A small tool to view real-world ActivityPub objects as JSON! Enter a URL
or username from Mastodon or a similar service below, and we'll send a
request with
the right
Accept
header
to the server to view the underlying object.
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"id": "https://mathstodon.xyz/users/varkor/outbox?min_id=0&page=true",
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"next": "https://mathstodon.xyz/users/varkor/outbox?max_id=109479305369084244&page=true",
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"id": "https://mathstodon.xyz/users/varkor/statuses/110474588808548742/activity",
"type": "Create",
"actor": "https://mathstodon.xyz/users/varkor",
"published": "2023-06-02T12:20:47Z",
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"id": "https://mathstodon.xyz/users/varkor/statuses/110474588808548742",
"type": "Note",
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"published": "2023-06-02T12:20:47Z",
"url": "https://mathstodon.xyz/@varkor/110474588808548742",
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"atomUri": "https://mathstodon.xyz/users/varkor/statuses/110474588808548742",
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"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@jonmsterling\" class=\"u-url mention\">@<span>jonmsterling</span></a></span> <span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@ltchen\" class=\"u-url mention\">@<span>ltchen</span></a></span> My impression is that the more set-theoretic aspects of locally presentable categories arise, at least primarily, when one does not fix the cardinality of presentability. In contrast, much of the theory of locally \\(\\lambda\\)-presentable categories, for a fixed \\(\\lambda\\), can be derived 2-category theoretically without set-theoretic assumptions. In my experience, many of the applications of locally presentable categories do not require a change of cardinal (particularly in computer science). So perhaps the obstructions to constructivity are not as severe as might be feared at first. (On the other hand, the theory of accessible categories seems much more reliant on freely changing cardinality.)</p>",
"contentMap": {
"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@jonmsterling\" class=\"u-url mention\">@<span>jonmsterling</span></a></span> <span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@ltchen\" class=\"u-url mention\">@<span>ltchen</span></a></span> My impression is that the more set-theoretic aspects of locally presentable categories arise, at least primarily, when one does not fix the cardinality of presentability. In contrast, much of the theory of locally \\(\\lambda\\)-presentable categories, for a fixed \\(\\lambda\\), can be derived 2-category theoretically without set-theoretic assumptions. In my experience, many of the applications of locally presentable categories do not require a change of cardinal (particularly in computer science). So perhaps the obstructions to constructivity are not as severe as might be feared at first. (On the other hand, the theory of accessible categories seems much more reliant on freely changing cardinality.)</p>"
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"id": "https://mathstodon.xyz/users/varkor/statuses/110467854389426772/activity",
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"published": "2023-06-01T07:48:08Z",
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"id": "https://mathstodon.xyz/users/varkor/statuses/110467854389426772",
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"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@ltchen\" class=\"u-url mention\">@<span>ltchen</span></a></span> Which aspects of the theory of locally presentable categories do not work well constructively?</p>",
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"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@ltchen\" class=\"u-url mention\">@<span>ltchen</span></a></span> Which aspects of the theory of locally presentable categories do not work well constructively?</p>"
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"id": "https://mathstodon.xyz/users/varkor/statuses/110414024602207819/activity",
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"actor": "https://mathstodon.xyz/users/varkor",
"published": "2023-05-22T19:38:30Z",
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"id": "https://mathstodon.xyz/users/varkor/statuses/110414024602207819",
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"inReplyTo": "https://mathstodon.xyz/users/stringdiagram/statuses/110413852343159524",
"published": "2023-05-22T19:38:30Z",
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"atomUri": "https://mathstodon.xyz/users/varkor/statuses/110414024602207819",
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"conversation": "tag:mathstodon.xyz,2023-05-22:objectId=51393972:objectType=Conversation",
"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@stringdiagram\" class=\"u-url mention\">@<span>stringdiagram</span></a></span> Another nice observation is that the same characterisation works for relative monads: a functor is (up to isomorphism) the Kleisli inclusion for a \\(j\\)-relative monad if and only if it is bijective-on-objects and has a right \\(j\\)-relative adjoint.</p>",
"contentMap": {
"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@stringdiagram\" class=\"u-url mention\">@<span>stringdiagram</span></a></span> Another nice observation is that the same characterisation works for relative monads: a functor is (up to isomorphism) the Kleisli inclusion for a \\(j\\)-relative monad if and only if it is bijective-on-objects and has a right \\(j\\)-relative adjoint.</p>"
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"published": "2023-05-18T19:25:39Z",
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"id": "https://mathstodon.xyz/users/varkor/statuses/110391324804515868",
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"published": "2023-05-18T19:25:39Z",
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"atomUri": "https://mathstodon.xyz/users/varkor/statuses/110391324804515868",
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"conversation": "tag:types.pl,2023-05-15:objectId=11223478:objectType=Conversation",
"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@zyang\" class=\"u-url mention\">@<span>zyang</span></a></span> In case you didn't see, Dima Szamozvancev responded to the question on the Category Theory Zulip (<a href=\"https://categorytheory.zulipchat.com/#narrow/stream/229136-theory.3A-category-theory/topic/Comonadicity.20of.20Presheaf.20Categories\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"ellipsis\">categorytheory.zulipchat.com/#</span><span class=\"invisible\">narrow/stream/229136-theory.3A-category-theory/topic/Comonadicity.20of.20Presheaf.20Categories</span></a>) :)</p>",
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"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@zyang\" class=\"u-url mention\">@<span>zyang</span></a></span> In case you didn't see, Dima Szamozvancev responded to the question on the Category Theory Zulip (<a href=\"https://categorytheory.zulipchat.com/#narrow/stream/229136-theory.3A-category-theory/topic/Comonadicity.20of.20Presheaf.20Categories\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"ellipsis\">categorytheory.zulipchat.com/#</span><span class=\"invisible\">narrow/stream/229136-theory.3A-category-theory/topic/Comonadicity.20of.20Presheaf.20Categories</span></a>) :)</p>"
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"id": "https://mathstodon.xyz/users/varkor/statuses/110335346740846669/activity",
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"published": "2023-05-08T22:09:42Z",
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"published": "2023-05-08T22:09:42Z",
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"conversation": "tag:mathstodon.xyz,2023-05-08:objectId=49904045:objectType=Conversation",
"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@MartinEscardo\" class=\"u-url mention\">@<span>MartinEscardo</span></a></span> Another relevant reference is Marmolejo–Wood's "Monads as Extension Systems - No Iteration is Necessary" (<a href=\"http://www.tac.mta.ca/tac/volumes/24/4/24-04abs.html\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">http://www.</span><span class=\"ellipsis\">tac.mta.ca/tac/volumes/24/4/24</span><span class=\"invisible\">-04abs.html</span></a>). However, they work in the setting of an arbitrary 2-category, so the definition there still requires some specialisation.</p>",
"contentMap": {
"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@MartinEscardo\" class=\"u-url mention\">@<span>MartinEscardo</span></a></span> Another relevant reference is Marmolejo–Wood's "Monads as Extension Systems - No Iteration is Necessary" (<a href=\"http://www.tac.mta.ca/tac/volumes/24/4/24-04abs.html\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">http://www.</span><span class=\"ellipsis\">tac.mta.ca/tac/volumes/24/4/24</span><span class=\"invisible\">-04abs.html</span></a>). However, they work in the setting of an arbitrary 2-category, so the definition there still requires some specialisation.</p>"
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"id": "https://mathstodon.xyz/users/varkor/statuses/110335338510976587/activity",
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"actor": "https://mathstodon.xyz/users/varkor",
"published": "2023-05-08T22:07:36Z",
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"cc": [
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"published": "2023-05-08T22:07:36Z",
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"atomUri": "https://mathstodon.xyz/users/varkor/statuses/110335338510976587",
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"conversation": "tag:mathstodon.xyz,2023-05-08:objectId=49904045:objectType=Conversation",
"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@MartinEscardo\" class=\"u-url mention\">@<span>MartinEscardo</span></a></span> <span class=\"h-card\" translate=\"no\"><a href=\"https://mastodon.social/@jaklt\" class=\"u-url mention\">@<span>jaklt</span></a></span> Walters's thesis is freely available (<a href=\"https://openresearch-repository.anu.edu.au/handle/1885/133321\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"ellipsis\">openresearch-repository.anu.ed</span><span class=\"invisible\">u.au/handle/1885/133321</span></a>). He defines algebras in extension form. However, he also considers a generalisation of monads to relative monads (there called "devices").</p>",
"contentMap": {
"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@MartinEscardo\" class=\"u-url mention\">@<span>MartinEscardo</span></a></span> <span class=\"h-card\" translate=\"no\"><a href=\"https://mastodon.social/@jaklt\" class=\"u-url mention\">@<span>jaklt</span></a></span> Walters's thesis is freely available (<a href=\"https://openresearch-repository.anu.edu.au/handle/1885/133321\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"ellipsis\">openresearch-repository.anu.ed</span><span class=\"invisible\">u.au/handle/1885/133321</span></a>). He defines algebras in extension form. However, he also considers a generalisation of monads to relative monads (there called "devices").</p>"
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"published": "2023-05-08T21:45:13Z",
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"conversation": "tag:mathstodon.xyz,2023-05-08:objectId=49904045:objectType=Conversation",
"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mastodon.social/@jaklt\" class=\"u-url mention\">@<span>jaklt</span></a></span> <span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@MartinEscardo\" class=\"u-url mention\">@<span>MartinEscardo</span></a></span> The notion essentially first appeared in Walters's "An alternative approach to universal algebra" (<a href=\"https://link.springer.com/chapter/10.1007/BFb0059142\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"ellipsis\">link.springer.com/chapter/10.1</span><span class=\"invisible\">007/BFb0059142</span></a>), and was expanded upon in his thesis. However, Walters's notation can be a little difficult to read compared to more modern papers.</p>",
"contentMap": {
"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mastodon.social/@jaklt\" class=\"u-url mention\">@<span>jaklt</span></a></span> <span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@MartinEscardo\" class=\"u-url mention\">@<span>MartinEscardo</span></a></span> The notion essentially first appeared in Walters's "An alternative approach to universal algebra" (<a href=\"https://link.springer.com/chapter/10.1007/BFb0059142\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"ellipsis\">link.springer.com/chapter/10.1</span><span class=\"invisible\">007/BFb0059142</span></a>), and was expanded upon in his thesis. However, Walters's notation can be a little difficult to read compared to more modern papers.</p>"
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"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@MartinEscardo\" class=\"u-url mention\">@<span>MartinEscardo</span></a></span> "Monads need not be endofunctors" (<a href=\"https://arxiv.org/abs/1412.7148\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"\">arxiv.org/abs/1412.7148</span><span class=\"invisible\"></span></a>) is a good reference. (The definitions are presented in the additional generality of relative monads, but specialising to monads relative to the identity functor will give you what you are looking for.)</p>",
"contentMap": {
"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@MartinEscardo\" class=\"u-url mention\">@<span>MartinEscardo</span></a></span> "Monads need not be endofunctors" (<a href=\"https://arxiv.org/abs/1412.7148\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"\">arxiv.org/abs/1412.7148</span><span class=\"invisible\"></span></a>) is a good reference. (The definitions are presented in the additional generality of relative monads, but specialising to monads relative to the identity functor will give you what you are looking for.)</p>"
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"published": "2023-05-02T07:02:21Z",
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"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://types.pl/@maxsnew\" class=\"u-url mention\">@<span>maxsnew</span></a></span> Zhen Lin has a nice characterisation of the finitary algebraic theories whose categories of algebras have finite biproducts (<a href=\"https://math.stackexchange.com/a/1102762\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"ellipsis\">math.stackexchange.com/a/11027</span><span class=\"invisible\">62</span></a>), which shows that commutativity is not necessary.</p>",
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"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://types.pl/@maxsnew\" class=\"u-url mention\">@<span>maxsnew</span></a></span> Zhen Lin has a nice characterisation of the finitary algebraic theories whose categories of algebras have finite biproducts (<a href=\"https://math.stackexchange.com/a/1102762\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"ellipsis\">math.stackexchange.com/a/11027</span><span class=\"invisible\">62</span></a>), which shows that commutativity is not necessary.</p>"
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"published": "2023-04-23T14:53:00Z",
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"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@ltchen\" class=\"u-url mention\">@<span>ltchen</span></a></span> Garner's "Combinatorial structure of type dependency" (<a href=\"https://arxiv.org/abs/1402.6799\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"\">arxiv.org/abs/1402.6799</span><span class=\"invisible\"></span></a>) may be relevant.</p>",
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"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@ltchen\" class=\"u-url mention\">@<span>ltchen</span></a></span> Garner's "Combinatorial structure of type dependency" (<a href=\"https://arxiv.org/abs/1402.6799\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"\">arxiv.org/abs/1402.6799</span><span class=\"invisible\"></span></a>) may be relevant.</p>"
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"published": "2023-04-23T14:48:55Z",
"url": "https://mathstodon.xyz/@varkor/110248678874267041",
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"conversation": "tag:mathstodon.xyz,2023-04-23:objectId=48248658:objectType=Conversation",
"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@stringdiagram\" class=\"u-url mention\">@<span>stringdiagram</span></a></span> Perhaps it's worth mentioning that, while actions/modules are less familiar than algebras from the perspective of ordinary category theory, they are the appropriate notion of algebra for monads in a 2-category (which is where much of their theory originates).</p>",
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"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@stringdiagram\" class=\"u-url mention\">@<span>stringdiagram</span></a></span> Perhaps it's worth mentioning that, while actions/modules are less familiar than algebras from the perspective of ordinary category theory, they are the appropriate notion of algebra for monads in a 2-category (which is where much of their theory originates).</p>"
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"published": "2023-04-23T14:46:31Z",
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"published": "2023-04-23T14:46:31Z",
"url": "https://mathstodon.xyz/@varkor/110248669463237176",
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"conversation": "tag:mathstodon.xyz,2023-04-23:objectId=48248658:objectType=Conversation",
"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://types.pl/@maxsnew\" class=\"u-url mention\">@<span>maxsnew</span></a></span> <span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@stringdiagram\" class=\"u-url mention\">@<span>stringdiagram</span></a></span> <span class=\"h-card\" translate=\"no\"><a href=\"https://types.pl/@jer_gib\" class=\"u-url mention\">@<span>jer_gib</span></a></span> Yes, this is precisely the universal property for an Eilenberg–Moore object in a 2-category.</p>",
"contentMap": {
"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://types.pl/@maxsnew\" class=\"u-url mention\">@<span>maxsnew</span></a></span> <span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@stringdiagram\" class=\"u-url mention\">@<span>stringdiagram</span></a></span> <span class=\"h-card\" translate=\"no\"><a href=\"https://types.pl/@jer_gib\" class=\"u-url mention\">@<span>jer_gib</span></a></span> Yes, this is precisely the universal property for an Eilenberg–Moore object in a 2-category.</p>"
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"published": "2023-04-11T20:19:35Z",
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"conversation": "tag:types.pl,2023-04-11:objectId=10320330:objectType=Conversation",
"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://types.pl/@maxsnew\" class=\"u-url mention\">@<span>maxsnew</span></a></span> If you're assuming that \\(C\\) is a small category with finite coproducts, then the category of finite product-preserving presheaves on \\(C\\) is equivalent to the cocompetion of \\(C\\) under sifted colimits, hence a locally strongly finitely presentable category. Such categories can be characterised as those reflective subcategories of presheaf categories for which the right adjoint preserves sifted colimits. However, if you do care about small products, rather than finite products, the situation is more subtle.</p>",
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"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://types.pl/@maxsnew\" class=\"u-url mention\">@<span>maxsnew</span></a></span> If you're assuming that \\(C\\) is a small category with finite coproducts, then the category of finite product-preserving presheaves on \\(C\\) is equivalent to the cocompetion of \\(C\\) under sifted colimits, hence a locally strongly finitely presentable category. Such categories can be characterised as those reflective subcategories of presheaf categories for which the right adjoint preserves sifted colimits. However, if you do care about small products, rather than finite products, the situation is more subtle.</p>"
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"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://types.pl/@maxsnew\" class=\"u-url mention\">@<span>maxsnew</span></a></span> What exactly would you like to be able to do?</p>",
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"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@Bryce\" class=\"u-url mention\">@<span>Bryce</span></a></span> The favicon was updated just under two weeks ago.</p>",
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"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@ohad\" class=\"u-url mention\">@<span>ohad</span></a></span> Hermida constructs this Yoneda structure on Multicat in the (unpublished) "Fibrations and Yoneda structure for multicategories".</p>",
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"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@ohad\" class=\"u-url mention\">@<span>ohad</span></a></span> Hermida constructs this Yoneda structure on Multicat in the (unpublished) "Fibrations and Yoneda structure for multicategories".</p>"
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"conversation": "tag:types.pl,2023-01-17:objectId=8051213:objectType=Conversation",
"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://types.pl/@maxsnew\" class=\"u-url mention\">@<span>maxsnew</span></a></span> I'm sure you've already come across it, but my impression was that Frey's "A 2-Categorical Analysis of the Tripos-to-Topos Construction" (<a href=\"https://arxiv.org/abs/1104.2776\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"\">arxiv.org/abs/1104.2776</span><span class=\"invisible\"></span></a>) was motivated by this question (though I've not yet had a chance to read it).</p>",
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"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://types.pl/@maxsnew\" class=\"u-url mention\">@<span>maxsnew</span></a></span> I'm sure you've already come across it, but my impression was that Frey's "A 2-Categorical Analysis of the Tripos-to-Topos Construction" (<a href=\"https://arxiv.org/abs/1104.2776\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"\">arxiv.org/abs/1104.2776</span><span class=\"invisible\"></span></a>) was motivated by this question (though I've not yet had a chance to read it).</p>"
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"conversation": "tag:mathstodon.xyz,2023-01-18:objectId=38018906:objectType=Conversation",
"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@OscarCunningham\" class=\"u-url mention\">@<span>OscarCunningham</span></a></span> Yes, when a 2-category admits Eilenberg–Moore objects, this assignment forms an algebra for Mnd.</p>",
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"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@OscarCunningham\" class=\"u-url mention\">@<span>OscarCunningham</span></a></span> Yes, when a 2-category admits Eilenberg–Moore objects, this assignment forms an algebra for Mnd.</p>"
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"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://types.pl/@maxsnew\" class=\"u-url mention\">@<span>maxsnew</span></a></span> They are called "pseudo-operads in Cat" in Day–Street's "Lax monoids, pseudo-operads, and convolution" (<a href=\"http://maths.mq.edu.au/~street/Multicats.pdf\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">http://</span><span class=\"ellipsis\">maths.mq.edu.au/~street/Multic</span><span class=\"invisible\">ats.pdf</span></a>).</p>",
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"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://types.pl/@maxsnew\" class=\"u-url mention\">@<span>maxsnew</span></a></span> They are called "pseudo-operads in Cat" in Day–Street's "Lax monoids, pseudo-operads, and convolution" (<a href=\"http://maths.mq.edu.au/~street/Multicats.pdf\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">http://</span><span class=\"ellipsis\">maths.mq.edu.au/~street/Multic</span><span class=\"invisible\">ats.pdf</span></a>).</p>"
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"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@ltchen\" class=\"u-url mention\">@<span>ltchen</span></a></span> While there is a standard definition of "category", there is no standard definition of "type theory"/"deductive system" (though there are many candidates). However, if you pick a specific framework (e.g. generalised algebraic theories), then one can give a general type theoretic definition of product just as easily as one gives the category theoretic definition of product.</p><p>Historically, studying "type theories" in general has not been popular, though it is becoming more fashionable now.</p>",
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"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@ltchen\" class=\"u-url mention\">@<span>ltchen</span></a></span> While there is a standard definition of "category", there is no standard definition of "type theory"/"deductive system" (though there are many candidates). However, if you pick a specific framework (e.g. generalised algebraic theories), then one can give a general type theoretic definition of product just as easily as one gives the category theoretic definition of product.</p><p>Historically, studying "type theories" in general has not been popular, though it is becoming more fashionable now.</p>"
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