Mixing
Exercise about map object (“Conceptual Mathematics Second Edition”, p.315, Article V Map objects,...
$begingroup$
I feel difficult to show the existence of a map $gamma$ which corresponds to the composition of two maps.
$gamma$ is the following map.
$$gamma: B^A times C^B to C^A$$
And I want to show that for any two maps $f: A to B$ and $g: B to C$, there exists a map $gamma$ satisfying the following equation
$$gamma(widetilde{f}, widetilde{g}) = widetilde{g circ f}$$
where I think $widetilde{f}$, $widetilde{g}$, and $widetilde{g circ f}$ are the maps from terminal object
$$widetilde{f}: 1 to B^A$$
$$widetilde{g}: 1 to C^B$$
$$widetilde{g circ f}: 1 to C^A.$$
I tried to show the existence of $gamma$ using the existence of evaluation maps $e1$, $e2$, and $e3$ satisfying the following commutative diagrams, but can't proceed the next step.
category-theory
asked Jan 20 at 2:47
konyonyokonyonyo
175
$endgroup$
add a comment |
$begingroup$
I feel difficult to show the existence of a map $gamma$ which corresponds to the composition of two maps.
$gamma$ is the following map.
$$gamma: B^A times C^B to C^A$$
And I want to show that for any two maps $f: A to B$ and $g: B to C$, there exists a map $gamma$ satisfying the following equation
$$gamma(widetilde{f}, widetilde{g}) = widetilde{g circ f}$$
where I think $widetilde{f}$, $widetilde{g}$, and $widetilde{g circ f}$ are the maps from terminal object
$$widetilde{f}: 1 to B^A$$
$$widetilde{g}: 1 to C^B$$
$$widetilde{g circ f}: 1 to C^A.$$
I tried to show the existence of $gamma$ using the existence of evaluation maps $e1$, $e2$, and $e3$ satisfying the following commutative diagrams, but can't proceed the next step.
category-theory
asked Jan 20 at 2:47
konyonyokonyonyo
175
$endgroup$
$begingroup$
Is there a reason a map "from" the terminal object would be unique/well-defined?
$endgroup$
– Randall
Jan 20 at 2:49
$begingroup$
In cartesian closed category, there is a terminal object. But a map from the terminal object don't necessarily exists?
$endgroup$
– konyonyo
Jan 20 at 14:02
1
$begingroup$
By adjunction, you need to produce a map $Atimes B^Atimes C^Bto C$. Using your maps $e_1$ and $e_2$, can you do it ?
$endgroup$
– Roland
Jan 20 at 14:59
$begingroup$
It is $e2 circ e1 times 1_{C^B }$ ?
$endgroup$
– konyonyo
Jan 20 at 15:30
$begingroup$
Yes. If you think about it, in the category of sets, this is exactly $(x,f,g)mapsto (f(x),g)mapsto g(f(x))$. Hence, it adjoint is $(f,g)mapsto (xmapsto g(f(x))=gcirc f$.
$endgroup$
– Roland
Jan 20 at 16:06
add a comment |
$begingroup$
I feel difficult to show the existence of a map $gamma$ which corresponds to the composition of two maps.
$gamma$ is the following map.
$$gamma: B^A times C^B to C^A$$
And I want to show that for any two maps $f: A to B$ and $g: B to C$, there exists a map $gamma$ satisfying the following equation
$$gamma(widetilde{f}, widetilde{g}) = widetilde{g circ f}$$
where I think $widetilde{f}$, $widetilde{g}$, and $widetilde{g circ f}$ are the maps from terminal object
$$widetilde{f}: 1 to B^A$$
$$widetilde{g}: 1 to C^B$$
$$widetilde{g circ f}: 1 to C^A.$$
I tried to show the existence of $gamma$ using the existence of evaluation maps $e1$, $e2$, and $e3$ satisfying the following commutative diagrams, but can't proceed the next step.
category-theory
asked Jan 20 at 2:47
konyonyokonyonyo
175
$endgroup$
I feel difficult to show the existence of a map $gamma$ which corresponds to the composition of two maps.
$gamma$ is the following map.
$$gamma: B^A times C^B to C^A$$
And I want to show that for any two maps $f: A to B$ and $g: B to C$, there exists a map $gamma$ satisfying the following equation
$$gamma(widetilde{f}, widetilde{g}) = widetilde{g circ f}$$
where I think $widetilde{f}$, $widetilde{g}$, and $widetilde{g circ f}$ are the maps from terminal object
$$widetilde{f}: 1 to B^A$$
$$widetilde{g}: 1 to C^B$$
$$widetilde{g circ f}: 1 to C^A.$$
I tried to show the existence of $gamma$ using the existence of evaluation maps $e1$, $e2$, and $e3$ satisfying the following commutative diagrams, but can't proceed the next step.
category-theory
category-theory
asked Jan 20 at 2:47
konyonyokonyonyo
175
asked Jan 20 at 2:47
konyonyokonyonyo
175
asked Jan 20 at 2:47
konyonyokonyonyo
175
asked Jan 20 at 2:47
konyonyokonyonyo
175
asked Jan 20 at 2:47
konyonyokonyonyo
175
175
$begingroup$
Is there a reason a map "from" the terminal object would be unique/well-defined?
$endgroup$
– Randall
Jan 20 at 2:49
$begingroup$
In cartesian closed category, there is a terminal object. But a map from the terminal object don't necessarily exists?
$endgroup$
– konyonyo
Jan 20 at 14:02
1
$begingroup$
By adjunction, you need to produce a map $Atimes B^Atimes C^Bto C$. Using your maps $e_1$ and $e_2$, can you do it ?
$endgroup$
– Roland
Jan 20 at 14:59
$begingroup$
It is $e2 circ e1 times 1_{C^B }$ ?
$endgroup$
– konyonyo
Jan 20 at 15:30
$begingroup$
Yes. If you think about it, in the category of sets, this is exactly $(x,f,g)mapsto (f(x),g)mapsto g(f(x))$. Hence, it adjoint is $(f,g)mapsto (xmapsto g(f(x))=gcirc f$.
$endgroup$
– Roland
Jan 20 at 16:06
add a comment |
$begingroup$
Is there a reason a map "from" the terminal object would be unique/well-defined?
$endgroup$
– Randall
Jan 20 at 2:49
$begingroup$
In cartesian closed category, there is a terminal object. But a map from the terminal object don't necessarily exists?
$endgroup$
– konyonyo
Jan 20 at 14:02
1
$begingroup$
By adjunction, you need to produce a map $Atimes B^Atimes C^Bto C$. Using your maps $e_1$ and $e_2$, can you do it ?
$endgroup$
– Roland
Jan 20 at 14:59
$begingroup$
It is $e2 circ e1 times 1_{C^B }$ ?
$endgroup$
– konyonyo
Jan 20 at 15:30
$begingroup$
Yes. If you think about it, in the category of sets, this is exactly $(x,f,g)mapsto (f(x),g)mapsto g(f(x))$. Hence, it adjoint is $(f,g)mapsto (xmapsto g(f(x))=gcirc f$.
$endgroup$
– Roland
Jan 20 at 16:06
$begingroup$
Is there a reason a map "from" the terminal object would be unique/well-defined?
$endgroup$
– Randall
Jan 20 at 2:49
$begingroup$
Is there a reason a map "from" the terminal object would be unique/well-defined?
$endgroup$
– Randall
Jan 20 at 2:49
$begingroup$
In cartesian closed category, there is a terminal object. But a map from the terminal object don't necessarily exists?
$endgroup$
– konyonyo
Jan 20 at 14:02
$begingroup$
In cartesian closed category, there is a terminal object. But a map from the terminal object don't necessarily exists?
$endgroup$
– konyonyo
Jan 20 at 14:02
1
1
$begingroup$
By adjunction, you need to produce a map $Atimes B^Atimes C^Bto C$. Using your maps $e_1$ and $e_2$, can you do it ?
$endgroup$
– Roland
Jan 20 at 14:59
$begingroup$
By adjunction, you need to produce a map $Atimes B^Atimes C^Bto C$. Using your maps $e_1$ and $e_2$, can you do it ?
$endgroup$
– Roland
Jan 20 at 14:59
$begingroup$
It is $e2 circ e1 times 1_{C^B }$ ?
$endgroup$
– konyonyo
Jan 20 at 15:30
$begingroup$
It is $e2 circ e1 times 1_{C^B }$ ?
$endgroup$
– konyonyo
Jan 20 at 15:30
$begingroup$
Yes. If you think about it, in the category of sets, this is exactly $(x,f,g)mapsto (f(x),g)mapsto g(f(x))$. Hence, it adjoint is $(f,g)mapsto (xmapsto g(f(x))=gcirc f$.
$endgroup$
– Roland
Jan 20 at 16:06
$begingroup$
Yes. If you think about it, in the category of sets, this is exactly $(x,f,g)mapsto (f(x),g)mapsto g(f(x))$. Hence, it adjoint is $(f,g)mapsto (xmapsto g(f(x))=gcirc f$.
$endgroup$
– Roland
Jan 20 at 16:06
add a comment |
0
active
oldest
votes
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%2f3080109%2fexercise-about-map-object-conceptual-mathematics-second-edition-p-315-artic%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3080109%2fexercise-about-map-object-conceptual-mathematics-second-edition-p-315-artic%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
$begingroup$
Is there a reason a map "from" the terminal object would be unique/well-defined?
$endgroup$
– Randall
Jan 20 at 2:49
$begingroup$
In cartesian closed category, there is a terminal object. But a map from the terminal object don't necessarily exists?
$endgroup$
– konyonyo
Jan 20 at 14:02
1
$begingroup$
By adjunction, you need to produce a map $Atimes B^Atimes C^Bto C$. Using your maps $e_1$ and $e_2$, can you do it ?
$endgroup$
– Roland
Jan 20 at 14:59
$begingroup$
It is $e2 circ e1 times 1_{C^B }$ ?
$endgroup$
– konyonyo
Jan 20 at 15:30
$begingroup$
Yes. If you think about it, in the category of sets, this is exactly $(x,f,g)mapsto (f(x),g)mapsto g(f(x))$. Hence, it adjoint is $(f,g)mapsto (xmapsto g(f(x))=gcirc f$.
$endgroup$
– Roland
Jan 20 at 16:06