A category-like structure without composition?Is there a free digraph associated to a graph?Can the inner...
A category-like structure without composition?
Is there a free digraph associated to a graph?Can the inner structure of an object be systematically deduced from its position in the category?Do non-associative objects have a natural notion of representation?Why does Hom need an identity in the definition of the category?Sets = structured sets without structureWhat is the composition in SesquiAlg?What kind of category is a cyclically ordered set?random category theoryA model category of abelian categories?Cartesian liftings in double categories
$begingroup$
Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $fin Hom(A,B)$ and $g in Hom(B,C)$ then $gcirc f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. 'Category-like' structures derived from directed graphs with at most one edge in each direction, where the vertices are the objects and the edges are the morphisms, provide plentiful examples, as do (equivalently) not-necessarily-transitive relations on a set $X$. Could anyone provide references which discuss this from a categorical perspective? Thanks in advance!
reference-request ct.category-theory
asked 3 hours ago
APRAPR
724
$endgroup$
add a comment |
$begingroup$
Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $fin Hom(A,B)$ and $g in Hom(B,C)$ then $gcirc f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. 'Category-like' structures derived from directed graphs with at most one edge in each direction, where the vertices are the objects and the edges are the morphisms, provide plentiful examples, as do (equivalently) not-necessarily-transitive relations on a set $X$. Could anyone provide references which discuss this from a categorical perspective? Thanks in advance!
reference-request ct.category-theory
asked 3 hours ago
APRAPR
724
$endgroup$
4
$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
3 hours ago
$begingroup$
In that case, you still necessarily have compositions, or am I misunderstanding something?
$endgroup$
– APR
3 hours ago
1
$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
2 hours ago
add a comment |
$begingroup$
Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $fin Hom(A,B)$ and $g in Hom(B,C)$ then $gcirc f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. 'Category-like' structures derived from directed graphs with at most one edge in each direction, where the vertices are the objects and the edges are the morphisms, provide plentiful examples, as do (equivalently) not-necessarily-transitive relations on a set $X$. Could anyone provide references which discuss this from a categorical perspective? Thanks in advance!
reference-request ct.category-theory
asked 3 hours ago
APRAPR
724
$endgroup$
Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $fin Hom(A,B)$ and $g in Hom(B,C)$ then $gcirc f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. 'Category-like' structures derived from directed graphs with at most one edge in each direction, where the vertices are the objects and the edges are the morphisms, provide plentiful examples, as do (equivalently) not-necessarily-transitive relations on a set $X$. Could anyone provide references which discuss this from a categorical perspective? Thanks in advance!
reference-request ct.category-theory
reference-request ct.category-theory
asked 3 hours ago
APRAPR
724
asked 3 hours ago
APRAPR
724
edited 2 hours ago
APR
asked 3 hours ago
APRAPR
724
asked 3 hours ago
APRAPR
724
asked 3 hours ago
APRAPR
724
724
4
$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
3 hours ago
$begingroup$
In that case, you still necessarily have compositions, or am I misunderstanding something?
$endgroup$
– APR
3 hours ago
1
$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
2 hours ago
add a comment |
4
$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
3 hours ago
$begingroup$
In that case, you still necessarily have compositions, or am I misunderstanding something?
$endgroup$
– APR
3 hours ago
1
$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
2 hours ago
4
4
$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
3 hours ago
$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
3 hours ago
$begingroup$
In that case, you still necessarily have compositions, or am I misunderstanding something?
$endgroup$
– APR
3 hours ago
$begingroup$
In that case, you still necessarily have compositions, or am I misunderstanding something?
$endgroup$
– APR
3 hours ago
1
1
$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
2 hours ago
$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
2 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.
A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.
answered 2 hours ago
Mike ShulmanMike Shulman
37.8k485235
$endgroup$
add a comment |
$begingroup$
Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.
edited 2 hours ago
Dan Petersen
26.1k277142
answered 2 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
29.8k881252
$endgroup$
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: "504"
};
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%2fmathoverflow.net%2fquestions%2f327093%2fa-category-like-structure-without-composition%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.
A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.
answered 2 hours ago
Mike ShulmanMike Shulman
37.8k485235
$endgroup$
add a comment |
$begingroup$
As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.
A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.
answered 2 hours ago
Mike ShulmanMike Shulman
37.8k485235
$endgroup$
add a comment |
$begingroup$
As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.
A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.
answered 2 hours ago
Mike ShulmanMike Shulman
37.8k485235
$endgroup$
As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.
A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.
answered 2 hours ago
Mike ShulmanMike Shulman
37.8k485235
answered 2 hours ago
Mike ShulmanMike Shulman
37.8k485235
answered 2 hours ago
Mike ShulmanMike Shulman
37.8k485235
answered 2 hours ago
Mike ShulmanMike Shulman
37.8k485235
37.8k485235
add a comment |
add a comment |
$begingroup$
Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.
edited 2 hours ago
Dan Petersen
26.1k277142
answered 2 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
29.8k881252
$endgroup$
add a comment |
$begingroup$
Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.
edited 2 hours ago
Dan Petersen
26.1k277142
answered 2 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
29.8k881252
$endgroup$
add a comment |
$begingroup$
Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.
edited 2 hours ago
Dan Petersen
26.1k277142
answered 2 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
29.8k881252
$endgroup$
Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.
edited 2 hours ago
Dan Petersen
26.1k277142
answered 2 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
29.8k881252
edited 2 hours ago
Dan Petersen
26.1k277142
edited 2 hours ago
Dan Petersen
26.1k277142
edited 2 hours ago
Dan Petersen
26.1k277142
26.1k277142
answered 2 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
29.8k881252
answered 2 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
29.8k881252
answered 2 hours ago
Theo Johnson-FreydTheo Johnson-Freyd
29.8k881252
29.8k881252
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- 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%2fmathoverflow.net%2fquestions%2f327093%2fa-category-like-structure-without-composition%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
4
$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
3 hours ago
$begingroup$
In that case, you still necessarily have compositions, or am I misunderstanding something?
$endgroup$
– APR
3 hours ago
1
$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
2 hours ago