Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

April 28, 2014

On Two-Dimensional Monad Theory

Posted by Emily Riehl

Guest post by Sam van Gool

Monads provide a categorical setting for studying sets with additional structure. Similarly, 2-monads provide a 2-categorical setting for studying categories with additional structure. While there is really only one natural notion of algebra morphism in the context of monads, there are several choices of algebra morphism in the context of 2-monads. The interplay between these different kinds of morphisms is the main focus of the paper that I discuss in this post:

I will give an overview of the results and methods used in this paper. Also, especially towards the end of my post, I will also indicate some points that I think could still be clarified further by formulating some questions, which will hopefully lead to fruitful discussions below.

This post forms the 9th instalment of the series of posts written by participants of the Kan Extension Seminar, of which I’m very glad to be a part. In preparing the post I have greatly benefited from discussions with the other participants in the seminar, and of course with the seminar’s organizer, Emily Riehl. I am very grateful for the enthusiasm, encouragement and guidance that you all offered.

2-monads, their algebras, and their morphisms

Two-dimensional universal algebra goes beyond the Cat\mathbf{Cat}-enriched setting in that it allows for non-strict morphisms. Consider the following (very) simple example.

Example. For a category AA, let TAT A be the category AA provided freely with a terminal object. This assignment can be extended to a 2-monad TT on Cat\mathbf{Cat}. Then:

  • an algebra for TT is (entirely determined by giving) a pair (A,t A)(A,t_A) where AA is a category and t At_A is a designated terminal object in AA;
  • a strict morphism (A,t A)(B,t B)(A,t_A) \to (B,t_B) is a functor ff for which f(t A)=t Bf(t_A) = t_B;
  • a pseudo morphism is a functor ff such that f(t A)f(t_A) is isomorphic to t Bt_B;
  • a lax morphism is just any functor from AA to BB, with no additional requirement on the terminal object.

If you didn’t know them already, you will probably have guessed the general definitions of strict, pseudo and lax morphisms by now, as well as the definition of 2-cells between them. Note that, in this post, all 2-monads and algebras for them will be strict, as in [BKP].

For any 2-monad TT, we thus get the following inclusions of 2-categories:

T-Alg sT-Alg pT-Alg l. T\text{-}\mathrm{Alg}_s \to T\text{-}\mathrm{Alg}_p \to T\text{-}\mathrm{Alg}_l.

(In [BKP], the category T-Alg pT\text{-}\mathrm{Alg}_p is denoted by T-AlgT\text{-}\mathrm{Alg}.) Roughly the first half of the paper [BKP] is devoted to the construction of left adjoints (in the 2-categorical sense) to these inclusion functors.

Note that T-Alg sT\text{-}\mathrm{Alg}_s is simply the Eilenberg-Moore 𝒱\mathcal{V}-category of the 𝒱\mathcal{V}-enriched monad TT in the case where 𝒱=Cat\mathcal{V} = \mathbf{Cat}, in the sense of the second paper that we read in this seminar. The categories T-Alg pT\text{-}\mathrm{Alg}_p and T-Alg lT\text{-}\mathrm{Alg}_l, on the other hand, are special to the Cat\mathbf{Cat}-enriched setting.

Limits in TT-Algp_p

The category T-Alg sT\text{-}\mathrm{Alg}_s has all 2-limits that the base 2-category 𝒦\mathcal{K} has. For T-Alg pT\text{-}\mathrm{Alg}_p, the situation is more subtle.

Example (c’t’d). In the example where TAT A is AA provided freely with a terminal object, let 1={*}1 = \{\ast\} be the terminal category and II the category with two objects 00, 11 and a unique isomorphism between them. There are two pseudo-morphisms (1,*)(I,0)(1,\ast) \to (I,0), one sending *\ast to 00, the other sending *\ast to 11. However, if C1C \to 1 is any functor which equalizes these two morphisms, then CC is empty, and so it does not admit a TT-algebra structure. Thus, the category T-Alg pT\text{-}\mathrm{Alg}_p does not admit equalizers in general.

Assuming that the 22-category 𝒦\mathcal{K} is complete, it is however possible to construct the following limits in T-Alg pT\text{-}Alg_p:

  • Products,
  • Inserters and iso-inserters,
  • Equifiers,

and they are created by the forgetful functor T-Alg p𝒦T\text{-}Alg_p \to \mathcal{K}. As we saw in last week’s post, these PIE-limits allow for the construction of many other limits. In particular, from the results discussed last week, we see that T-Alg pT\text{-}\mathrm{Alg}_p also has inverters and co-tensors, and hence also lax and pseudo limits.

It is also worth noting that each of the results on existence of limits “restricts to strict” (for lack of a better name), by which I mean that, for each of these limits, there exists a limiting cone such that the algebra 1-cells in the limiting cone:

  1. are strict, and

  2. detect strictness.

For example, for any parallel pair f,g:BCf, g : B \to C in T-Alg pT\text{-}\mathrm{Alg}_p there is an inserter p:ABp : A \to B such that (1) pp is strict, and (2) if phph is strict for some algebra morphism h:DAh : D \to A, then hh is strict.

The pseudomorphism classifier

Example (c’t’d). In the example where TAT A is AA provided freely with a terminal object, note that pseudo-morphisms can be mimicked using strict morphisms: for any algebra (A,t A)(A,t_A), consider the algebra (A,t A)(A',t_{A'}), defined by adding one new object t At_{A'} and an isomorphism t At At_{A} \cong t_{A'} to AA. It is then clear that, for any algebra (B,t B)(B,t_B), strict morphisms (A,t A)(B,t B)(A',t_{A'}) \to (B,t_B) correspond to pseudo-morphisms (A,t A)(B,t B)(A,t_A) \to (B,t_B). In fact, this correspondence is an isomorphism between the categories of morphisms and natural transformations between them.

The following theorem, which is arguably at the heart of the paper [BKP], says that the above phenomenon in fact occurs for any reasonably well-behaved 2-monad.

Theorem. Let TT be an accessible 2-monad on a 2-category 𝒦\mathcal{K} that is complete and cocomplete. Then the inclusion 2-functor T-Alg sT-Alg pT\text{-}\mathrm{Alg}_s \to T\text{-}\mathrm{Alg}_p has a left adjoint.

Proof (Sketch). The proof of the theorem consists of three steps:

  1. A general fact: in order to find a left adjoint to a 2-functor G:𝒦G : \mathcal{K} \to \mathcal{L}, it suffices to find a left adjoint to its underlying ordinary functor G oG_o, provided that 𝒦\mathcal{K} has cotensors with the walking arrow category 22 and GG preserves them.
  2. Using (1), one shows that there exists a left adjoint, () o()^o, to the inclusion functor T-Alg sT/𝒦, T\text{-}\mathrm{Alg}_s \to T/{\mathcal{K}}, where T/𝒦T/{\mathcal{K}} is the comma 2-category.
  3. The hardest part: pseudo-morphisms out of a TT-algebra (A,a)(A,a) can be mimicked by T/𝒦T/\mathcal{K}-morphisms out of a certain object (C,c,Z)(C,c,Z) of T/𝒦T/{\mathcal{K}}.

Now, composing (2) and (3), one associates to any TT-algebra (A,a)(A,a) the TT-algebra (C,c,Z) o(C,c,Z)^o and observes that this gives an ordinary (1-categorical) left adjoint to T-Alg sT-Alg pT\text{-}\mathrm{Alg}_s \to T\text{-}\mathrm{Alg}_p. Then, by (1) and the fact that cotensors exist in T-Alg pT\text{-}\mathrm{Alg}_p, it is also a 2-categorical left adjoint. \qed

The image under the left adjoint of an algebra AA, seen as an object of T-Alg pT\text{-}\mathrm{Alg}_p, is denoted by AA' and called the pseudo-morphism classifier of AA. Under the conditions of the Theorem, there is also a lax morphism classifier.

There are more conceptual proofs of these facts, using the concept of codescent objects; see, for example, this paper (which will be discussed in these series in a month or so) and Section 4 of the 2-categories companion by Stephen Lack. The latter paper, by the way, has been an indispensable source for me in preparing this post, and those who are familiar with it will probably recognize its influence throughout the post.


We denote by the letters pp and qq the unit and co-unit of the adjunction

J:T-Alg sT-Alg p:() J : T\text{-}\mathrm{Alg}_s \leftrightarrows T\text{-}\mathrm{Alg}_p : ()'

from the above theorem. For any algebra AA in T-Alg pT\text{-}\mathrm{Alg}_p, the morphism q A:AAq_A : A' \to A is in fact always a surjective equivalence in the 2-category T-Alg pT\text{-}\mathrm{Alg}_p, but in general q Aq_A does not even need to be an equivalence in T-Alg sT\text{-}\mathrm{Alg}_s, as we will see shortly. If q A:AAq_A : A' \to A is an equivalence in T-Alg sT\text{-}\mathrm{Alg}_s, then AA is called semi-flexible, and AA is called flexible if q Aq_A is a surjective equivalence in T-Alg sT\text{-}\mathrm{Alg}_s. The flexible objects are the cofibrant objects in a model structure on T-Alg sT\text{-}\mathrm{Alg}_s lifted from the model structure on 𝒦\mathcal{K}, and the pseudomorphism classifier AA' is then a special cofibrant replacement of AA (see Section 7.3 of the 2-categories companion for more details about this).

Several equivalent characterizations of flexibility and semi-flexibility are given in Theorems 4.4 and 4.7, respectively, of [BKP]. One useful equivalent way to say that a TT-algebra AA is semi-flexible is that every pseudo-morphism out of AA is isomorphic to a strict morphism out of AA. With this definition, we can see that not every TT-algebra is semi-flexible:

Example. Let TT be the 2-monad on Cat\mathbf{Cat} whose algebras are small categories with assigned finite limits. Let AA be the terminal category, with finite limits assigned in the only possible way. Let BB be any category with assigned finite limits in which t Bt_B is the assigned terminal object and the assigned product t B×t Bt_B \times t_B is not equal to t Bt_B (the two objects will of course be isomorphic). Then the functor ABA \to B which sends the unique object of AA to t Bt_B is a pseudo-morphism, but it is clearly not isomorphic to any strict morphism.

The following example shows that flexibility and semi-flexibility are really different concepts.

Example. Categories whose objects are functors can also often be represented as the TT-algebras for an appropriate monad TT on an appropriate base 2-category 𝒦\mathcal{K}. For instance, there is a 2-monad TT on Cat×Cat\mathbf{Cat} \times \mathbf{Cat}, given on objects by T(X,Y):=(X,X+Y)T(X,Y) := (X,X+Y), such that TT-algebras are functors, a pseudomorphism from f:ABf : A \to B to g:CDg : C \to D is a diagram of the form A f B u α v C g D \begin{matrix} A & \overset{f}{\to} & B \\ u \downarrow & \overset{\alpha}{\cong} & \downarrow v \\ C & \overset{g}{\to} & D \end{matrix} and such a pseudomorphism is strict exactly when α\alpha is the identity. Now, letting 11 denote the terminal category, it is easy to describe the pseudomorphism classifier of the TT-algebra a:A1a : A \to 1: this is the inclusion functor j:AA¯j : A \to \overline{A}, where A¯\overline{A} is the indiscrete category on objects ob(A)+{*}\mathrm{ob}(A) + \{\ast\} (As a simple but nice exercise, you may check that, indeed, any pseudomorphism out of the algebra a:A1a : A \to 1 corresponds uniquely to a strict morphism out of the algebra j:AA¯j : A \to \overline{A}.) Now, letting II again denote the category with two objects 00, 11 and a unique isomorphism between them, one may check that the algebra I1I \to 1 is equivalent in T-Alg sT\text{-}\mathrm{Alg}_s to the algebra 111 \to 1, which is flexible, and therefore I1I \to 1 is semi-flexible. However, I1I \to 1 is not flexible. (See example 4.11 in [BKP]).

Biadjunctions and bicolimits in T-Alg pT\text{-}\mathrm{Alg}_p

So far, we have only considered limits, which one would expect to exist in a category of algebras. On the other hand, we wouldn’t generally expect colimits to exist in a category of algebras, but as it turns out, in the last section of [BKP], the authors prove that:

  1. the category T-Alg pT\text{-}\mathrm{Alg}_p admits bicolimits, and
  2. any strict map of 2-monads θ:ST\theta : S \to T induces a map T-Alg pS-Alg pT\text{-}\mathrm{Alg}_p \to S\text{-}\mathrm{Alg}_p that has a left biadjoint.

Both of these results are consequences of the following more technical fact:

Theorem. If G:T-Alg pG : T\text{-}\mathrm{Alg}_p \to \mathcal{L} is a 2-functor so that the composite 2-functor

TAlg s J TAlg p G L \begin{matrix} T-Alg_s & \overset{J}{\to} & T-Alg_p & \overset{G}{\to} & L \end{matrix}

has a left adjoint HH, then HH maps into flexible algebras, and JHJ \circ H is left biadjoint to GG.

From the above theorem and the relation between biadjoints and bicolimits that we discussed last week, bicolimits can now be constructed in T-Alg pT\text{-}\mathrm{Alg}_p, as claimed in (1) above. To prove (2), one first notices that 2-functor θ *:T-Alg sS-Alg s\theta^* : T\text{-}\mathrm{Alg}_s \to S\text{-}\mathrm{Alg}_s extends to a 2-functor θ #:T-Alg pS-Alg p\theta^\# : T\text{-}\mathrm{Alg}_p \to S\text{-}\mathrm{Alg}_p making the diagram

T-Alg s θ * S-Alg s J J T-Alg p θ S-Alg p \begin{matrix} T\text{-}\mathrm{Alg}_s & \overset{\theta^*}{\to} & S\text{-}\mathrm{Alg}_s \\ J\downarrow & \quad & \downarrow J \\ T\text{-}\mathrm{Alg}_p & \overset{\theta^\sharp}{\to} & S\text{-}\mathrm{Alg}_p \end{matrix}

commute. One may then apply the Theorem in the case G=θ #G = \theta^\#.

More examples of 2-monads

Above I motivated the concepts and theorems in [BKP] with some simple examples of 2-monads. The last section of [BKP] contains many more examples. About the general method for constructing such examples, the authors make the following interesting comment.

“In practice one is seldom presented with a 2-monad and invited to consider its algebras; more commonly one contemplates some structure borne by a category (…) and one concludes in certain cases that the structure is given by an action of a 2-monad (…)”

With this comment in mind, one may now construct 2-monads whose algebras are monoidal categories, symmetric monoidal closed categories (here the 2-monad is over the 2-category Cat g\mathbf{Cat}_g, where the 2-cells are only taken to be natural isomorphisms), and even finitary 2-monads themselves (they are the algebras for a certain 2-monad RR on the functor 2-category [Cat f,Cat][\mathbf{Cat}_f,\mathbf{Cat}], where Cat f\mathbf{Cat}_f is the full subcategory of Cat\mathbf{Cat} consisting of the finitely presentable objects. This perspective was exploited in a later paper by Kelly and Power on presentations of 2-monads.

A final point of interest is that one may distinguish a special kind of 2-monad TT, namely those for which the TT-algebra structure on an object AA is unique if it exists. Such 2-monads TT define a property of rather than a structure on the objects of the base 2-category 𝒦\mathcal{K}, and may thus be called property-like (as they are in this later paper by Kelly and Lack). As the authors of [BKP] remark, it “may well be a hard problem” how to distinguish the property-like 2-monads TT from, say, a presentation for them. A particular class of 2-monads which are ‘property-defining’ are the lax-idempotent 2-monads (which also go by the names “quasi-idempotent” and “Kock-Zöberlein” 2-monads).


Let me finish with a (non-exhaustive) list of questions that may be interesting to discuss below.

  1. Can the fact that limits in T-Alg pT\text{-}\mathrm{Alg}_p can be chosen with a fair amount of “strictness” be understood using this account of lax / pseudo limits for morphisms between TT-algebras using \mathcal{F}-enrichment?

  2. The flexible algebras are exactly the strict retracts of pseudomorphism classifiers. The latter are “free algebras”, in some sense (at least in the sense that they are the images of a left adjoint). This suggests that one could think of the concept ‘flexible algebra’ as a 2-categorical version of the familiar concept ‘projective algebra’ in the 1-categorical setting. Is this a good intuition, and if so, can it be (or has it already been) made more precise?

  3. In order to better understand the concept of flexible algebra and the biadjunctions in the later part of [BKP], it would probably be useful to study different examples of 2-monads, and in particular, answer the following questions in such examples:

(a) is there a concrete construction of the pseudomorphism classifier?

(b) which algebras are (semi-)flexible?

(c) (for a strict map between 2-monads) what does the biadjunction do?

Posted at April 28, 2014 9:31 PM UTC

TrackBack URL for this Entry:

0 Comments & 0 Trackbacks

Post a New Comment