## ct.category theory – On \$mathbb{E}_{n-k}\$-monoidal structures on \$mathbb{E}_{n-m}\$-algebras in \$mathbb{E}_{n}\$-monoidal \$infty\$-categories

For ordinary categories, the assignment $$mathcal{C}mapstomathsf{Mon}(mathcal{C})$$ defines a functor $$mathsf{Mon}colonmathsf{Alg}_{mathbb{E}_{k}}(mathsf{Cats})tomathsf{Alg}_{mathbb{E}_{k-1}}(mathsf{Cats})$$, which is to say that:

• If $$mathcal{C}$$ is monoidal ($$mathbb{E}_{1}$$), then $$mathsf{Mon}(mathcal{C})$$ exists;
• If $$mathcal{C}$$ is braided ($$mathbb{E}_{2}$$), then $$mathsf{Mon}(mathcal{C})$$ is a monoidal category ($$mathbb{E}_{1}$$) in two different ways, related by replacing $$beta_{A,B}$$ by $$beta^{-1}_{B,A}$$
• If $$mathcal{C}$$ is symmetric ($$mathbb{E}_{3}=mathbb{E}_{4}=cdots$$), then $$mathsf{Mon}(mathcal{C})$$ is braided ($$mathbb{E}_{2}$$), and also symmetric ($$mathbb{E}_{3}$$), since $$mathbb{E}_{3}=mathbb{E}_{4}=cdots$$.

Morevoer, if one replaces $$mathsf{Mon}(mathcal{C})$$ by $$mathsf{CMon}(mathcal{C})$$ (i.e. $$mathsf{Alg}_{mathbb{E}_{1}}(mathcal{C})$$ by $$mathsf{Alg}_{mathbb{E}_{2}}(mathcal{C})congmathsf{Alg}_{mathbb{E}_{3}}(mathcal{C})congcdots$$), then $$mathsf{CMon}(mathcal{C})$$ is still monoidal ($$mathbb{E}_{1}$$… and braided ($$mathbb{E}_{2}$$) and symmetric ($$mathbb{E}_{3}$$)) when $$mathcal{C}$$ is symmetric ($$mathbb{E}_{3}$$… as again $$mathbb{E}_{3}=mathbb{E}_{4}=cdots$$), but having $$mathcal{C}$$ be braided fails to endow $$mathsf{CMon}(mathcal{C})$$ with a monoidal structure. So now the assignment $$mathcal{C}mapstomathsf{CMon}(mathcal{C})$$ gives a functor $$mathsf{CMon}colonmathsf{Alg}_{mathbb{E}_{k}}(mathsf{Cats})tomathsf{Alg}_{mathbb{E}_{k-2}}(mathsf{Cats})$$.

This whole situation made me wonder what happens in the $$infty$$-setting, where now we have not only monoidal, braided, and symmetric structures, but the whole array of $$mathbb{E}_{k}$$-monoidal structures for $$1leq kleqinfty$$, starting with $$mathbb{E}_{1}$$ (i.e. monoidal $$infty$$-categories) all the way up to $$mathbb{E}_{infty}$$ (i.e. symmetric monoidal $$infty$$-categories):

• Given an $$mathbb{E}_{n}$$-monoidal $$infty$$-category $$mathcal{C}$$, is there a sensible way to “count” how many natural induced $$mathbb{E}_{n-k}$$-structures are there on $$mathsf{Alg}_{mathbb{E}_{n-m}}(mathcal{C})$$, where $$1leq k,mleq n-1$$?
• Does the “space” of these induced structures have some kind of symmetry, such as in the case mentioned above where having a braided monoidal structure on $$mathcal{C}$$ gave $$mathsf{Mon}(mathcal{C})$$ two different monoidal structures related by exchanging $$beta_{A,B}$$ with $$beta^{-1}_{B,A}$$?

¹This was pointed out by Amar Hadzihasanovic on Zulip in reply to a question of David Roberts.

## ct.category theory – Let A be an Artin algebra. What happens if the limit and inverse limit are the same in mod A?

Let $$A$$ be an Artin algebra and $$text{mod},A$$ the category of finite length modules. Further, let $$X_0 longrightarrow X_1 longrightarrow X_2 longrightarrow …$$ and $$Y_0 longleftarrow Y_1 longleftarrow Y_2 longleftarrow …$$ be chains of morphisms in $$text{mod},A$$ such that $$underset{longrightarrow}{lim} X_i cong underset{longleftarrow}{lim} Y_j$$. What can we say about the module $$Z:=underset{longrightarrow}{lim} X_i cong underset{longleftarrow}{lim} Y_j$$? Does $$Z$$ have any nice properties? Does someone know an example, where $$Z$$ is not of finite length?

## ct.category theory – Dense subcategory of measurable spaces

Recall the notion of a dense subcategory $$mathcal{D}$$ of a category $$mathcal{C}$$. It means that the restricted Yoneda functor $$mathcal{C} to mathrm{Hom}(mathcal{D}^{op},mathbf{Set})$$, $$A mapsto mathrm{Hom}(-,A)|_{mathcal{D}}$$ is fully faithful. Roughly, it means that $$mathcal{D}$$ “detects morphisms” in $$mathcal{C}$$.

One can show that $$mathbf{Meas}$$, the category of measurable spaces$$^1$$, has no small dense subcategory. Trivially, $$mathbf{Meas}$$ is a dense subcategory of $$mathbf{Meas}$$, but that is not very interesting.

Question. What is an example of a “quite small” proper dense full subcategory of $$mathbf{Meas}$$?

By “quite small” I mean that we are not just removing a bunch of measurable spaces, but rather that the objects of the dense subcategory are parametrized by a very simple structure. Imagine, very informally, there was a measure on $$mathbf{Meas}$$, then I want the dense subcategory to be of measure $$0$$.

We can assume that the one-point measurable space belongs to the subcategory. If $$mathcal{K}$$ denotes the rest, we have the following characterization of density: If $$X,Y$$ are measurable spaces, then a map $$f : X to Y$$ is measurable iff for every measurable map $$a : A to X$$ for $$A in mathcal{K}$$ the composition $$f circ a : A to Y$$ is measurable. (This is what I meant above with “detecting morphisms”). The question asks for such a class of measurable spaces.

At first you might think that this is completely impossible. I had the same suspicion for $$mathbf{Top}$$, but it turns out that for $$mathbf{Top}$$ it is possible: take the one-point-space and the topological spaces of the form $$P cup {infty}$$ for directed sets $$P$$, where the sets $$P_{geq p} cup {infty}$$ form a local base at $$infty$$. This subcategory is dense: This is just a fancy way of saying that a map is continuous iff it preserves net convergence. Maybe there is some similar theory of “net convergence” for measurable spaces? I found the related discussion What properties are preserved under a measurable mapping?, but I am not sure if Eric Wofsey’s answer settles my question, because convergent filters cannot be seen as maps.

$$^1$$ Since Dmitri Pavlov’s notion of a measurable space has become quite prominent on mathoverflow, let me mention that I use the “classical” definition here. It’s just a set with a $$sigma$$-algebra. However, if there was a very good answer for Pavlov’s measurable spaces, I would be happy to hear about that too.

## ct.category theory – What is a cotopos?

This question is comprised of two parts, one particular and one general.

In the nLab page on base change, one finds the following statement:

For $$fcolon Xto Y$$ a morphism in a category $$mathcal{C}$$ with pullbacks, there is an induced functor $$f^{*}colonmathcal{C}_{/Y}tomathcal{C}_{/X}$$ of over-categories.
This is the base change morphism. If $$mathcal{C}$$ is a topos, then this refines to an essential geometric morphism $$(f_{!}dashv f^{*}dashv f_{*})colonmathcal{C}_{/X}tomathcal{C}_{/Y}.$$ The dual concept is cobase change.

In the dual case, if $$mathcal{C}$$ has pushouts, then we have an induced functor
$$f^{*} colon mathcal{C}_{X/} to mathcal{C}_{Y/}$$
$$(f_{*}dashv f^{*}) colon mathcal{C}_{Y/} rightleftarrows mathcal{C}_{X/}.$$
Particular Question. What is the dual condition on $$mathcal{C}$$ for which this adjunction extends to a triple adjunction
$$(f_{*}dashv f^{*}dashv f_{!}) colon mathcal{C}_{Y/} underset{to}{rightleftarrows} mathcal{C}_{X/}$$
between $$mathcal{C}_{Y/}$$ and $$mathcal{C}_{X/}$$?

General Question. What is the dual notion of a topos? Such a notion, call it a cotopos for brevity, might perhaps behave in the following ways:

• For cotopoi, Cartesian closedness is replaced by coCartesian coclosedness.
• Slices of cotopoi may fail to be cotopoi, but coslices of cotopoi are always cotopoi.
• Sheaves are to topoi as cosheaves are to cotopoi (Maybe. Should this indeed be the case?)
• (Etc.)

## ct.category theory – Under what hypotheses can a limit of presheaf categories, in \$mathsf{CAT}\$, be computed as presheaves on a colimit

Suppose

$$mathsf{X}:mathsf{J} longrightarrow mathsf{CAT}$$

factors through the functor

$$mathsf{Cat}^{mathsf{op}} longrightarrow mathsf{CAT}$$

which sends a small category $$mathsf{A}$$ to the category of presheaves $$widehat{mathsf{A}}$$ and sends a functor $$f:mathsf{A} rightarrow mathsf{B}$$ to the inverse image functor $$f^*:widehat{mathsf{B}} rightarrow widehat{mathsf{A}}$$. What further hypotheses are necessary so that the conical limit, in $$mathsf{CAT}$$ of the diagram $$mathsf{X}$$ may be computed as presheaves on the colimit of the factoring of $$mathsf{X}$$ through $$mathsf{Cat}^{mathsf{op}}$$?

I know from http://tac.mta.ca/tac/reprints/articles/25/tr25.pdf that this works for some limits taken in the category of toposes, but the forgetful functor$$mathsf{Topos} longrightarrow mathsf{CAT}$$ admits a right 2-adjoint, so in general those limits of toposes do not agree with the limits of the underlying categories.

However, this does work sometimes, for example:

-) products (definitely) (take the coproduct before taking presheaves)
-) op-lax limits of arrows (I think) (take the collage before taking presheaves)

but I’m really not clear on what’s known about this question which seems like the sort of thing which is “well known”.

## ct.category theory – Realizing a fusion category as endomorphisms of an algebra

Consider the two statements:

1. “Any unitary fusion category can be realised as a category of endomorphisms on a hyperfinite von Neumann algebra”, as stated in 1506.03546 page 4. The above paper refers to (I think) Theorem 7.6 of this paper. In the next paragraph of 1506.03546, they say that this is true for non-unitary fusion categories as well.

2. Not every unitary fusion category is strict, e.g. the unitary fusion category associated to any finite group G and a nontrivial 3-cocycle of G.

Now, I was previously under the impression that any category of endomorphisms is strict, but the two statements above show that this is wrong. What is a simple example that illustrates that a category of endomorphisms can have nontrivial associator?

## ct.category theory – Examples of (co)lax idempotent pseudocomonads on Cat

A lax idempotent pseudomonad, also called a KZ doctrine or KZ monad, is a pseudomonad $$(T, mu, eta)$$ with the property that $$T eta dashv mu dashv eta T$$. Lax idempotent pseudomonads were introduced to axiomatise colimit structure on categories, and it turns out that there is a strong sense in which lax idempotent pseudomonads exactly characterise colimit structure: Power–Cattani–Winskel prove in Theorem 16 of A representation result for free cocompletions that a 2-monad on $$mathrm{Cat}$$ is lax idempotent and dense if and only if there is a class of weights $$Phi$$ for which the 2-monad is the free $$Phi$$-cocompletion. (Density is a technical condition that serves to exclude pathological counterexamples, like the terminal 2-monad.)

Dually, colax idempotent pseudomonads axiomatise limit structure on categories.

I would like to know whether (co)lax idempotent pseudocomonads (i.e. KZ codoctrines, or KZ comonads) may be characterised analogously. Since I do not expect a general classification result exists, like that for lax idempotent pseudomonads, I am really looking for a few (nontrivial) examples of (co)lax idempotent pseudocomonads on $$mathrm{Cat}$$, to get an intuitive for what their coalgebras look like.

## ct.category theory – Reference request: the fixed category of an adjunction

Let $$F: A to B$$ and $$G: B to A$$ be adjoint functors, with $$F dashv G$$. There is a full subcategory $$A’$$ of $$A$$ consisting of those objects $$a$$ for which the unit map $$a to GF(a)$$ is an isomorphism, and there is a dually-defined full subcategory $$B’$$ of $$B$$. It is an elementary exercise to show that $$F$$ and $$G$$ restrict to an equivalence $$A’ simeq B’$$.

Either of the equivalent categories $$A’$$ and $$B’$$ is called the invariant part or fixed category of the adjunction. There are other names too; the terminology hasn’t settled down.

Q. Where did this general construction first appear in print?

Adjoint functors were introduced by Kan in 1958. I don’t see this construction in his paper. But I guess someone must have mentioned or used it quite soon thereafter. I want to know who I should cite.

(Let me make clear that I’m not asking about particular instances of this construction. It’s the general construction I’m after.)

## ct.category theory – Algebras for general transfors

Algebras for endofunctors bridge the gap between functors acting on a category and structures defined in it. An algebra for an endofunctor $$F$$ is instantiated by some morphism $$Fa to a$$, and more crucially a morphism of algebras is a map $$ato b$$ between their codomains making the evident diagram commute. This construction is intrinsically homotopical, so it makes good sense to loosen the restriction from just endofunctors.

Do you know of any writing about algebras for more general transfors? Seems that many 2-categorical results about algebras over monads, lax morphisms, et cetera, would enjoy a more full account of transfors and their algebras.

## ct.category theory – Reference request for Linton’s theorems on equational theories

This is a reference request for the following “well-known” theorems in category theory:

1. There is an equivalence of categories between finitary monads on $$mathbf{Set}$$ and finitary Lawvere theories (i.e. single-sorted finite product theories).

2. There is an equivalence of categories between monads on $$mathbf{Set}$$ and infinitary Lawvere theories (i.e. singled-sorted product theories).

3. The category of models of a finitary/infinitary Lawvere theory is equivalent to the category of algebras of the corresponding monad.

I am not looking for proofs (I have written them up). I am looking for references which actually give proofs, so that I can cite them in a paper (instead of writing up the proof in the paper). Ideally, they should be classical references. I have scrolled through lots of papers which mention the theorems, and most of the time one (or several) of Linton’s papers (let’s give them letters) are cited:

• E. Some aspects of equational theories, Proceedings of the Conference on Categorical
Algebra, Springer, 1966
• F. An outline of functorial semantics, Seminar on triples and categorical homology theory,
Springer, 1969
• A. Applied functorial semantics, Seminar on triples and categorical homology theory,
Springer, 1969

Sometimes, also the book “Toposes, Triples and Theories” by Barr & Wells is cited.

In E Linton only mentions 2) in the end of section 6, without proof. In fact, Linton proves a characterization theorem of the concrete categories of models of varietal theories (his name for infinitary Lawvere theories), via a version of the first isomorphism theorem, and he only mentions in passing that a combination with Beck’s monadicity theorem yields the
equivalence to monads (but this detour is actually not necessary to get the equivalence; so this is not the simplest proof anyway).

I do not understand much what is going on in F (I find it very hard to read, also because of the typesetting and the chaotic structure), but it seems to deal with a much more general situation, and therefore I don’t see where 1) or 2) is proven either. This is funny because both in the introduction of the Lecture Notes and in the introduction of A it is claimed that Linton proves 2) in F. Can perhaps someone help me to “decipher” F and explain where 2) is actually proved? The only result which looks similar is Lemma 10.2, but its proof is omitted… Maybe 3) follows from Theorem 9.3, but I don’t see how.

The paper A focusses on monadicity criterions, and just mentions 2) in the introduction.

I could not find a proof in the book by Barr-Wells either. They talk about the history of these theorems in section 4.5 and attribute 1) and 2) to Linton’s E and F.

I have found references with proofs of more general versions of 1), for example in the enriched case (Nishizawa, Power, Lawvere theories enriched over a general base), but it is probably awkward to cite such a paper for a classical result. I haven’t found a published proof of 2) so far. The only thing which comes very close to 2) and 3) is the nlab article on algebraic theories: https://ncatlab.org/nlab/show/algebraic+theory, but the proof is sketchy.