# Tagged Questions

**2**

votes

**1**answer

137 views

### Loop defects in Walker-Wang model

My question is about the description of general defects (specially loop defects) in the Walker-Wang (WW) model.
Elementary excitations in the WW model can be point particles, loop defects and more ...

**10**

votes

**1**answer

169 views

### What sort of W-types follow from existence of an NNO?

A W-type is an initial algebra for a polynomial endofunctor $P$ on a category $C$. A well-known example is that of a natural numbers object (NNO). Usually it is assumed that $C$ is locally cartesian ...

**5**

votes

**4**answers

610 views

### Are abelian non-degenerate tensor categories semisimple?

A pivotal monoidal category is called non-degenerate if the inner product $\left(x,y\right) = Tr\left(xy^{*}\right)$ (where $y^{*}$ is the dual map) is non-degenerate. As a rule of thumb ...

**5**

votes

**0**answers

119 views

### Is there a higher, “orientalish” version of geometric realisation?

Geometric realisation of simplicial sets can be roughly thought of like this:
In some category $\mathcal{C}$, we choose an object for every abstract $n$-simplex. In topological spaces, we would ...

**5**

votes

**1**answer

101 views

### What homomorphisms $G \to BrPic(\mathcal{C})$ correspond to group-theoretical $G$-extensions of $\mathcal{C}$?

For a fusion category $\mathcal{C}$ the Brauer-Picard group $\text{BrPic}(\mathcal{C})$ is the group of all invertible $\mathcal{C}$-bimodule categories under multplication $\boxtimes_\mathcal{C}$.
...

**1**

vote

**0**answers

131 views

### Merging / combining categories

Given a category $\mathcal X$ (that will be used as an underlying category) and collection of categories $\mathcal C(i)$ for $i \in I$ with a faithful functor from each $\mathcal C(i)$ to $\mathcal ...

**6**

votes

**0**answers

133 views

### The bifunctoriality of co/limits

I recently noticed that there are two senses in which colimits are functorial, and I'm curious about their interplay.
Let $C$ be a cocomplete category. Then, on the one hand, for any diagram ...

**7**

votes

**1**answer

510 views

### What are $( \infty , n)$-categories useful for?

I know that mathematicians are trying to construct adequate models for $( \infty, n)$-categories. Although, it seems to be an interesting task, I would like to know some explicity examples where this ...

**7**

votes

**1**answer

226 views

### Separation condition for higher Deligne-Mumford stacks

Let $X$ be a stack of $n$-groupoids on the site of affine schemes over a fixed base, with the etale topology. If $n=1$ then for $X$ to be Deligne-Mumford, aside from having an etale atlas from an ...

**3**

votes

**1**answer

289 views

### Notation: Categories of measur(abl)e spaces

Is there a common notation in the literature for
the category of measurable spaces and measurable maps?
the category of measure spaces and measure-preserving maps?
The nlab suggests ...

**17**

votes

**5**answers

1k views

### What's special about the Simplex category?

I have been wondering lately what makes simplicial sets 'tick'.
Edited
The category $\Delta$can be viewed as the category of standard $n$-simplices and order preserving simplicial maps. The goal of ...

**1**

vote

**0**answers

70 views

### is sufficient cohesion equivalent to the connectedness of subobject classifier?

I'm following Lawvere article Axiomatic Cohesion.
He states (Proposition VI.4) that sufficient cohesion is equivalent to the connectedness of subject classifier, but I can't follow the proof. I can't ...

**2**

votes

**0**answers

117 views

### Relationship between coherent toposes/coherent logic and coherent sheaves

I've heard it claimed that the adjective "coherent" in logic/topos theory (i.e. coherent logic, coherent toposes, coherent categories) was adopted to fit in with the terminology of coherent sheaves in ...

**2**

votes

**0**answers

57 views

### Notions of/References for freely generated (symmetric) monoidal categories

We often describe a category by giving a (directed, multi-)graph and freely generating a category of paths. I would like to know to what degree this intuition generalizes to monoidal categories, and ...

**5**

votes

**2**answers

334 views

+50

### Primitive Recursive Arithmetic via Universal Algebra

From the wikipedia article on Primitive Recursive Arithmetic (see http://en.wikipedia.org/wiki/Primitive_recursive_arithmetic):
"Primitive recursive arithmetic, or PRA, is a quantifier-free ...

**2**

votes

**0**answers

138 views

### Continuous maps to fat geometric realizations of simplicial spaces

The nLab page on partitions of unity mentions the application of partitions of unity as a way to construct continuous maps to geometric realizations of simplicial spaces. However I often feel ...

**0**

votes

**1**answer

87 views

### tree derived from monad is itself a monad

I have constructed a functor from a monad that appears (based on computer experiments to test the monad laws) to also have monad properties but I am having trouble proving it.
Here is the idea: M[A] ...

**0**

votes

**0**answers

61 views

### Is this quasi-coherent sheaf a subsheaf of $\ker f$?

Let $f: \mathcal{F}\to \mathcal{G}$ be a morphism of quasi-coherent sheaves over a scheme $X$. Let also $T_U$ be a submodule of $\ker f_U$ with $|T_U|\leq \kappa$ for each open subset $U$ of $X$ ...

**99**

votes

**10**answers

5k views

### What non-categorical applications are there of homotopical algebra?

(To be honest, I actually mean something more general than 'homotopical algebra' - topos theory, $\infty$-categories, operads, anything that sounds like its natural home would be on the nLab.)
More ...

**3**

votes

**0**answers

96 views

### Proofs in monoidal categories [closed]

I have to do some pretty ugly proofs in monoidal categories. Basically, I have some long identities that I would like to prove. A random example:
$$(a\otimes b)\circ (c\otimes d) \circ q = q $$
Are ...

**2**

votes

**1**answer

236 views

### Recollement of multiple $t$-structures

Given a recollement
$$
\mathbf{D}^0 \underset{\underset{i_R}\leftarrow}{\overset{\overset{i_L}\leftarrow}\to} \mathbf{D} \underset{\underset{q_R}\leftarrow}{\overset{\overset{q_L}\leftarrow}\to} ...

**3**

votes

**1**answer

89 views

### Is the image of a idempotent morphism in $\mathcal{K}(\mathcal{A})$ defined in the naive way?

Let $\mathcal{A}$ be an abelian category and $\mathcal{K}(\mathcal{A})$ be the homotopy category of chain complexes in $\mathcal{A}$. It is well-known that $\mathcal{K}(\mathcal{A})$ is idempotent ...

**36**

votes

**7**answers

3k views

### Is the ultraproduct concept fundamentally category-theoretic?

Once again, I would like to take advantage of the large number of knowledgable category theorists on this site for a question I have about category-theoretic aspects of a fundamental logic concept.
...

**5**

votes

**1**answer

355 views

### Functor generalisation

In an article I am writing, I am led to the following generalization of the notion of functor. Let $C$ and $D$ and be two categories. A generalized functor $F : C \to D$ is given by:
a function $f : ...

**15**

votes

**4**answers

2k views

### Is there a categorical treatment of dynamical systems?

Let $X$ be a set and $(T,\cdot)$ an abelian group. Is there a category of $T$-dynamical systems on $X$ which yields useful information about $X$ and $T$?
More precisely, is there a category whose ...

**5**

votes

**2**answers

563 views

### Internal equivalence implies weak equivalence for Frechet Lie groupoids?

It is a known theorem that an internal equivalence of Lie groupoids (finite dimensional manifolds!) - that is an equivalence in the 2-category of Lie groupoids, smooth functors and transformations - ...

**3**

votes

**1**answer

92 views

### Regarding the definition of S-flows over a category (given a monoid S)

(This question was originally directed to Simone Virili, referring to the answer http://mathoverflow.net/a/103840/2926, but could also be addressed to the greater community.)
I was wondering if you ...

**11**

votes

**4**answers

674 views

### Brandt's definition of groupoids (1926)

The definition of a category is usually attributed to Mac Lane and Eilenberg (1945). What seems to be less known is that the german mathematician Heinrich Brandt has developed the notion of a groupoid ...

**1**

vote

**0**answers

125 views

### Récollement of stable $t$-structures

Given a recollement
$$
\mathbf{D}^0 \underset{\underset{i_R}\leftarrow}{\overset{\overset{i_L}\leftarrow}\to} \mathbf{D} \underset{\underset{q_R}\leftarrow}{\overset{\overset{q_L}\leftarrow}\to} ...

**-1**

votes

**1**answer

76 views

### Is this apushout diagram [closed]

Let $A, B, C, E$ and $F$ be some objects in an abeleian category $\mathcal{C}$. Let we have a commutative diagram
\begin{array}{ccccccccc}
0 & \xrightarrow{} & A & \xrightarrow{f} & ...

**0**

votes

**0**answers

33 views

### How is the monoidal product defined on the functor category between symmetric monoidal dagger cats

I have found a quote in a paper by Abramsky and Heunen
If C and D are symmetric monoidal dagger categories, then so is the category
[C, D] of functors F : C → D that preserve the dagger. ...

**0**

votes

**1**answer

64 views

### Canonical colimit and cartesian product of simplicial sets

Let $K$ be a simplicial set and let $\Delta K$ be the category of simplices, i.e the category where the objects are simplicial maps
$$
\Delta[n]\to K
$$
and the maps $\phi\: : \: (\Delta[n]\to K)\to ...

**2**

votes

**0**answers

35 views

### Lattice of subobjects of a particular coproduct

I have the following situation: $\mathcal C$ is a (good enough, say Grothendieck) Abelian category and $F:\mathcal C\to \mathcal C$ is self-equivalence. Given an object $C$ in $\mathcal C$, what can I ...

**9**

votes

**1**answer

400 views

### Non-unique splittings of homotopy idempotents

By a homotopy idempotent I mean a map $f:X\to X$, where $X$ is a space, equipped with a homotopy $f\circ f \sim f$. In contrast to the situation in stable homotopy theory (where $X$ would be a ...

**1**

vote

**0**answers

66 views

### When is the category of endofunctors braided? When is it ribbon? Fusion? Modular?

Given a category $\mathcal{C}$, we can define the category of endofunctors $\operatorname{Cat}(\mathcal{C})$, with objects functors $F: \mathcal{C} \to \mathcal{C}$ and morphisms natural ...

**5**

votes

**2**answers

379 views

### Explicit description of a fibered category

I found the following exercise in Vistoli's notes. He proves a theorem stating that any category $\mathcal{F}$ fibered over $\mathcal{C}$ is equivalent, as a fibered category, to a split one. Namely ...

**21**

votes

**3**answers

707 views

### Possible categorical reformulation for the usual definition of compactness

Let $X$ be a compact topological space, $f_i:Y_i\to X$ a family of continuous maps such that the topology on $X$ is final for it (i.e., $U\subset X$ is open iff $f_i^{-1}(U)$ is open for each $i$, for ...

**8**

votes

**4**answers

691 views

### Topological Grothendieck Construction

Let $C$ be a small category and $F\colon C^{op}\rightarrow Set$ a functor. The Grothendieck construction is the category $F\wr C$ with objects being pairs $(c,x)$ where $c$ is a object of $C$ and ...

**0**

votes

**0**answers

61 views

### The first lemma in Auslander's functors and morphisms determined by objects

[lemma 1.1] Let $\mathcal{C}$ be a preadditive category. Suppose G is in ($\mathcal{C^{op}}$, $\mathcal{Ab}$). If for each $X$ in $\mathcal{C}$ we are given a subgroup $A_x$ of $G(X)$ such that ...

**5**

votes

**1**answer

136 views

### Relations between functors in a recollement

Consider a recollement situation like the following
by the very definition of the various functors it follows that $i^* j_*=0$, and $j^! i_* = 0 = j^* i_!$. Also, $j^! i_! = 0 = j^* i_*$ by ...

**1**

vote

**1**answer

304 views

### (Homotopy) limits and colimits in a dg-category

It is known that differential graded categories (or, also, $A_\infty$-categories) are 'incarnations', in some sense, of (stable) $(\infty,1)$-categories. I'm not used to the theory of ...

**6**

votes

**3**answers

472 views

### Category of Gödel Codings? [Reference Request]

Consider computation with the integers $\mathbb{Q}$. The traditional theory of recursive functions on $\mathbb{N}$ applies to $\mathbb{Q}$ by the identification of $\frac{a}{b} \in \mathbb{Q}$ with ...

**0**

votes

**0**answers

48 views

### Intersection and union of torsion classes

One of the main result in
Cassidy, C., M. Hébert, and G. M. Kelly. "Reflective subcategories, localizations and factorization systems." Journal of the Australian Mathematical Society (Series A) ...

**4**

votes

**1**answer

186 views

### Establishing Duality in Tannakian Categories

I sometimes need to prove a category is Tannakian. Part of the definition of a Tannakian category is that it is rigid.
However, I find the definition of rigid categories somewhat difficult. I don't ...

**5**

votes

**3**answers

440 views

### Why are pushouts the right tool in these setups

$\newcommand{\cat}[1]{\mathcal{#1}}$
$\newcommand{\cod}{\operatorname{cod}}$
$\DeclareMathOperator{\dom}{dom}$
$\DeclareMathOperator{\colim}{colim}$
The question is about two pushout constructions ...

**3**

votes

**1**answer

474 views

### Opposite Symmetric Monoidal Structure on an Infinity Category

Given an $\infty$-category (in the sense of Lurie) $C$, and a symmetric monoidal structure on $C$ associated to a coCartesian fibration $p:C^\otimes\to N(Fin_\ast)$, Lurie says in Remark 2.4.2.7 of ...

**8**

votes

**2**answers

278 views

### Definition of internal field objects

Let $\mathcal{C}$ be a category with finite limits and a strictly initial object $\mathbf{0}$. The final object is denoted by $\mathbf{1}$. I propose the following definition of a field object ...

**0**

votes

**2**answers

226 views

### Serre's Theorem for Coherent Sheaves

I recently heard a discussion about a certain of Serre which reconstructs the category of coherent sheaves of a variety $V$ as the category of modules over the homogeneous space of $V$ modulo modules ...

**5**

votes

**2**answers

339 views

### A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category

I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea.
Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ ...

**4**

votes

**0**answers

112 views

### When does the canonical model structure on $\mathcal V$-$\mathbf{Cat}$ give a structure of monoidal model category?

Let $\mathcal V$ be a closed symmetric monoidal model category. It is well known that the category $\mathcal V$-$\mathbf{Cat}$ of $\mathcal V$-enriched categories is itself a closed symmetric monoidal ...