## ct.category theory – In a category with a projective generator, do morphisms from the generator determine the object?

I have a cocomplete abelian category $$mathcal C$$ and two objects $$X$$, $$Y$$ in $$mathcal C$$.
Further, $$mathcal C$$ has a projective generator $$P$$. I have an isomorphism

$$mathcal C(P,X) cong mathcal C(P,Y)$$

It feels like this should be enough to show that $$Xcong Y$$ in $$mathcal C$$. Because any object $$Z$$ in $$mathcal C$$ can be written as a cokernel $$P^Jlongrightarrow P^Ilongrightarrow Zlongrightarrow 0$$ and so we have $$mathcal C(Z,X)cong mathcal C(Z,Y)$$. It feels like the isomorphisms $$mathcal C(Z,X)cong mathcal C(Z,Y)$$ are natural in $$Z$$ and do not depend on the choice of projective resolutions.

But I cannot be sure. Is this true? Any reference or pointers would be appreciated.

## 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 – Equivalence of definitions tensor functor

Let $$(mathcal{C}, otimes , I)$$ and $$(mathcal{C}, otimes’, I’)$$ be tensor categories. A tensor functor $$F: (mathcal{C}, otimes , I)to (mathcal{C}’, otimes’ , I’)$$ consists of a functor $$F: mathcal{C}to mathcal{C’}$$ together with natural isomorphisms $$J_{X,Y}: F(X)otimes’ F(Y) to F(Xotimes Y)$$ and an isomorphism $$varphi: F(I)to I’$$ such that three compatibility diagrams with respect to the associators commute.

Next, consider the following two definitions:

(1) A tensor functor $$F$$ is called an equivalence of tensor categories if it is an equivalence of ordinary categories.

(2) A tensor functor $$F: (mathcal{C}, otimes , I)to (mathcal{C}’, otimes’ , I’)$$ is called an equivalence of tensor categories if there exists a tensor functor $$F’: (mathcal{C}’, otimes’ , I’)to (mathcal{C}, otimes , I)$$ together with natural tensor isomorphisms $$eta: operatorname{id}_{mathcal{C’}}to FF’$$ and $$theta: F’Fto operatorname{id}_{mathcal{C}}$$.

Definition (1) is the definition in Etingof’s book “Tensor categories” and definition (2) is in Kassel’s book “Quantum groups”. Clearly definition (2) implies definition (1). Is the converse true?

## ct.category theory – A tensor category need not be isomorphic to a strict tensor category

This question was originally posted on MSE, but got no answer even after putting a bounty on it, so I’ll try here.

I’m reading the book “Tensor categories” by Etingoff (and others). In remark 2.8.6 (posted below), it is claimed that the category $$mathcal{C}_G^omega$$ (defined in example 2.3.8, also below) is not isomorphic to a strict tensor category whenever $$omega$$ is not cohomologically trivial. Can someone explain this a bit more?

If someone knows another example of a tensor category that is not isomorphic to a strict tensor category, I’m also interested in that.

## 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 – The category of connected ribbon graph and its connected component

Let $$RG$$ be a category of connected ribbon graph, the morphisms are admissible epimorphism or finite composition of contraction. By a ribbon graph we mean a connected graph $$Gamma$$ with fixed cyclic orders on the set $$edge(v)$$ of half-edges attached to each vertex v. we assume that all vertices are at least ternary. We denote cyclic order on $$Gamma$$ is $$sigma_0 : edge(v) rightarrow edge(v)$$ and the involution of graph is $$sigma_1$$. Then the boundary operator $$sigma_{infty} = sigma_0 circ sigma_1$$. The orbit of $$sigma_{infty}$$ is called boundary. We denote $$n(Gamma)$$ as the number of boundaries. Now we can define Euler characteristic $$chi(Gamma) = V – E + n(Gamma) = 2 – 2g$$ where V is the number of vertices, E is the number of orbits of $$sigma_1$$.Since the admissible epimorphism or contraction preserve number of boundaries and number of genus( preserve Euler characteristic.

We have following equation $$RG = bigsqcup RG_{g,n}$$. I don’t know $$RG_{g,n}$$ is connected or not. I already know one vertex graph in $$RG_{g,n}$$ is not initial object, because there exist for same g and n, it have two different type of boundary cycles. If we assign a positive real number to the edge of ribbon graph, we got $$RG^{met}_{g,n}$$. which is fibration of $$RG_{g,n}$$. we take $$RBG_{g,n}$$ is the subcategory where the morphism preserve boundaries cycle i.e.it does not permutation boundary cycles. we also have $$RBG^{met}_{g,n}$$ as fibration of $$RBG_{g,n}$$. By Strebel theory we already know that $$RBG^{met}_{g,n} simeq M_{g,n} times mathbb{R}_+^n$$. And since $$M_{g,n}$$ is connected so $$RBG^{met}_{g,n}$$.

Does this means that $$RBG_{g,n}$$ is connected?(By fibration?) so $$RG_{g,n}$$? If it is true, Could I proof $$RG_{g,n}$$ only use the information from ribbon graph?

Soft question. Could we use the path category of ribbon graph which can help us read some information from Riemann surface, And we build ribbon structure as a sheaf of Nerve of path category of graph?

## ct.category theory – Cotensor products (in monoidal categories) without regularity

In Internal Categories and Quantum Groups, Aguiar defines the cotensor product of two bicomodules as follows. Let

• $$(mathcal{V},otimes_{mathcal{V}},mathbf{1}_{mathcal{V}})$$ be a monoidal category;
• $$(C,Delta_{C},epsilon_{C})$$, $$(D,Delta_{D},epsilon_{D})$$, and
$$(E,Delta_{E},epsilon_{E})$$ be comonoids in $$mathcal{V}$$;
• $$(M,alpha^{mathrm{L}}_{M},alpha^{mathrm{R}}_{M})$$ be a $$(C,D)$$-bicomodule in $$mathcal{V}$$;
• $$(N,alpha^{mathrm{L}}_{N},alpha^{mathrm{R}}_{N})$$ be a $$(D,E)$$-bicomodule in $$mathcal{V}$$.

Then the cotensor product of $$M$$ and $$N$$ is the $$(C,E)$$-bicomodule $$Mboxtimes_{D}N$$ defined by

To define the left $$C$$-and right $$E$$-coactions of $$Mboxtimes_{D}N$$, however, Aguiar assumes that $$mathcal{V}$$ is regular, i.e. that it has all equalisers and, for each parallel pair of morphisms $$f,gcolon Xrightrightarrows Y$$ of $$mathcal{V}$$ and each $$A,Binmathrm{Obj}(mathcal{V})$$, we have

$$Aotimes_{mathcal{V}}mathrm{Eq}(f,g)otimes_{mathcal{V}}B cong mathrm{Eq}(mathrm{id}_{A}otimes_{mathcal{V}}fotimes_{mathcal{V}}mathrm{id}_{B},mathrm{id}_{A}otimes_{mathcal{V}}gotimes_{mathcal{V}}mathrm{id}_{B}).$$

This in principle cuts out some desirable examples, as e.g. $$mathsf{Mod}_{R}$$ is regular iff all $$R$$-modules are flat (i.e. iff $$R$$ is von Neumann regular). On the other hand, I have seen it claimed elsewhere that, in the case of $$mathsf{Mod}_{R}$$, one can endow $$Mboxtimes_{D}N$$ with the structure of a $$(C,E)$$-bicomodule without any mention of $$R$$ being regular.

Question. Can one prove the following basic properties of bicomodules that Aguiar uses in his thesis without assuming $$mathcal{V}$$ is regular (or perhaps under weaker hypotheses)?

• Left and Right Coactions (Proposition 2.2.1 of Aguiar’s Thesis): The object $$Mboxtimes_{D}N$$ has the structure of a $$(C,E)$$-bicomodule in that we have coactions of the form
begin{align*} alpha^{mathrm{L}}_{Mboxtimes_{D}N} &colon Mboxtimes_{D}N longrightarrow Cotimes_{mathcal{V}}(Mboxtimes_{D}N),\ alpha^{mathrm{R}}_{Mboxtimes_{D}N} &colon Mboxtimes_{D}N longrightarrow (Mboxtimes_{D}N)otimes_{mathcal{V}}E end{align*}
satisfying the axioms of a $$(C,E)$$-bicomodule in $$mathcal{V}$$.
• Associators (Proposition 2.2.2 of Aguiar’s Thesis): For every triple $$(M,N,P)$$, we have a coherent isomorphism
$$alpha colon (Mboxtimes_{D}N)boxtimes_{E}P overset{{!sim}}{dashrightarrow} Mboxtimes_{D}(Nboxtimes_{E}P).$$
• (Unitors (Proposition 2.2.3 of Aguiar’s Thesis): We have coherent isomorphisms $$Cboxtimes_{C}Mcong M$$ and $$Mboxtimes_{D}Dcong M$$. This holds without regularity already.)

## ct.category theory – Is there a monoidal analogue of equalizers?

There are three different kinds of finite limits in categories: terminal objects, binary products, and equalizers. In a category $$C$$, these define functors $$1_{C},times,mathrm{Eq}colonmathrm{Fun}(I,C)to C$$ where $$I=emptyset,{bullet bullet}$$, and $${bulletrightrightarrowsbullet}$$ respectively.

Monoidal categories generalise the first two in that we now have a functor $$1_{C}$$ from $$mathcal{C}^{emptyset}=*$$ to $$mathcal{C}$$ and a functor $$otimes_C$$ from $$C^{{bullet bullet}}=C$$ to $$C$$, i.e. functors
begin{align*} 1_C &colon * to mathcal{C}\ otimes_C &colon mathcal{C}timesmathcal{C} to mathcal{C} end{align*}
together with associativity and unitality natural isomorphisms satisfying compatibility conditions.

What about equalizers? Has the notion of a category $$C$$ equipped with a functor $$rm{Eq}colonrm{Fun}({bulletrightrightarrowsbullet},C)to C$$, a unit functor, and unitality/associativity natural isomorphisms satisfying coherence conditions been studied before? Moreover, are there any examples of such structures “found in nature”?