**9**

votes

**0**answers

208 views

### Goodwillie calculus and morphisms of functors

Let $F,G: \mathcal{T}\to \mathcal{S}$ be two functors from topological spaces to spectra (or topological spaces) and let $s: F\to G$ be a morphism between them.
Suppose $F$ and $G$ are analytic and ...

**1**

vote

**0**answers

127 views

### Axioms for a symmetric monoidal bicategory

I start reading the axioms for a symmetric monoidal bicategory. The axioms include so many diagrams to be satisfied. I am wondering if people really use these axioms directly to check a given data is ...

**5**

votes

**1**answer

174 views

### formally smooth functor

Let $p$ be a prime number, $\mathcal{O}$ the integers of a finite extension of $\mathbb{Q}_p$ with residue field $k$. Let $\mathcal{C}$ be the category of complete, local, noetherian ...

**1**

vote

**0**answers

66 views

### Can a category's partial monoid of arrows be completed to a total monoid?

Categories can be presented in the language of arrows only, without reference to objects (as discussed, say, here: Categories presented with Arrows only, no objects: partial monoids), and this is a ...

**8**

votes

**1**answer

214 views

### What is this operad-like structure called?

I'd like to know what's the name (if any) of the following categorical structure, and also references where it has been considered.
Given a category $C$, let $O=\{O(n)\}_{n\geq 0}$ be a sequence of ...

**5**

votes

**2**answers

331 views

### Homotopy factorization of morphisms of chain complexes

This is a sort of follow up to this MO question.
Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). ...

**6**

votes

**1**answer

322 views

### What information is lost in $X \to \mathrm{Sh}(X)$?

Given a topological space or site $X$. Construct $\mathrm{Sh}(X)$ - the sheaves on $X$ with values in $\mathrm{Set}$. Is it known what information is lost in this procedure?
Thanks, Adrian.

**10**

votes

**1**answer

333 views

### Categorical interpretation of disjoint cycle notation for tracing permutations

For a natural number $n\in\mathbb{N}$, let $\underline{n}$ denote the finite set $$\underline{n}:=\{1,2,\ldots,n\}.$$
A permutation $\sigma\in Aut(\underline{n})$ can be uniquely written (up to order) ...

**0**

votes

**1**answer

130 views

### Gauss Sums over “semisimple spherical tensor category”?

I read on the arXiv the following:
Let $\mathcal{\mathbf{C}}$ be a semisimple spherical tensor category with simple unit and let
$\mathbf{\Gamma}$ be the set of isomorphism classes of simple ...

**2**

votes

**0**answers

95 views

### Is the following a sufficient condition for being a primal algebra?

I have a question regarding universal algebra and, in particular, primal algebras:
Suppose that A is a finite simple algebra with no proper subalgebra, no automorphism except the identity map, with a ...

**1**

vote

**1**answer

199 views

### Inducing a Monoidal Structure using an Equivalence of Categories [closed]

Given an equivalence of categories $C \equiv D$, such that $C$ has a monoidal structure, is it clear that we can use the equivalence to induce a monoidal structure on $D$. Is there a standard ...

**3**

votes

**0**answers

98 views

### Stable $\infty$ categories as a 2-category

Is there a treatment in the literature of stable $\infty$ categories as a 2-category? I.e. with non invertible 2-morphisms.
Mostly I am interested in the behavior of the tensor product with respect ...

**2**

votes

**0**answers

28 views

### Dualities between varieties and quasivarieties at the finite level

Suppose one has two locally finite quasivarieties $\mathcal{V}$ and $\mathcal{W}$.
Further suppose that:
$\mathcal{V}$ is a variety.
The finite algebras $\mathcal{V}_f$ are dually equivalent to ...

**7**

votes

**1**answer

125 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 ...

**4**

votes

**2**answers

146 views

### Is there a purely module theoretic characterization of semiprimitive rings?

A ring (say unital for simplicity) is semiprimitive (or Jacobson semisimple) if its Jacobson radical is trivial, or equivalently it has faithful semisimple module. Semiprimitivity is a Morita ...

**0**

votes

**0**answers

60 views

### Monoid action on an uncountably infinite set

The action of a monoid on a finite set is equivalent to a finite state machine, however I would like a categorical way to think about an uncountably infinite state machine (a state transition ...

**5**

votes

**1**answer

302 views

### A categorical characterization of ordinal numbers

It's rather easy to notice that the operation of join of categories reproduces the ordinal sum once restricted to act on (iso classes of) well-ordered set; it's rather easy to see that $\alpha\star ...

**10**

votes

**1**answer

122 views

### Understanding the computation of the center of Tambara-Yamagami fusion categories when realized as C* categories

Recall that the Tambara-Yamagami categories are those with fusion ring $\mathbb{Z}[A \cup m]$ where $A$ is an abelian group and $m$ is a non-invertible (simple) object such that $ma = am = m$ for all ...

**15**

votes

**4**answers

796 views

### Expressing the Lebesgue integral using categories + the difficulty of describing estimates in category theory

In this question of mine in a comment to the accepted answer, someone remarked:
There are ways to express even basic things in analysis, such as the
spectral theorem or the Lebesgue integral, ...

**1**

vote

**1**answer

131 views

### How can one define the direct limit of classes?

If we have a family of classes $(\mathfrak{M}_\alpha)_{\alpha\in D}$ of $\in$-structures with $D$ being a limit ordinal or the class of ordinals, and a family ...

**1**

vote

**0**answers

220 views

### Functors with Mayer-Vietoris Sequences

Let $F$ be a contravariant functor from some category of spaces (e.g. smooth manifolds or (compact?) topological Hausdorff spaces), to Abelian groups. Assume that for any open sets $U, V \subseteq X$ ...

**2**

votes

**0**answers

79 views

### A natural simplicial object in the simplicial category (?)

In several works (es. [CS]) the study of the properties of the simplicial category $\Delta$ reveals fundamental aspects of universal properties (eg monoid generator) or basic constructions (eg ...

**2**

votes

**1**answer

90 views

### When is an Eilenberg-Moore category or Kleisli category braided monoidal? When semisimple?

I have a braided monoidal, semisimple linear category $\mathcal{C}$. (Imagine representations of a semisimple quasitriangular Hopf algebra.) I also have a monad $(T,\mu,\eta)$ on it, however, $T$ is ...

**4**

votes

**2**answers

170 views

### Universal property of gluing [collage, cograph] of dg-categories

In some recent works, such as this one (3.2, page 15), a definition of "gluing of dg-categories along a dg-bimodule" is given. It is obviously the analogue of the notion of collage (or cograph) of a ...

**0**

votes

**1**answer

210 views

### What is the delooping of a looping?

What is $\mathbf{B}\Omega A$, where $A$ is a pointed object of an $(\infty,1)$ category with point $*\to A$, $\Omega A$ is the loop space of $A$, and $\mathbf{B}X$ is the delooping of $X$?
The ...

**3**

votes

**1**answer

193 views

### Postnikov towers in bounded t-structures

If $\mathcal{H}$ is the heart of a bounded t-structure in a triangulated category $\mathcal{T}$, then for every object $E$ in $\mathcal{T}$ there exists a finite sequence of integers ...

**1**

vote

**0**answers

98 views

### Flat and injective quasi-coherent sheaves

Let $X$ be a quasi-compact semi-separated scheme and
$$\varepsilon: 0\to A \to B \to C \to 0$$
be a short exact sequence of quasi-coherent sheaves. $\varepsilon$ is called a (categorical) pure exact ...

**6**

votes

**4**answers

1k views

### 'Category-theory'-free areas of pure math, 'category-theory'-loaded areas of applied math

To put it short: In which active research areas of (pure) mathematics no (or only minimal) knowledge in category theory is required ?
To put it long: I know almost nothing about category theory - but ...

**2**

votes

**2**answers

137 views

### When is a triangulated category uniquely determined by its generators?

Notation: Let $\mathcal T$ be a triangulated category, and let $\mathcal E$ be a full subcategory of $\mathcal T$. I write $\langle \mathcal E \rangle$ to indicate the smallest strictly full ...

**8**

votes

**1**answer

333 views

### Modern versions of Verdier's hypercovering theorem?

Let $\mathcal{C}$ be a small category equipped with a terminal object $1$ and a Grothendieck topology. (Assume $\mathcal{C}$ also has pullbacks, if it is more convenient.) The following is a ...

**0**

votes

**0**answers

85 views

### Path objects in projective model structure

I want to know how path objects look like in the presheaf category $[\mathcal{B}(\mathbb{Z}/2\mathbb{Z})^{\text{op}}, \text{Gpd}]$. Note that this category is just groupoids equipped with involutions. ...

**4**

votes

**2**answers

370 views

### Isomorphisms and higher homotopy

It is well known that a simply connected groupoid is already contractible. Thus, isomorphisms cannot model higher homotopy. But I wonder, is this a global phenomenon (because we consider categories ...

**3**

votes

**2**answers

242 views

### When are automorphisms in categories homotopically trivial?

First, let $\mathcal{G}$ be a groupoid. Then an automorphism $\gamma\colon X\rightarrow X$ in $\mathcal{G}$ considered as a loop in the nerve of $\mathcal{G}$ is homotopic to the point $X$ if and only ...

**1**

vote

**0**answers

185 views

### Factorization systems in a triangulated/stable category

Google is of little help when questioned about "factorization systems in a triangulated category". Are there informations about them? Are there "natural" (meaning: "obviously defined") OFS/WFS in a ...

**0**

votes

**1**answer

104 views

### What is the universal property of being the maximal common subobject of two objects in a semisimple category?

Imagine a semisimple abelian category $\mathcal{C}$, for example representations of a finite group.
Take two (nonsimple) objects $X, Y$ that are subobjects of a common object $Z$, and decompose them ...

**7**

votes

**1**answer

145 views

### Generalizing indexed coproduct from $\mathrm{Set}$ to other monoidal categories

Consider the diagonal functor $\Delta_\mathcal{J} : \mathrm{Set} \to \mathrm{Set}^\mathcal{J}$, given by $\Delta_{\mathcal{J}}(X) = J \mapsto X$. This has left and right adjoints, which in the case ...

**2**

votes

**0**answers

159 views

### Banach space interpolation theory in terms of categories

I have recently learned a bit about higher category theory.
And there is a question I would like to ask. Can we enrich banach space interpolation theory by using higher category theory?
Is it ...

**5**

votes

**1**answer

150 views

### Reference request: maps between moduli spaces

I want to understand the relationship between moduli spaces as we vary the different parameters. I'll focus on the moduli space ${\mathcal M}_{g,n}(X,\beta)$ of stable maps from genus $g$ curves with ...

**5**

votes

**2**answers

349 views

### What is the co-form of Grothendieck construction?

For simplicity, let us consider only a functor out of a small category $\mathcal{C}$ to $Set$,
$$
f:\mathcal{C}\to Set,
$$
The Grothendieck construction produces a category (category of elements) ...

**0**

votes

**0**answers

97 views

### Continuity of Kan extension along the Yoneda embedding

Let $\mathcal{C}$ be a category and $h_-: \mathcal{C} \to \mathrm{Set}^{\mathcal{C}^{op}}$ be the Yoneda embedding. Let $\mathcal{A}$ be a cocomplete category and $F: \mathcal{C} \to \mathcal{A}$ a ...

**6**

votes

**2**answers

236 views

### Weakening simplicial identities

The generators $d_i, s_i$ for morphisms of the simplicial category satisfy simplicial identities:
$d_jd_i = d_id_{j−1}$ for $i < j$
$s_jd_i = d_is_{j−1}$ for $i < j$
$s_jd_i = id$ for $i = ...

**4**

votes

**2**answers

143 views

### symmetric monoidal double categories?

Let me preface this by saying that I don't know much category theory.
I am running into a situation where I have a double category and additionally there is a multiplication. Moreover, choosing ...

**7**

votes

**0**answers

150 views

### What's the Hochschild homology of the category of constructible sheaves?

Let $X$ be a manifold. Does the Hochschild homology/cohomology of the category of constructible sheaves on $X$ have a more familiar name?

**6**

votes

**2**answers

275 views

### Unexpected interaction between limits and colimits

Let $D$ be a limit-complete category. My vague question is: given two diagrams in $D$, what comparisons between them induce a morphism of their limits? I'm especially interested in the case that the ...

**11**

votes

**0**answers

179 views

### Is there a quantum group or loop group description of a braided monoidal 2-category giving Khovanov homology?

Recall that there are (at least) two ways to describe the modular tensor category that $3$-dimensional Chern-Simons (with gauge group $G$ and level $k$) assigns to a circle: one involving ...

**9**

votes

**1**answer

575 views

### I think I have a category enriched in $(\infty,n-1)$-categories. Is it an $(\infty,n)$-category?

I have recently been thinking about some mathematical gadgetry that should together combine into an $(\infty,n)$-category (actually, an $(n,n)$-category) for $n = 4$. I don't know what axioms I need ...

**6**

votes

**1**answer

219 views

### $\Omega$ and $B$ as adjoints between symmetric monoidal $(\infty,n)$- and $(\infty,n-1)$-categories

Given a symmetric monoidal $(\infty,n)$-category $\mathcal{D}$, one obtains a symmetric monoidal $(\infty,n-1)$-category $\Omega \mathcal{D}$ by taking $\Omega \mathcal{D}= ...

**2**

votes

**2**answers

266 views

### Morita equivalence via Kan extension

Are there necessary/sufficient conditions for a functor $f\colon \mathcal C\to \widehat{\mathcal A}$ to induce an equivalence $\text{Lan}_yf=F\colon \widehat{\mathcal C}\leftrightarrows ...

**4**

votes

**1**answer

254 views

### Universal property of module categories over monads

Let $T$ be a monad on a cocomplete category $\mathcal{C}$. Let's assume that $T$ preserves reflexive coequalizers (or something weaker?). Then the category of $T$-modules $\mathsf{Mod}(T)$ is ...

**2**

votes

**1**answer

200 views

### Follow up: Is continuity preserved when taking comma object?

Here, I asked wether taking lax pullback preserves continuity, but got no precise answer.
However, I have found this recent article by Riehl and Verity which proves something very similar, but I ...