user122424
Nov 14 at 20:32
Just try Googling the phrase. It's a very standard concept.
– Kevin Carlson
Nov 14 at 22:09
Yes, but could you please possibly draw the few commuting squares with labels so that I would easily understand which compositions are meant here? For whiskering as well as for vertical composition?
– user122424
Nov 14 at 14:55
Yes, but could you please possibly draw the few commuting squares with labels so that I would easily understand which compositions are meant here? For whiskering as well as for vertical composition?
– user122424
Nov 14 at 14:55
@user122424 I would prefer you just look up the definition of pseudonatural transformation. It's a bit annoying to type diagrams on here.
– Kevin Carlson
Nov 14 at 17:34
@user122424 I would prefer you just look up the definition of pseudonatural transformation. It's a bit annoying to type diagrams on here.
– Kevin Carlson
Nov 14 at 17:34
OK, and can you please help by giving me a reference at least?
– user122424
Nov 14 at 20:32
OK, and can you please help by giving me a reference at least?
– user122424
Nov 14 at 20:32
Just try Googling the phrase. It's a very standard concept.
– Kevin Carlson
Nov 14 at 22:09
Just try Googling the phrase. It's a very standard concept.
– Kevin Carlson
Nov 14 at 22:09
add a comment |
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There seems to be no reason to create the tag "var." It seems to be much too specific. It doesn't even have a wikipedia page. See here for guidelines on tag creation: meta.stackexchange.com/questions/146949/…
– jgon
Nov 13 at 21:09