## June 2, 2014

### Codescent Objects and Coherence

#### Posted by Emily Riehl

Guest post by Alex Corner

This is the 11th post in the Kan Extension Seminar series, in which we will be looking at Steve Lack’s paper

A previous post in this series introduced us to two-dimensional monad theory, where we were told about $2$-monads, their strict algebras, and the interplay of the various morphisms that can be considered between them. The paper of Lack has a slightly different focus in that not only are we interested in morphisms of varying levels of strictness but also in the weaker notions of algebra for a $2$-monad, namely the pseudoalgebras and lax algebras.

An example that we will consider is that of the free monoid $2$-monad on the $2$-category $\mathbf{Cat}$ of small categories, functors, and natural transformations. The strict algebras for this $2$-monad are strict monoidal categories, whilst the lax algebras are (unbiased) lax monoidal categories. Similarly, the pseudoalgebras are (unbiased) monoidal categories. The classic coherence theorem of Mac Lane is then almost an instance of saying that the pseudoalgebras for the free monoid $2$-monad are equivalent to the strict algebras. We will see conditions for when this can be true for an arbitrary $2$-monad.

Thanks go to Emily, my supervisor Nick Gurski, the other participants of the Kan extension seminar, as well as all of the participants of the Sheffield category theory seminar.

Algebras for $2$-monads

When doing $2$-category theory, we often look at weakening familiar notions. We generally do this by replacing axioms that required commutativity of certain diagrams with (possibly invertible) $2$-cells, which themselves are required to satisfy coherence axioms. For instance, given a $2$-monad $T$ (with multiplication $\mu$ and unit $\eta$) on a $2$-category $\mathcal{K}$, a lax algebra for $T$ consists of an object $A$ of $\mathcal{K}$, a $1$-cell $x : TX \rightarrow X$ of $\mathcal{K}$ and $2$-cells $\begin{matrix} T^2X & \overset{Tx}{\longrightarrow} & TX & & X & \overset{1_X}{\longrightarrow} & X \\ {}_{\mu_X}\downarrow & \Downarrow {\chi} & \downarrow^x & & {}_{\eta_X}\searrow & \Downarrow {\chi_0} & \nearrow_x & \\ TX & \underset{x}{\longrightarrow} & X & & \quad & TX & \\ \end{matrix}$ in $\mathcal{K}$ which satisfy suitable axioms. A pseudoalgebra is defined as above but with invertible $2$-cells.

Example We’ll see what’s going on by looking at the free monoid $2$-monad again, call it $M$. A lax algebra for $M$ is a category $X$ and a functor $x : MX \rightarrow X$ with natural transformations $\chi$, $\chi_0$ as above. Now $MX$ is the coproduct $\coprod_{n \in \mathbb{N}} X^n$ meaning that objects in $MX$ are finite lists of objects in $X$, and similarly for morphisms. The functor $x : MX \rightarrow X$ is a functor out of a coproduct so in fact corresponds to a family of functors $(x_n : X^n \rightarrow X)_{n \in \mathbb{N}}$ which we can view as being the $n$-ary tensors of an unbiased lax monoidal category. The natural transformation $\chi$ then has components which are morphisms $\left(\left(a_{11} \otimes \ldots \otimes a_{1k_1}\right) \otimes \ldots \otimes \left(a_{n1} \otimes \ldots \otimes a_{nk_n}\right)\right) \rightarrow \left(a_{11} \otimes \ldots \otimes a_{nk_n}\right)$ in $X$. These are what correspond to the associators in a biased monoidal category. The associativity and unit axioms can then be found to be expressed by the lax algebra axioms.

These differing levels of strictness offer us a whole host of $2$-categories to look at. For our purposes we will be looking at the following $2$-categories:

• $T\text{-Alg}_s$, of strict algebras, strict morphisms, and transformations;
• $\text{Ps-}T\text{-Alg}$, of pseudoalgebras, pseudomorphisms, and transformations;
• $\text{Lax-}T\text{-Alg}_l$, of lax algebras, lax morphisms, and transformations.

Lax codescent objects

The second section of the paper begins by considering lax morphisms of the form $(f, \overline{f}) : (X, x, \chi, \chi_0) \rightarrow (Y,y),$ between a lax algebra $X$ and a strict algebra $Y$. The idea is that lax morphisms of this form in $\text{Ps-}T\text{-Alg}$ can be recast as strict morphisms $(g = y \cdot Tf, \overline{g} = 1_{y} \ast T\overline{f}) : (TX, \mu_X) \rightarrow (Y,y)$ in $T\text{-Alg}_s$. There is an inclusion 2-functor $U : T\text{-Alg}_s \rightarrow \text{Lax-}T\text{-Alg}_l$ and the aim is to construct a left adjoint. To this end, Lack describes a universal property related to $1$-cells in $T\text{-Alg}_s$ of the form $TX \rightarrow X'$ so that there is an isomorphism $T\text{-Alg}_s(X',Y) \cong \text{Lax-}T\text{-Alg}_l(X,Y)$ which is natural in Y. This tells us that if such an object $X'$ exists for every lax algebra $X$, then the left adjoint also exists.

The universal property in question turns out to be that of a lax codescent object in a $2$-category. First we define lax coherence data to be diagrams $\begin{array}{ccccc} \quad & \overset{p}{\rightarrow} & \quad & \overset{d}{\rightarrow} & \\ X_3 & \overset{q}{\rightarrow} & X_2 & \overset{e}{\leftarrow} & X_1\\ \quad & \overset{r}{\rightarrow} & \quad & \overset{c}{\rightarrow} & \end{array}$ accompanied by $2$-cells $\begin{array}{cc} \delta : de \Rightarrow 1_{X_1}, & \gamma : 1_{X_1} \Rightarrow ce, \\ \kappa : dp \Rightarrow dq, & \lambda : cr \Rightarrow cq, \\ \rho : cp \Rightarrow dr. \end{array}$ A lax codescent object is then an object $X$, a $1$-cell $x : X_1 \rightarrow X$, and a $2$-cell $\chi : xd \Rightarrow xc$, all interacting with the $1$-cells and $2$-cells of the lax coherence data. These then also satisfy universal properties of a $2$-categorical nature, much like those we saw in a previous post.

Consider for a moment, an algebra $(A,a)$ for a $1$-monad $S$ on a $1$-category $\mathcal{C}$. We know that this can be expressed as the reflective coequaliser of the diagram $\begin{array}{ccc} \quad & \overset{\mu_A}{\longrightarrow} & \quad \\ S^2A & \overset{S\eta_A}{\longleftarrow} & SA \\ \quad & \overset{Sa}{\longrightarrow} & \quad \\ \end{array}$ in the category $S\text{-Alg}$ of $S$-algebras. However in the case of a lax algebra $(X, x, \chi, \chi_0)$ for a $2$-monad $T$, this won’t be the case. Instead we can form lax coherence data $\begin{array}{ccccc} \quad & \overset{\mu_{TA}}{\rightarrow} & \quad & \overset{\mu_A}{\rightarrow} & \\ T^3X & \overset{T\mu_X}{\rightarrow} & T^2X & \overset{T\eta_A}{\leftarrow} & TX\\ \quad & \overset{T^2x}{\rightarrow} & \quad & \overset{Tx}{\rightarrow} & \end{array}$ in $T\text{-Alg}_s$ when we accompany it with $2$-cells $T\chi_0$ and $T\chi$, where the rest of the $2$-cells are just identities arising from the $2$-monad axioms. The universal property alluded to above is then that the lax codescent object of this lax coherence data is the same as that of the replacement (strict) algebra $X'$ which would give the adjunction previously described.

If all of the mentions of $2$-cells in the above description of a lax codescent object were replaced with invertible $2$-cells, then we would have the notion of a codescent object. This is the analogous situation in the case of pseudoalgebras, where the aim is to find a left adjoint to the inclusion to the inclusion $2$-functor $T\text{-Alg}_s \rightarrow \text{Ps-}T\text{-Alg}.$

A useful observation is that lax codescent objects may be defined using weighted colimits and can be built from coinserters and coequifiers. Also worthy of note is that codescent objects can be built from co-iso-inserters and coequifiers. Now co-iso-inserters exist whenever coinserters and coequifiers do, so that anything we want to prove about lax algebras by utilising such colimits, will also be true for pseudoalgebras.

This section of the paper also includes a number of results concerning adjunctions between the various $2$-categories of algebras, with the following theorem then being the basis for the first characterisation of a coherence theorem.

Theorem: (Lack, 2.4) For a $2$-monad $T$ on a $2$-category $\mathcal{K}$, the inclusion $T\text{-Alg}_s \rightarrow \text{Lax-}T\text{-Alg}_l$ has a left adjoint if any of the following conditions holds:

1. $\mathcal{K}$ admits lax codescent objects and $T$ preserves them;
2. $\mathcal{K}$ admits coinserters and coequifiers and $T$ preserves them;
3. $\mathcal{K}$ is cocomplete and $T$ preserves $\alpha$-filtered colimits for some regular cardinal $\alpha$.

Conditions $2$ and $3$ also give us a left adjoint to the inclusion $T\text{-Alg}_s \rightarrow \text{Ps-}T\text{-Alg}$. Furthermore, we also find that a left adjoint to the inclusion $T\text{-Alg}_s \rightarrow T\text{-Alg}$, which we saw in the paper of Blackwell, Kelly, and Power, also exists under these conditions. Something else that we saw in that paper is the reason for needing $T$ to preserve these colimits - the colimits exist in $T\text{-Alg}_s$ just when $T$ preserves them.

Coherence

The simplest possible characterisation of coherence for $2$-monads would be:

Theorem-Schema: The inclusion $T\text{-Alg}_s \rightarrow \text{Ps-}T\text{-Alg}$ has a left adjoint, and the components of the unit are equivalences in $\text{Ps-}T\text{-Alg}$.

Now this is certainly not true in general. A counter-example (3.1) is given in the paper, whilst Mike Shulman also shows that not every pseudoalgebra is equivalent to a strict one.

Something that is rather nice, though, is that we already have some conditions under which the theorem-schema is satisfied.

Theorem: (Lack, 3.2) If $T$ is a $2$-monad on a $2$-category $\mathcal{K}$ admitting codescent objects, and $T$ preserves them, then the inclusion $T\text{-Alg}_s \rightarrow \text{Ps-}T\text{-Alg}$ has a left adjoint, and the components of the unit are equivalences in $\text{Ps-}T\text{-Alg}$. In particular this is the case if $\mathcal{K}$ has coinserters and coequifiers, and $T$ preserves them.

The proof of this is rather simple and falls out of the two-dimensional universal property of the codescent objects.

I’m going to roll the latter two sections of the paper together now and talk about the other characterisation of coherence, which concerns a general coherence result of Power. That paper looks at $2$-monads on $\mathbf{Cat}^X$ and $\mathbf{Cat}^X_g$, where $X$ is a small set and the latter $2$-category is attained from the first by only considering invertible $2$-cells. Power then shows that if $T$ is a $2$-monad on one of these $2$-categories which preserves bijective-on-objects functors, then every pseudoalgebra for $T$ is equivalent to a strict one.

Some $2$-monads which satisfy these conditions include $\mathbf{Set}$-based clubs, whose strict algebras give such structures as monoidal categories (see the scope of the results below for more monoidal examples) or categories with strictly associative finite products or coproducts. Also described in Power’s paper is a $2$-monad on $\mathbf{Cat}^{X \times X}$ for which the pseudoalgebras are unbiased bicategories with object set $X$. The coherence result then tells us that every bicategory is biequivalent to a $2$-category with the same set of objects.

Comparing Power’s statement to the theorem-schema, we see that they are not quite the same. The schema asks for there to be an adjunction for which the components of the unit give the equivalences we are concerned with. As it turns out, the conditions which Power proposes are indeed enough to give what we desire, and this is what the latter characterisation of Lack looks at.

Recall that every functor can be factored as a bijective-on-objects functor followed by a full and faithful functor. This gives an orthogonal factorisation system $(bo,ff)$ on $\mathbf{Cat}$. However, the $(bo,ff)$ factorisation system has an extra two-dimensional property concerning $2$-cells. If we are given a natural isomorphism $\begin{matrix} A & \overset{R}{\longrightarrow} & C \\ {}_{F}{\downarrow} & {\Downarrow}_{\alpha} & \downarrow^G \\ B & \underset{S}{\longrightarrow} & D \\ \end{matrix}$ where $F$ is bijective-on-objects and $G$ is full and faithful, then there is a unique pair $(H,\beta)$ consisting of a functor $H:B \rightarrow C$ and a natural isomorphism $\beta:GH \Rightarrow S$ such that $HF = R$ and the whiskering of $\beta$ with $F$ gives back $\alpha$. For an arbitrary $2$-category $\mathcal{K}$, an orthogonal factorisation system with such a property is deemed an enhanced factorisation system.

Theorem: (Lack, 4.10) If $\mathcal{K}$ is a $2$-category with an enhanced factorisation system $(\mathcal{L},\mathcal{R})$ having the property that if $j \in \mathcal{R}$ and $jk \cong 1$ then $kj \cong 1$, and if $T$ is a $2$-monad on $\mathcal{K}$ for which $T$ preserves $\mathcal{L}$-maps, then the inclusion $T\text{-Alg}_s \rightarrow \text{Ps-}T\text{-Alg}$ has a left adjoint, and the components of the unit of the adjunction are equivalences in $\text{Ps-}T\text{-Alg}$.

The proof starts by noting that if we have a pseudoalgebra $(X, x, \chi, \chi_0)$ then we can factorise $x:TX \rightarrow X$ as $TX \overset{e}{\longrightarrow} X' \overset{m}{\longrightarrow} X,$ where $e \in \mathcal{L}$ and $m \in \mathcal{R}$. Thus we have an invertible $2$-cell $\begin{array}{ccccc} T^2X & \overset{\mu_X}{\longrightarrow} & TX & \overset{e}{\longrightarrow} & X' \\ {}_{Te}{\downarrow} & \quad & \Downarrow^{\chi} & \quad & \downarrow^{m} \\ TX' & \underset{Tm}{\longrightarrow} & TX & \underset{x}{\longrightarrow} & X \\ \end{array}$ and, since $T$ preserves $\mathcal{L}$-maps, we can use the enhanced factorisation system to get a strict algebra $X'$ which is equivalent to $X$. (See Power’s coherence result for the details on this.)

It is interesting to see the scope of these results and the places in which people have considered this type of coherence problem before.

• Dunn proved the theorem-schema when $\mathcal{K}$ is the $2$-category of based topological categories and for which $T$ is a $2$-monad induced by a braided $\mathbf{Cat}$-operad.
• The theorem-schema was also proved by Hermida, though required much more of both the $2$-category $\mathcal{K}$ and the $2$-monad $T$, such as requiring existence and preservation of various limits and colimits, exactness properties relating these, as well as further conditions on the unit and multiplication of the $2$-monad. Something that does fall out of this alternative setup is that $T$ can be replaced by a new $2$-monad, on a different $2$-category, which is lax-idempotent.
• Rather more recently Nick Gurski and I wrote about operads with general groups of equivariance. Therein we showed that the $2$-monads which arise from $\mathbf{Cat}$-operads in this way satisfy the coherence conditions following the enhanced factorisation system route. These $2$-monads capture many different structures, including monoidal categories, braided monoidal categories, symmetric monoidal categories, and ribbon braided monoidal categories. Thus we can say, for example, that every unbiased braided monoidal category is equivalent to a braided strict monoidal category, and similarly for the other variations.
• The first theorem we mentioned above has three conditions, the third being the requirement that $\mathcal{K}$ is cocomplete and $T$ preserves $\alpha$-filtered colimits for some regular cardinal $\alpha$. We mentioned aboe that it was proved by Blackwell, Kelly, and Power that this is also sufficient to give a left adjoint to the inclusion $U : T\text{-Alg}_s \rightarrow T\text{-Alg}$. They also proved further that if $\mathcal{K}$ is locally $\alpha$-presentable then there is a $2$-monad $T'$ which preserves $\alpha$-filtered colimits and where $T'\text{-Alg}_s = \text{Ps-}T\text{-Alg}$. The result of the theorem we discussed then follows when $\mathcal{K}$ is locally presentable and $T$ preserves $\alpha$-filtered colimits. Lack comments that it is a major unsolved problem as to whether the entire theorem-schema can be shown to be true under these asumptions - and further whether it is true when $\mathcal{K}$ is only cocomplete.
Posted at June 2, 2014 12:46 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2745

### Re: Codescent Objects and Coherence

Another interesting appearance of codescent objects, as touched on in this paper, is in the bo-ff factorisation of a functor. Just as how in Set (or more generally a regular category) we may form the image factorisation of a function by taking the coequaliser of its kernel pair (the equivalence relation it induces on its domain), in Cat (or more generally) we get the bo-ff factorisation of a functor by taking the codescent object of its “higher kernel”. See for instance John Bourke’s thesis, where this 2-dimensional exactness play continues, featuring cateads in the role of equivalence relations.

Posted by: Alexander Campbell on June 2, 2014 9:42 AM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

Nick pointed something out to me this morning regarding the last bullet point above - some discussion related to this also came up between the seminar participants as well. Turns out I should have looked more closely at Mike Shulman’s paper showing that not every pseudoalgebra is equivalent to a strict one and not just looked at what the counterexample was. In fact, in the abstract, he says that the $2$-monad giving the counterexample is finitary, i.e., preserves $\alpha$-filtered colimits for some regular cardinal $\alpha$. This is the the third condition of Theorem 2.4 in the post. In the last bullet point of the post I repeated what Steve had said in that it would be ideal to be able to prove the coherence theorem under the condition that $\mathcal{K}$ be cocomplete and $T$ preserve $\alpha$-filtered colimits but Mike’s counterexample clearly shows this wouldn’t be sufficient.

Posted by: Alex Corner on June 2, 2014 12:25 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

Right: that was the whole point of that paper, to show that the answer to Steve’s unsolved problem is (perhaps surprisingly) “no”.

Posted by: Mike Shulman on June 3, 2014 7:01 AM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

I thought I’d create a thread similar the one we had before, in order to collect some more examples of where the coherence theorems apply.

A few neat examples come from some stuff I was looking at to do with $T$-multicategories. I’ll try to describe them here.

A $T$-multicategory is a multicategory defined relative to a cartesian monad $T$, from which we recover plain multicategories as those defined relative to the free monoid monad on $\mathbf{Set}$. This can also be done for $2$-cartesian $2$-monads - replace every mention of pullback by $2$-pullback. For now I just want to run with the identity $2$-monad $I$ on $\mathbf{Cat}$, so we can see what’s going on.

To define an $I$-multicategory we need a category of objects, $C_0$, and a category of arrows, $C_1$, such that (along with some maps) these form a monad in the bicategory of spans, $Span(\mathbf{Cat})$, in $\mathbf{Cat}$. For our example take these to be the discrete categories $C_0 = \lbrace s, t\rbrace$ and $C_1 = C_0 \times C_0$. Once we have this data we can define a monad $I_c$ on $\mathbf{Cat}/C_0$. Note that an object $(X, r_X : X \rightarrow C_0$ can be seen as the disjoint union of two categories, $X_s$ and $X_t$, each lying over the respective object. Now we define $I_c(X, r_X)$ as being the composite running along the top of the following diagram, defined by using a $2$-pullback. $\begin{array}{ccccc} I_cX & \longrightarrow & C_1 & \overset{c}{\longrightarrow} & C_0 \\ \downarrow & \quad & \downarrow_d & & \\ X & \underset{r_X}{\longrightarrow} & C_0 & & \\ \end{array}$ So the category $I_cX$ consists of triples $(a, i_1, i_2)$ where $a \in X$ and $i_1, i_2 \in C_0$.

Now a strict algebra $x : I_c(X,r_X) \rightarrow (X,r_X)$ is a functor (which I’ll also call $x$) $x : I_cX \rightarrow X$, making the usual diagrams commute. It is also required to be a $1$-cell in $\mathbf{Cat}/C_0$ which tells us that, for example, the object $(a, s, t)$ will end up in the category $X_t$. Similarly for the other possibilities $(ss)$, $(ts)$, and $(tt)$. What this actually does is define four functors $x_s : X_s \rightarrow X_s$, $x_t : X_t \rightarrow X_t$, $x_{st} : X_s \rightarrow X_t$, and $x_{ts} : X_t \rightarrow X_s$, where the axioms tell us that $x_s$ and $x_t$ are both identities. We also see that $x_{st} x_{ts} = 1_{X_t}$ and $x_{ts} x_{st} = 1_{X_s}$, so that a strict $I_c$-algebra is in fact an isomorphism of categories. It is then easy to see that a pseudoalgebra for $I_c$ is an equivalence of categories.

Now $\mathbf{Cat}/C_0$ inherits the $(bo,ff)$ enhanced factorisation system and $I_c$ preserves bijective-on-objects functors, so $I_c$ satisfies the coherence theorem. Thus every equivalence of categories is ‘equivalent’ to an isomorphism of categories. To spell this out in more detail we actually have a commuting square $\begin{array}{ccc} X_s & \overset{\cong}{\longrightarrow} & X_t \\ {\simeq}_{\downarrow} & \quad & \downarrow^{\simeq} \\ X_s' & \underset{\simeq}{\longrightarrow} & X_t' \\ \end{array}$ where the bottom horizontal and both vertical functors are equivalences, and the top horizontal arrow is an isomorphism.

Now we can do a similar thing to this for monoidal functors, which we acquire by using the free monoid $2$-monad $M$ on $\mathbf{Cat}$. However, instead of setting $C_1 = C_0 \times C_0$ we now have it be a subcategory of $MC_0 \times C_0$ generated by tuples of the form $(s, \ldots, s ; s)$, $(t, \ldots, t; t)$ and $(s, \ldots, s ; t)$, as well as those with empty ‘domain’. After a bit of a slog it’s possible to see that strict algebras for the induced $2$-monad $M_c$ are in fact strict monoidal functors (just one functor $X_s \rightarrow X_t$ this time, rather than getting one in each direction) and further that pseudoalgebras are unbiased monoidal functors. Again the $2$-monad satisfies the coherence conditions, which gives a similar commuting square as that one above where the vertical functors are unbiased monoidal equivalences, saying that every unbiased monoidal functor is equivalent to a strict monoidal functor.

Even niftier is that all of the $\mathbf{Cat}$-operads mentioned above, that give the various flavours of monoidal category, are also $2$-cartesian and we can go through the same process with those.

Posted by: Alex Corner on June 2, 2014 1:17 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

Street’s 420 paper ‘Fibrations and Yoneda’s lemma in a 2-category’ describes fibrations in a 2-category as the pseudoalgebras of a 2-monad; the strict algebras are the split fibrations. Power’s coherence theorem then gives the result that in Cat every fibration is equivalent to a split fibration. Do the results of this week’s paper apply to fibrations in more general 2-categories? When does the 2-monad preserve codescent objects? The 2-monad acts by composition of spans / by pullback, so this is perhaps some kind of exactness property.

Posted by: Alexander Campbell on June 2, 2014 2:12 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

a monad in the bicategory of spans in $Cat$

… otherwise known as an internal category in $Cat$, i.e. a double category. (-:

Posted by: Mike Shulman on June 3, 2014 7:04 AM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

A nice example to play around with is the 2-monad whose strict algebras are 2-functors from $C$ to $Cat$.

Ie. given a small 2-category $C$ one has the set of objects $obC$ and inclusion $obC\to C$ which gives rise to restriction $U:[C,Cat]\to [obC,Cat]$. Because $obC$ is discrete the left adjoint, left Kan extension, has the easy formula $FX(c)=\Sigma_{j}C(j,c) \times Xj$.

$U$ is monadic and the monad $T=UF$ is cocontinuous and so preserves codescent objects and satisfies the various coherence theorems.

What is nice about this particular 2-monad is that you can easily check that the pseudoalgebras for $T$ are precisely pseudofunctors, and not something unbiased, so that the coherence result then asserts exactly that each pseudofunctor from $C$ to $Cat$ is equivalent to a 2-functor.

Posted by: John Bourke on June 5, 2014 6:59 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

Another interesting thing about that example is that $U$ is also comonadic, via a continuous comonad. Thus, the dual of the first type of coherence theorem is also true, i.e. the inclusion from pseudofunctors into 2-functors also has a right adjoint.

Posted by: Mike Shulman on June 5, 2014 11:58 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

Here’s an exercise: does the theory of this paper imply that every symmetric monoidal category is equivalent to a strict symmetric monoidal category, where associators and symmetries are all identities? Lack references an argument of Isbell reported in Mac Lane’s book (on p. 160) which should show the underlying category of the strict replacement can’t be isomorphic to the one we started with in general.

So the question is: does the 2-monad for commutative monoids in $\mathbf{Cat}$ preserve coisoinserters and coequifiers?

If this does work out, what does the strict replacement of a symmetric monoidal category look like? An interesting test case would be the monoidal category of graded vector spaces, which admits at least two interesting symmetries: one has $\sigma(x\otimes y) = y \otimes x$ while the other has $\sigma(x\otimes y) = (-1)^{\mathrm{deg} x \, \mathrm{deg} y} y \otimes x$. It would be interesting to compare the strictifications for these two different symmetries.

Posted by: Tim Campion on June 2, 2014 1:32 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

Ah! This should have been a reply to Alex’s comment above!

Posted by: Tim Campion on June 2, 2014 1:34 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

Isn’t the category of graded vector spaces (with the “non-trivial” symmetry) a counterexample? Consider $\sigma_{V,V} : V \otimes V \to V \otimes V$. If the category in question were equivalent (as a symmetric monoidal category) to a strict symmetric monoidal category, then by naturality, $\sigma_{V,V} = \mathrm{id}$. But this is not the case for a general $V$.

Posted by: Zhen Lin on June 2, 2014 2:03 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

This should mean that there is no coherence theorem of the second type for the commutative monoid monad $T$ on $\mathbf{Cat}$, i.e. not every pseudoalgebra is equivalent to a strict one. It doesn’t rule out a coherence theorem of the first type, i.e. having a left 2-adjoint to the forgetful functor $T-\mathrm{Alg}_s \to \mathrm{Ps}-T-\mathrm{Alg}$.

But of course, Lack shows that if $T$ preserves codescent objects then a coherence theorem of the second type holds. So $T$ must fail to preserve codescent objects; in particular it must fail to preserve either coisoinserters or coequifiers. Which one is it?

The natural next question is: does $T$ preserve filtered colimits? If it does, then Lack’s Thm 2.4 shows that $T$ satisfies a coherence theorem of the first type, and as a bonus, we get a simpler counterexample to Lack’s “missing coherence theorem”. It seems to me that $T$ must preserve filtered colimits because it is the monad for certain algebras with finite arities, right?

Posted by: Tim Campion on June 2, 2014 10:06 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

One should not confuse strict symmetric (strict) monoidal categories with symmetric strict monoidal categories. The former are literally internal commutative monoids whereas the latter are the things that symmetric monoidal categories are pseudo-versions of. (To reiterate, a pseudoalgebra for the commutative monoid monad is not a symmetric monoidal category!)

But your intuition is correct: the symmetric strict monoidal category monad does indeed preserve filtered colimits, since it is given by the formula $T X = \coprod_{n \ge 0} X^n // S^n$, where $X^n // S_n$ is the pseudo-quotient of $X^n$ by the canonical $S_n$-action, which is just a certain weighted colimit.

Posted by: Zhen Lin on June 2, 2014 10:26 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

It’s been a while since I thought about this, but I believe there is a coherence theorem of the second type for the commutative monoid monad on $Cat$. But, as Zhen says, even if that’s the case, it doesn’t tell you that symmetric monoidal categories can be strictified to commutative monoids in $Cat$, since the pseudoalgebras for the commutative monoid monad are not symmetric monoidal categories. It’s a fun exercise to work out what they are — they’re kind of weird-looking!

Posted by: Mike Shulman on June 3, 2014 6:58 AM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

Nice post Alex!

I have a question related to the motivation of lax coherence data. For 1-monads every algebra has a standard presentation as quotient of a free algebra. So is there some way to present a lax algebra for a 2-monad as some kind of colimit of a standard diagram?

Posted by: Sean Moss on June 2, 2014 9:19 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

To answer your question: sort of! I imagine what I’m about to say could be done by taking some form of lax limit, rather than a pseudo-limit. As you say, an algebra $(A,a)$ for a monad $T$ can be seen as the colimit of the reflexive coequalizer $\begin{array}{ccc} \quad & \overset{\mu_A}{\longrightarrow} & \quad \\ T^2A & \overset{T\eta_A}{\longleftarrow} & TA. \\ \quad & \overset{Ta}{\longrightarrow} & \quad \\ \end{array}$ Now if we have a pseudoalgebra (or lax algebra) then we obviously can’t do this as we know that $Ta \circ T\eta_A \neq 1$, there is a $2$-cell in there. This is why we have the coherence data. What we can do with this is look at a certain kind of colimit called a pseudocoequalizer - this is seen in the paper Beck’s theorem for pseudo-monads by Le Creuer, Marmalejo, and Vitale. The relevant definition is given near the start of section $2$ in that paper, whilst Lemma $2.3$ shows that the morphism $a : TA \rightarrow A$ is the pseudocoequalizer of the pseudoalgebra’s coherence data.

Posted by: Alex Corner on June 3, 2014 11:27 AM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

Unfortunately, calling that kind of colimit a “pseudo-coequalizer” is incorrect, since it is not the pseudo-fication of a coequalizer. A pseudo-coequalizer would be given $f,g:A\;\rightrightarrows\; B$ and consider the object $C$ universally equipped with maps $p:A\to C$ and $q:B\to C$ and isomorphisms $q f \cong p$ and $q g \cong p$. The correct name is… “codescent object”. (-:

Posted by: Mike Shulman on June 3, 2014 5:33 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

The “coherence theorem of the first type” that Steve proves, describing a bijection between pseudomorphisms $A \to B$ and strict morphisms $A' \to B$ can be understood as construction some sort of cofibrant replacement – at least in the case where $A$ is also a strict algebra.

Can this analogy between “cofibrancy” and “coherence” be pushed any further?

Posted by: Emily Riehl on June 3, 2014 1:04 AM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

The categorification of various coherence theorems into the ones ‘of first type’ (correspondence between weak and strict morphisms) and ‘of second type’ (equivalence between strict and weak algebras) is very interesting. In particular, the addition of a coherence result of first type to Power’s coherence result of second type was achieved by proving that the established equivalences between pseudo and strict algebras constitute the unit of the pursued adjunction. Of course, the enhanced factorisation system plays an important role in this proof, but is this phenomenon something we would expect? Are the coherence theorems usually/always connected via such a relation?

Posted by: Christina Vasilakopoulou on June 3, 2014 3:44 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

That’s a very interesting question! Actually, I think it’s two questions:

1. Does there exist a 2-monad $T$ for which every pseudoalgebra is equivalent to a strict one, but for which the inclusion $T Alg \to Ps T Alg$ does not have a left adjoint?

2. Does there exist a 2-monad $T$ for which every pseudoalgebra is equivalent to a strict one, and the inclusion $T Alg \to Ps T Alg$ has a left adjoint, but the components of the unit are not equivalences?

I suspect the answer to the first question is yes, since without any cocompleteness hypotheses you can cook up all sorts of terrible things like Steve’s example (3.1). But I wouldn’t be surprised if the answer to the second were no.

Posted by: Mike Shulman on June 3, 2014 5:48 PM | Permalink | Reply to this

### Re: Codescent Objects and Coherence

With great consideration and apologizing for the occasion , a very valuable standpoint in descriptive universalization is defined in ’ Isomorphic formulae in classical propositional logic ’ by Dòsen and Petric in Mat Log Q 2012 ; 58 ; 1-2 : 5-17 , explaining the syntactic characteristics of pairs of isomorphic formulae in the spirit of coherence results in monoidal category .

Posted by: SABINO GUILLERMO ECHEBARRIA MENDIETA on June 27, 2014 11:18 AM | Permalink | Reply to this

Post a New Comment