**25**

votes

**0**answers

558 views

### Enriched Categories: Ideals/Submodules and algebraic geometry

While working through Atiyah/MacDonald for my final exams I realized the following:
The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...

**24**

votes

**0**answers

612 views

### Computer calculations in A_infinity categories?

Is there a good computer program for doing calculations in A-infinity categories?
Explicit calculations in A-infinity categories are an important, useful, yet very tedious task. One has to keep ...

**17**

votes

**0**answers

467 views

### categorical characterization of large cardinals

Question 1. Is there a categorical representation of Kunen's inconsistency result?
Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and ...

**17**

votes

**0**answers

626 views

### Pre-triangulated category that isn't triangulated

I've been working through some of the early parts of Neeman's book on triangulated categories, and he mentions that he does not know of a pre-triangulated category that is not triangulated. Is this ...

**16**

votes

**0**answers

597 views

### $\infty$-topos and localic $\infty$-groupoids?

It's known that every classical (Grothendieck) topos is equivalent to the topos of sheaves on a localic groupoid (a groupoid in the category of locales).
For the record, this is proved by, starting ...

**15**

votes

**0**answers

161 views

### Finite limits in the category of smooth manifolds?

The category of smooth manifolds does not have all finite limits. However, it does have some limits: it has finite products, it has splittings of idempotents, and it has certain other limits if we ...

**15**

votes

**0**answers

465 views

### Is there a category of topological spaces such that open surjections admit local sections?

The class of open surjections $Q \to X$ is a Grothendieck pretopology on the category $Top$ of spaces, and includes the class of maps $\amalg U_\alpha \to X$ where $\{U_\alpha\}$ is an open cover of ...

**14**

votes

**0**answers

281 views

### a naive question about the second dual of a vector space

Let $K$ be a field. Are there non-scalar endomorphisms of the endofunctor
$$
V\mapsto V^{**}/V
$$
of the category of $K$-vector spaces?
I asked a related question on Mathematics Stackexchange, but ...

**14**

votes

**0**answers

331 views

### Applications of Lawvere's fixed point theorem

Lawvere's fixed point theorem states that in a cartesian closed category, if there is a morphism $A \to X^A$ which is point-surjective (meaning that $\hom(1,A) \to \hom(1,X^A)$ is surjective), then ...

**14**

votes

**0**answers

569 views

### Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?

This is a followup to my previous question, which asked whether
the category of commutative or noncommutative C*-algebras or von Neumann algebras
is equivalent to the category of commutative or ...

**13**

votes

**0**answers

271 views

### The topologies for which a presheaf is a sheaf?

Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal.
Suppose that $Q$ is a presheaf on ...

**13**

votes

**0**answers

344 views

### Relative consistency of ETCS over the theory of a well-pointed topos with NNO

Gödel's well-known proof of the implication $Con(ZF) \Rightarrow Con(ZFC)$ used the construction of the inner model $L$ in $ZF$ to get a model of $ZFC$ (and in fact much more). However such a ...

**13**

votes

**0**answers

499 views

### Recognizing classifying toposes

Suppose $\mathbb{T}$ is a geometric theory, $\mathcal{E}$ is a topos, and $M$ is a model of $\mathbb{T}$ in $\mathcal{E}$. Is there any sort of elementary condition on $M$ and $\mathcal{E}$ (or, even ...

**12**

votes

**0**answers

337 views

### How much choice is required to prove concretizability theorems in category theory?

A concretization of a category is a faithful functor to the category of sets. A category is concretizable if there exists such a functor.
An evident necessary condition for concretizability is ...

**12**

votes

**0**answers

413 views

### What are some examples of weak ω-categories?

As is usual, let's say an (n, k)-category is something with
objects, morphisms, 2-morphisms, ..., n-morphisms, such that all
j-morphisms for j > k are invertible, everything meant in the
weak sense. ...

**12**

votes

**0**answers

330 views

### When can we tell if PROPs, Algebraic Theories, etc. are faithfully detected in a given category?

I am interested in understanding a certain phenomenon. I am hoping this sort of problem has been studied before, but I don't know the proper terminology and am having trouble finding answers. I am ...

**12**

votes

**0**answers

190 views

### Is there a common framework for Tannaka and Gabriel-Ulmer reconstruction theorems?

Gabriel-Ulmer duality is a biequivalence between the 2-category of finite limit categories and the 2-category of locally finitely presentable categories. It allows for the reconstruction of a theory ...

**12**

votes

**0**answers

654 views

### Homotopy flat DG-modules

A right DG-module $F$ over an associative DG-algebra (or DG-category) $A$ is said to be homotopy flat (h-flat for brevity) if for any acyclic left DG-module $M$ over $A$ the complex of abelian groups ...

**11**

votes

**0**answers

359 views

### Categorifications of Zorn's lemma

I'm wondering about categorifications of Zorn's lemma along the following lines.
Lemma: if $\mathbf{C}$ is a small category in which every directed diagram of monomorphisms has a cocone of ...

**11**

votes

**0**answers

581 views

### Fully dualizable objects in classical field theories

This is a follow up to this MO question: Free symmetric monoidal $(\infty,n)$-categories with duals
Freed-Hopkins-Lurie-Teleman define a classical field theory as a symmetric monoidal functor $I$ ...

**11**

votes

**0**answers

454 views

### Is there some way to see a Hilbert space as a C-enriched category?

The inner product of vectors in a Hilbert space has many properties in common with a hom functor. I know that one can make a projectivized Hilbert space into a metric space with the Fubini-Study ...

**10**

votes

**0**answers

130 views

### Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes.
Equivalently (i) every interpretation of ...

**10**

votes

**0**answers

174 views

### Smallest class of rings closed under familiar operations

Suppose I start out with the ring $\mathbb{Z}$, and call $\mathcal{C}$ the smallest collection of (commutative, unital) rings closed under the following list of operations (which I am aware has some ...

**10**

votes

**0**answers

341 views

### Higher holonomies for higher local systems

In Jacob Lurie's classification of tqfts, one finds a version of the cobordism hypothesis for $(X,\zeta)$-structure, where an $(X,\zeta)$ structure on a manifold $M$ is the datum of a continuous map ...

**10**

votes

**0**answers

370 views

### Can we describe equivariant vector bundles of free group action in terms of descent theory (Barr-Beck theorem)?

It is well known that for a compact topological group $G$ acts (say, from the right) freely on a compact space $X$. Then the category of equivariant complex vector bundles on $X$, $\text{Vect}_G(X)$, ...

**10**

votes

**0**answers

259 views

### What is the history of the notion of subdivision of categories?

A recent answer by Peter May prompts me to ask a question which I have been considering to ask for several months. (The reason why I have not asked it before is that it is not directly related to my ...

**10**

votes

**0**answers

442 views

### Can the similarity between the Riesz representation theorem and the Yoneda embedding lemma be given a formal undergirding?

For example, by viewing Hilbert spaces as enriched categories in some fashion? (I suppose the same idea of considering the inner product of a Hilbert space as a generalized Hom-set has also been ...

**10**

votes

**0**answers

543 views

### Categorical Schur's Lemma

In attempt to prove (and compute) a formula for the dimensions of the HOMFLY homology
of the (p,q)-torus knot one could try to follow original proof by Jones of a formula for
HOMFLY polynomial of ...

**9**

votes

**0**answers

211 views

### Unitary structures on fusion categories

A unitary fusion category is a fusion category with a $C^*$-tensor structure.
Hence, in principle, a fusion category could have more than one unitary structure. Does exist a fusion category with more ...

**9**

votes

**0**answers

441 views

### Is “being a full ring of quotients” a Morita invariant property?

Definition and context:
An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...

**9**

votes

**0**answers

340 views

### Can we categorify the formula for the quadratic Gauss sum?

Background
Fix an odd prime $p$ and set $\zeta=e^{2\pi i/p}$. We define the quadratic Gauss sum as
$$g=\sum_{n=0}^{p-1} \zeta^{n^2}.$$
It's pretty easy to show that
$$g^2=
\begin{cases}
p & ...

**9**

votes

**0**answers

161 views

### Are pivotal categories the algebras for a cartesian monad?

It seems to be "known" but not written down that the following are more-or-less equivalent:
...

**9**

votes

**0**answers

409 views

### The open problem of nth quantization

In trying to explain a quote by E. Nelson, "First quantization is a mystery, but second quantization is a functor!" Baez points out what follows (full text available in this week find; I'm also ...

**9**

votes

**0**answers

412 views

### Is there any elementary text unravelling the definitions of 2-category, lax functor and lax transformation, allowing people who do not know in the first place what these things are to really understand the definitions?

The question is in the title.
My current research subject is the homotopy theory of $2$-categories. It may be slightly unreasonable to expect my PhD thesis to be read or even looked at by people ...

**9**

votes

**0**answers

437 views

### Characterizing the surcomplex numbers

Conway showed that the Field of surreal numbers ("${\bf No}$")
is the maximal totally ordered Field.
Later Jacob Lurie showed that the Group of all partizan games ${\bf Pg}$ is
the universally ...

**9**

votes

**0**answers

603 views

### monomorphisms and epimorphisms of local rings

I want to understand the structure of monomorphisms/epimorphisms in the category of local rings (with local homomorphisms), or dually in the category of local schemes. Let $LR$ denote this category.
...

**8**

votes

**0**answers

186 views

### Quotient of a category by a group

Let $G$ be a group acting on a category $\mathcal{C}$; that is, a monoidal functor $G\rightarrow \mathrm{Aut}(\mathcal{C})$, where the latter is the 2-group of auto-equivalences of $G$. Explicitly, ...

**8**

votes

**0**answers

70 views

### What are filtered colimits in the category of complete semilattices?

My question is basically stated in the title. Does somebody know an explicit description of the filtered colimits in the category of complete semilattices?
I am happy to provide background explaining ...

**8**

votes

**0**answers

234 views

### Can a composition with itself of a universal self-map be non-universal?

I have formulated (and published) the notion of a universal map (and of a universal morphism), and the problems below, in the early 1960-ies.
DEFINITION A continuous map $u: ...

**8**

votes

**0**answers

318 views

### The category theory of $(\infty, 1)$-categories

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent ...

**8**

votes

**0**answers

239 views

### Twisted duality in a symmetric monoidal category

I would like to know if the following definition is already known or even well-known. Is there a reference in the literature? Do you know prominent examples?
Definition. Let $\mathcal{C}$ be a ...

**8**

votes

**0**answers

422 views

### Conceptual proofs for the computation of the structure sheaf

The following lemma in commutative algebra is important for the foundations of algebraic geometry:
If $A$ is a commutative ring, $U \subseteq A$ is a finite subset generating the unit ideal, then ...

**8**

votes

**0**answers

390 views

### Compact objects in triangulated and infinity categories

Hello,
I think of a compact object in a category as an object, $Hom$ from which commutes with filtered colimits.
I guess that in an infinity category, one also defines a compact object as an object, ...

**8**

votes

**0**answers

440 views

### Categorification of finite type invariants

Recall that finite type invariants are those numerical knots invariants vanishing on knots with enough singularities. The value of an invariant on a singular knots is computed using the Vassiliev ...

**8**

votes

**0**answers

233 views

### Adjoint functor theorem for infinity categories

In HTT, a version of the adjoint functor theorem for (locally) presentable infinity categories is proven (Corollary 5.5.2.9). Is there a more refined version of this somewhere, which more closely ...

**8**

votes

**0**answers

601 views

### Monads and Comonads that interact

Hi,
I am reading Awodey's text "Category Theory". In section 10.4 he talks about comonads and monads that occur together and interact and he says that "possibility" and "necessity" in propositional ...

**8**

votes

**0**answers

347 views

### Deducing properness from H^i(X,F) finitely generated over \Gamma(O_X)

Suppose that $X$ is a quasi-projective variety over a field $k$ and that we further know that for every coherent sheaf $\mathcal{F}$, $H^i(X,\mathcal{F})$ is finitely generated over $\Gamma(O_X)$. Is ...

**8**

votes

**0**answers

390 views

### Does the category of strict $2$-categories together with Dwyer-Kan equivalences provide a model for $(\infty,1)$-categories?

The question is the title.
In what follows, all $2$-categories and $2$-functors will be strict. Let $2-Cat$ denote the categories whose objects are $2$-categories and whose morphisms are ...

**8**

votes

**0**answers

279 views

### Bicommutative Hopf algebras have internal hom objects. What are they?

Let $k$ be a field and $\mathcal H$ the category of conilpotent bicommutative Hopf algebras over $k$. (Conilpotent means that every element eventually becomes zero if you apply the reduced ...

**8**

votes

**0**answers

243 views

### What are the logical morphisms from a topos E to Set?

If $E$ is a topos, is there a nice way to characterize the category of logical morphisms $E\to Set$? Is it complete and/or cocomplete?
The topos $Set$ geometrically represents a point; what does it ...