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1. • CommentRowNumber1.
   • CommentAuthorWilf Offord
   • CommentTimeFeb 6th 2023
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   Author: Wilf Offord
   Format: TextHi! I was recently looking for references on the horizontal categorification of the notion of a monad, maybe called something like a 'monadoid'. I was surprised that I couldn't find anything on nlab or elsewhere about this, given that monads are so well-documented and it it clear to see what the right definition of a monadoid should be. (Monads are internal monoids in some endofunctor category End(C), equivalently they're End(C) enriched categories with one object, and then you can immediately generalize this and define monadoids to be End(C)-enriched categories.) I've quickly put together a document listing the right definitions for monadoids and some related concepts like their algebras (algebroids?) and the counterpart of the Kleisi category. I've also written down some simple examples and how they might be interpreted as computational side effects with a twist. So my question is twofold: 1) Has anyone seen anything like this before? Is there already a reference somewhere on nlab for what I've been calling monadoids? 2) If not, would nlab be a good place to upload these definitions/make an nlab entry for them? I would be a first-time contributor here on nlab, so am hesitant to just make a page without checking first that the concept is relevant and has not already been documented! Thanks!

   Hi!
   
   I was recently looking for references on the horizontal categorification of the notion of a monad, maybe called something like a 'monadoid'. I was surprised that I couldn't find anything on nlab or elsewhere about this, given that monads are so well-documented and it it clear to see what the right definition of a monadoid should be. (Monads are internal monoids in some endofunctor category End(C), equivalently they're End(C) enriched categories with one object, and then you can immediately generalize this and define monadoids to be End(C)-enriched categories.) I've quickly put together a document listing the right definitions for monadoids and some related concepts like their algebras (algebroids?) and the counterpart of the Kleisi category. I've also written down some simple examples and how they might be interpreted as computational side effects with a twist. So my question is twofold:
   
   1) Has anyone seen anything like this before? Is there already a reference somewhere on nlab for what I've been calling monadoids?
   
   2) If not, would nlab be a good place to upload these definitions/make an nlab entry for them?
   
   I would be a first-time contributor here on nlab, so am hesitant to just make a page without checking first that the concept is relevant and has not already been documented!
   
   Thanks!
2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 6th 2023
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   Author: David_Corfield
   Format: MarkdownItexHi. There's an MO question [What are the internal categories in an endofunctor category](https://mathoverflow.net/q/306044/447) which may be of interest.

   Hi. There’s an MO question What are the internal categories in an endofunctor category which may be of interest.

3. • CommentRowNumber3.
   • CommentAuthorWilf Offord
   • CommentTimeFeb 6th 2023
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   Author: Wilf Offord
   Format: TextThanks for the reply! Yes, the 'straightforward generalisation of monads' mentioned by Peter Lumsdaine on that MO page is the exact structure I'm considering here, but I can't see that anyone's fleshed it out with any examples, etc.

   Thanks for the reply! Yes, the 'straightforward generalisation of monads' mentioned by Peter Lumsdaine on that MO page is the exact structure I'm considering here, but I can't see that anyone's fleshed it out with any examples, etc.
4. • CommentRowNumber4.
   • CommentAuthorvarkor
   • CommentTimeFeb 7th 2023
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   Author: varkor
   Format: MarkdownItexA monad in a bicategory $K$ is a lax functor $1 \to K$. This definition admits a natural generalisation to a set $X$ equipped with a lax functor $X_{\mathrm{ch}}} \to K$ from the chaotic bicategory on $X$ (i.e. having a single 1-cell between each pair of elements of $X$), which is exactly a category enriched in $K$. Thus [[categories enriched in a bicategory]] are many-object generalisations of monads. They were called "polyads" by Bénabou. In particular, your $End(C)$-enriched categories are bicategory-enriched categories where every object has the same extent (namely, $C$). The paper [Enriched categories as a free cocompletion](https://arxiv.org/abs/1301.3191) by Garner and Shulman takes this perspective on enriched categories, for instance.

   A monad in a bicategory is a lax functor . This definition admits a natural generalisation to a set equipped with a lax functor from the chaotic bicategory on (i.e. having a single 1-cell between each pair of elements of ), which is exactly a category enriched in . Thus categories enriched in a bicategory are many-object generalisations of monads. They were called “polyads” by Bénabou. In particular, your -enriched categories are bicategory-enriched categories where every object has the same extent (namely, ). The paper Enriched categories as a free cocompletion by Garner and Shulman takes this perspective on enriched categories, for instance.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 7th 2023
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   Author: David_Corfield
   Format: MarkdownItexThere's also work by Orchard et al. [here](https://ncatlab.org/nlab/show/graded+monad#OrchWadEad) on category-graded monads: > Our approach can be summarised as the horizontal categorification of monads.

   There’s also work by Orchard et al. here on category-graded monads:

   Our approach can be summarised as the horizontal categorification of monads.

6. • CommentRowNumber6.
   • CommentAuthorWilf Offord
   • CommentTimeFeb 7th 2023
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   Author: Wilf Offord
   Format: TextThanks for both of the comments - they're exactly the kind of thing I was looking for! (the first for the more higher-categorical flavour, and the second for the computational interpretation). The generalisation of Kleisi objects to Collages is fascinating in particular. Cheers!

   Thanks for both of the comments - they're exactly the kind of thing I was looking for! (the first for the more higher-categorical flavour, and the second for the computational interpretation). The generalisation of Kleisi objects to Collages is fascinating in particular. Cheers!