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arxiv: 2606.26755 · v1 · pith:OKPD4UABnew · submitted 2026-06-25 · 🧮 math.CT

On categories of monads and comonads in double categories

Vasileios Aravantinos-Sotiropoulos , Theofilos Tsantilas , Christina Vasilakopoulou This is my paper

Pith reviewed 2026-06-26 01:55 UTC · model grok-4.3

classification 🧮 math.CT
keywords double categoriesmonadscomonadsmonadicitycocompletenesslocal presentabilityendomorphisms
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The pith

Conditions on colimits in a double category ensure its monads category is monadic over endomorphisms and inherits cocompleteness or local presentability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Results about monoids in monoidal categories extend to monads and comonads inside double categories. The paper gives conditions on a double category D so that the category of its monads becomes monadic over the category of endomorphisms. Under the same conditions the monads category is cocomplete or locally presentable when D itself is. Dual statements are proved for the category of comonads. These inheritance properties let one construct free monads and work with limits and colimits in the monads category whenever the base double category satisfies the stated colimit assumptions.

Core claim

We identify conditions on a double category D under which one can show that the category Mnd(D) of monads in D is monadic over the category of endomorphisms End(D), is cocomplete or even locally presentable. We also tackle the issue of local presentability in the dual case Cmd(D) of comonads. In these results, our assumptions on the double category D revolve around notions of colimit, in particular those of parallel and stable local colimits, as well as a notion of local presentability of a double category.

What carries the argument

Parallel and stable local colimits on the double category D that transfer monadicity, cocompleteness and local presentability to the category Mnd(D) of its monads.

If this is right

  • Existence of free monads in D follows once the forgetful functor from Mnd(D) to End(D) has a left adjoint.
  • Cocompleteness of D implies cocompleteness of Mnd(D) when the colimit conditions hold.
  • Local presentability of D implies local presentability of Mnd(D).
  • The same inheritance statements hold for the category of comonads under the dual colimit assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same colimit conditions may allow analogous transfers in related 2-dimensional structures such as framed bicategories.
  • Verification of the parallel and stable local colimit assumptions in concrete double categories would immediately yield free monads in those settings.
  • The results open the possibility of iterating the construction to obtain categories of monads of monads with the same good properties.

Load-bearing premise

The double category must admit parallel and stable local colimits and satisfy the stated form of local presentability.

What would settle it

A concrete double category that possesses parallel and stable local colimits yet whose category of monads fails to be monadic over the category of endomorphisms.

read the original abstract

As is well known in the literature, the category Mon(V) of monoids in a monoidal category V satisfies various fundamental categorical properties, at least when the monoidal base V is correspondingly well-behaved. In particular, Mon(V) is monadic over V as soon as free monoids exist, while if V is cocomplete or locally presentable and its tensor b is sufficiently compatible with the appropriate colimits, then Mon(V) inherits the analogous property. In the present work, we extend such results to the context of double categories. More precisely, we identify conditions on a double category D under which one can show that the category Mnd(D) of monads in D is monadic over the category of endomorphisms End(D), is cocomplete or even locally presentable. We also tackle the issue of local presentability in the dual case Cmd(D) of comonads. In these results, our assumptions on the double category D revolve around notions of colimit, in particular those of parallel and stable local colimits, as well as a notion of local presentability of a double category which has been introduced in previous work.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript extends classical results on the category of monoids in a monoidal category to the setting of double categories. It identifies conditions on a double category D (involving parallel and stable local colimits together with a notion of local presentability imported from prior work) under which the category Mnd(D) of monads is monadic over the category End(D) of endomorphisms and inherits cocompleteness or local presentability from D; an analogous treatment is given for the category Cmd(D) of comonads.

Significance. If the stated conditions can be verified in concrete cases, the results supply a direct generalization of monadicity and presentability theorems that is likely to be useful in 2-category theory and the study of algebraic structures internal to double categories. The explicit linkage to local presentability notions is a positive feature that could facilitate applications in accessible-category contexts.

minor comments (1)
  1. The abstract refers to 'parallel and stable local colimits' and 'local presentability of a double category' without recalling their definitions or citing the precise prior work in which they appear; a brief reminder in §1 would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting its potential utility in 2-category theory and the study of algebraic structures internal to double categories. We are also pleased that the explicit connection to local presentability is viewed as a positive feature. The recommendation is listed as uncertain, which appears to stem from the question of whether the stated conditions can be verified in concrete cases; we address this below.

read point-by-point responses

  1. Referee: If the stated conditions can be verified in concrete cases, the results supply a direct generalization of monadicity and presentability theorems that is likely to be useful in 2-category theory and the study of algebraic structures internal to double categories. The explicit linkage to local presentability notions is a positive feature that could facilitate applications in accessible-category contexts.

    Authors: We agree that applicability depends on the conditions (parallel and stable local colimits together with local presentability of the double category) being verifiable in practice. The paper establishes the general theorems under these hypotheses, which are formulated precisely so that they can be checked in standard examples such as double categories arising from monoidal categories or from 2-categories with suitable colimits. While the manuscript does not contain an exhaustive catalogue of verifications (its focus being the abstract theory), the conditions are inherited from prior work on local presentability and are designed to hold in the motivating cases. If the referee considers it helpful, we can add a short remark or example section indicating how the hypotheses specialize in familiar settings. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends standard results on monoids in monoidal categories (Mon(V) monadic over V when free monoids exist, and inheritance of cocompleteness/local presentability under colimit compatibility) to the setting of monads/comonads in double categories. The derivation is conditioned on explicit external assumptions (existence of free monads, parallel/stable local colimits in D, and local presentability of D). The latter notion is imported from prior work rather than defined or fitted inside this paper, and no equation, prediction, or central claim reduces by construction to the paper's own inputs or to a self-citation chain. The work is self-contained against the stated hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The work relies on standard background axioms of category theory and on a previously introduced notion of local presentability for double categories.

axioms (3)
  • standard math Standard axioms of category theory, double categories, monads, and comonads.
    The results presuppose the usual definitions and properties from the field.
  • domain assumption Existence of parallel and stable local colimits under the stated conditions.
    These colimit notions are invoked as the key hypotheses on D.
  • domain assumption Local presentability of the double category as defined in previous work.
    The paper explicitly references a prior definition of local presentability for double categories.

pith-pipeline@v0.9.1-grok · 5747 in / 1359 out tokens · 26445 ms · 2026-06-26T01:55:51.690588+00:00 · methodology

discussion (0)

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