Mixing
Meaning of index of a multiplication symbol in a Cartesian product
$begingroup$
I've just started to read about Category Theory.
More precisely nLab, opposite category. I'm trying to understand the expression:
$$
circ_{C^{op}}
:
C_{mathrm{mor}}^{op} {}_{s^{op}}times_{t^{op}} C_{mathrm{mor}}^{op}
=
C_{mathrm{mor}} {}_{t} times_{s} C_{mathrm{mor}}
stackrel{simeq}{to}
C_{mathrm{mor}} {}_{s} times_{t} C_{mathrm{mor}}
stackrel{circ}{to}
C_{mathrm{mor}}
,,
$$
Are the indices of $times$ something that can be explained within the set theoretical Cartesian product or it is something strictly connected to Category Theory (it seems to occur near the word pullback)? In the first case I would appreciate a short explanation, in the second case I'll wait to get there.
category-theory products
asked Jan 12 at 10:20
PeptideChainPeptideChain
454311
$endgroup$
add a comment |
$begingroup$
I've just started to read about Category Theory.
More precisely nLab, opposite category. I'm trying to understand the expression:
$$
circ_{C^{op}}
:
C_{mathrm{mor}}^{op} {}_{s^{op}}times_{t^{op}} C_{mathrm{mor}}^{op}
=
C_{mathrm{mor}} {}_{t} times_{s} C_{mathrm{mor}}
stackrel{simeq}{to}
C_{mathrm{mor}} {}_{s} times_{t} C_{mathrm{mor}}
stackrel{circ}{to}
C_{mathrm{mor}}
,,
$$
Are the indices of $times$ something that can be explained within the set theoretical Cartesian product or it is something strictly connected to Category Theory (it seems to occur near the word pullback)? In the first case I would appreciate a short explanation, in the second case I'll wait to get there.
category-theory products
asked Jan 12 at 10:20
PeptideChainPeptideChain
454311
$endgroup$
add a comment |
$begingroup$
I've just started to read about Category Theory.
More precisely nLab, opposite category. I'm trying to understand the expression:
$$
circ_{C^{op}}
:
C_{mathrm{mor}}^{op} {}_{s^{op}}times_{t^{op}} C_{mathrm{mor}}^{op}
=
C_{mathrm{mor}} {}_{t} times_{s} C_{mathrm{mor}}
stackrel{simeq}{to}
C_{mathrm{mor}} {}_{s} times_{t} C_{mathrm{mor}}
stackrel{circ}{to}
C_{mathrm{mor}}
,,
$$
Are the indices of $times$ something that can be explained within the set theoretical Cartesian product or it is something strictly connected to Category Theory (it seems to occur near the word pullback)? In the first case I would appreciate a short explanation, in the second case I'll wait to get there.
category-theory products
asked Jan 12 at 10:20
PeptideChainPeptideChain
454311
$endgroup$
I've just started to read about Category Theory.
More precisely nLab, opposite category. I'm trying to understand the expression:
$$
circ_{C^{op}}
:
C_{mathrm{mor}}^{op} {}_{s^{op}}times_{t^{op}} C_{mathrm{mor}}^{op}
=
C_{mathrm{mor}} {}_{t} times_{s} C_{mathrm{mor}}
stackrel{simeq}{to}
C_{mathrm{mor}} {}_{s} times_{t} C_{mathrm{mor}}
stackrel{circ}{to}
C_{mathrm{mor}}
,,
$$
Are the indices of $times$ something that can be explained within the set theoretical Cartesian product or it is something strictly connected to Category Theory (it seems to occur near the word pullback)? In the first case I would appreciate a short explanation, in the second case I'll wait to get there.
category-theory products
category-theory products
asked Jan 12 at 10:20
PeptideChainPeptideChain
454311
asked Jan 12 at 10:20
PeptideChainPeptideChain
454311
asked Jan 12 at 10:20
PeptideChainPeptideChain
454311
asked Jan 12 at 10:20
PeptideChainPeptideChain
454311
asked Jan 12 at 10:20
PeptideChainPeptideChain
454311
454311
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
As you say, these indices are used to denote a pullback rather than a (cartesian) product. In a general category, given two arrows with the same codomain $Xstackrel{f}longrightarrow Ystackrel{g}longleftarrow Z$, their pullback is an object, denoted $X {_ftimes_g } Y$ (or more frequently, $Xtimes_Z Y$), and equipped with two maps $pi_1 :X {_ftimes_g } Yto X$ and $pi_2:X {_ftimes_g } Yto Y$ such that $fcircpi_1=gcircpi_2$, wich is universal among such objects; this means that for every object $W$ and arrows $u:Wto X$ and $v:Wto Y$ such that $fcirc u=gcirc v$, there exist a unique map $w:Wto X {_ftimes_g } Y$ such that $pi_1 circ w=u$ and $pi_2circ w=v$.
By contrast, the product of $X$ and $Y$ is defined by the same data of an object with two maps, but without the condition $fcircpi_1=gcircpi_2$, and it has the same property, but without the conditions $fcirc u=gcirc v$; so the pullback is a bit like a product but with an additional restriction. Incidentally, the pullback is often called "fibered product".
In the set-theoretical case, the pullback can be easily obtained from the product : whereas the product is the set of pairs $(x,y)$ such that $xin X$ and $yin Y$, the pullbacks is the set of such pairs with the additional condition that $f(x)=g(y)$. The two maps $pi_1$ and $pi_2$ are then just the restriction of the product projections to this set (i.e. $pi_1(x,y)=x$ and $pi_2(x,y)=y$).
In this specific case, the pullback $C_{mathrm{mor}} {_{s} times_{t}} C_{mathrm{mor}}$ is then the set of pairs of arrows $(alpha,beta)$ such that the source/domain of $alpha$ is the target/codomain of $beta$, so in other words it is the set of pairs of arrows that can be composed in your category $C$.
answered Jan 12 at 11:37
Arnaud D.Arnaud D.
15.9k52443
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3070766%2fmeaning-of-index-of-a-multiplication-symbol-in-a-cartesian-product%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
As you say, these indices are used to denote a pullback rather than a (cartesian) product. In a general category, given two arrows with the same codomain $Xstackrel{f}longrightarrow Ystackrel{g}longleftarrow Z$, their pullback is an object, denoted $X {_ftimes_g } Y$ (or more frequently, $Xtimes_Z Y$), and equipped with two maps $pi_1 :X {_ftimes_g } Yto X$ and $pi_2:X {_ftimes_g } Yto Y$ such that $fcircpi_1=gcircpi_2$, wich is universal among such objects; this means that for every object $W$ and arrows $u:Wto X$ and $v:Wto Y$ such that $fcirc u=gcirc v$, there exist a unique map $w:Wto X {_ftimes_g } Y$ such that $pi_1 circ w=u$ and $pi_2circ w=v$.
By contrast, the product of $X$ and $Y$ is defined by the same data of an object with two maps, but without the condition $fcircpi_1=gcircpi_2$, and it has the same property, but without the conditions $fcirc u=gcirc v$; so the pullback is a bit like a product but with an additional restriction. Incidentally, the pullback is often called "fibered product".
In the set-theoretical case, the pullback can be easily obtained from the product : whereas the product is the set of pairs $(x,y)$ such that $xin X$ and $yin Y$, the pullbacks is the set of such pairs with the additional condition that $f(x)=g(y)$. The two maps $pi_1$ and $pi_2$ are then just the restriction of the product projections to this set (i.e. $pi_1(x,y)=x$ and $pi_2(x,y)=y$).
In this specific case, the pullback $C_{mathrm{mor}} {_{s} times_{t}} C_{mathrm{mor}}$ is then the set of pairs of arrows $(alpha,beta)$ such that the source/domain of $alpha$ is the target/codomain of $beta$, so in other words it is the set of pairs of arrows that can be composed in your category $C$.
answered Jan 12 at 11:37
Arnaud D.Arnaud D.
15.9k52443
$endgroup$
add a comment |
$begingroup$
As you say, these indices are used to denote a pullback rather than a (cartesian) product. In a general category, given two arrows with the same codomain $Xstackrel{f}longrightarrow Ystackrel{g}longleftarrow Z$, their pullback is an object, denoted $X {_ftimes_g } Y$ (or more frequently, $Xtimes_Z Y$), and equipped with two maps $pi_1 :X {_ftimes_g } Yto X$ and $pi_2:X {_ftimes_g } Yto Y$ such that $fcircpi_1=gcircpi_2$, wich is universal among such objects; this means that for every object $W$ and arrows $u:Wto X$ and $v:Wto Y$ such that $fcirc u=gcirc v$, there exist a unique map $w:Wto X {_ftimes_g } Y$ such that $pi_1 circ w=u$ and $pi_2circ w=v$.
By contrast, the product of $X$ and $Y$ is defined by the same data of an object with two maps, but without the condition $fcircpi_1=gcircpi_2$, and it has the same property, but without the conditions $fcirc u=gcirc v$; so the pullback is a bit like a product but with an additional restriction. Incidentally, the pullback is often called "fibered product".
In the set-theoretical case, the pullback can be easily obtained from the product : whereas the product is the set of pairs $(x,y)$ such that $xin X$ and $yin Y$, the pullbacks is the set of such pairs with the additional condition that $f(x)=g(y)$. The two maps $pi_1$ and $pi_2$ are then just the restriction of the product projections to this set (i.e. $pi_1(x,y)=x$ and $pi_2(x,y)=y$).
In this specific case, the pullback $C_{mathrm{mor}} {_{s} times_{t}} C_{mathrm{mor}}$ is then the set of pairs of arrows $(alpha,beta)$ such that the source/domain of $alpha$ is the target/codomain of $beta$, so in other words it is the set of pairs of arrows that can be composed in your category $C$.
answered Jan 12 at 11:37
Arnaud D.Arnaud D.
15.9k52443
$endgroup$
add a comment |
$begingroup$
As you say, these indices are used to denote a pullback rather than a (cartesian) product. In a general category, given two arrows with the same codomain $Xstackrel{f}longrightarrow Ystackrel{g}longleftarrow Z$, their pullback is an object, denoted $X {_ftimes_g } Y$ (or more frequently, $Xtimes_Z Y$), and equipped with two maps $pi_1 :X {_ftimes_g } Yto X$ and $pi_2:X {_ftimes_g } Yto Y$ such that $fcircpi_1=gcircpi_2$, wich is universal among such objects; this means that for every object $W$ and arrows $u:Wto X$ and $v:Wto Y$ such that $fcirc u=gcirc v$, there exist a unique map $w:Wto X {_ftimes_g } Y$ such that $pi_1 circ w=u$ and $pi_2circ w=v$.
By contrast, the product of $X$ and $Y$ is defined by the same data of an object with two maps, but without the condition $fcircpi_1=gcircpi_2$, and it has the same property, but without the conditions $fcirc u=gcirc v$; so the pullback is a bit like a product but with an additional restriction. Incidentally, the pullback is often called "fibered product".
In the set-theoretical case, the pullback can be easily obtained from the product : whereas the product is the set of pairs $(x,y)$ such that $xin X$ and $yin Y$, the pullbacks is the set of such pairs with the additional condition that $f(x)=g(y)$. The two maps $pi_1$ and $pi_2$ are then just the restriction of the product projections to this set (i.e. $pi_1(x,y)=x$ and $pi_2(x,y)=y$).
In this specific case, the pullback $C_{mathrm{mor}} {_{s} times_{t}} C_{mathrm{mor}}$ is then the set of pairs of arrows $(alpha,beta)$ such that the source/domain of $alpha$ is the target/codomain of $beta$, so in other words it is the set of pairs of arrows that can be composed in your category $C$.
answered Jan 12 at 11:37
Arnaud D.Arnaud D.
15.9k52443
$endgroup$
As you say, these indices are used to denote a pullback rather than a (cartesian) product. In a general category, given two arrows with the same codomain $Xstackrel{f}longrightarrow Ystackrel{g}longleftarrow Z$, their pullback is an object, denoted $X {_ftimes_g } Y$ (or more frequently, $Xtimes_Z Y$), and equipped with two maps $pi_1 :X {_ftimes_g } Yto X$ and $pi_2:X {_ftimes_g } Yto Y$ such that $fcircpi_1=gcircpi_2$, wich is universal among such objects; this means that for every object $W$ and arrows $u:Wto X$ and $v:Wto Y$ such that $fcirc u=gcirc v$, there exist a unique map $w:Wto X {_ftimes_g } Y$ such that $pi_1 circ w=u$ and $pi_2circ w=v$.
By contrast, the product of $X$ and $Y$ is defined by the same data of an object with two maps, but without the condition $fcircpi_1=gcircpi_2$, and it has the same property, but without the conditions $fcirc u=gcirc v$; so the pullback is a bit like a product but with an additional restriction. Incidentally, the pullback is often called "fibered product".
In the set-theoretical case, the pullback can be easily obtained from the product : whereas the product is the set of pairs $(x,y)$ such that $xin X$ and $yin Y$, the pullbacks is the set of such pairs with the additional condition that $f(x)=g(y)$. The two maps $pi_1$ and $pi_2$ are then just the restriction of the product projections to this set (i.e. $pi_1(x,y)=x$ and $pi_2(x,y)=y$).
In this specific case, the pullback $C_{mathrm{mor}} {_{s} times_{t}} C_{mathrm{mor}}$ is then the set of pairs of arrows $(alpha,beta)$ such that the source/domain of $alpha$ is the target/codomain of $beta$, so in other words it is the set of pairs of arrows that can be composed in your category $C$.
answered Jan 12 at 11:37
Arnaud D.Arnaud D.
15.9k52443
answered Jan 12 at 11:37
Arnaud D.Arnaud D.
15.9k52443
answered Jan 12 at 11:37
Arnaud D.Arnaud D.
15.9k52443
answered Jan 12 at 11:37
Arnaud D.Arnaud D.
15.9k52443
15.9k52443
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3070766%2fmeaning-of-index-of-a-multiplication-symbol-in-a-cartesian-product%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
