**2**

votes

**2**answers

220 views

### Example(s) of monoidal symmetric closed category with NNO without infinite coproducts?

The question is in the title, here is my motivation:
$\require{AMScd}$Let $(\mathcal C,\otimes,I)$ be a monoidal symmetric closed category. Then, the tensor product commutes with colimits, and if ...

**3**

votes

**1**answer

136 views

### Infinite Dimensional Weil Restriction and Ind-scheme

I'm studying the Weil Restriction $\mbox{Res}_{k/\textbf{F}_p}$ where $\textbf{F}_p$ is the field with $p$ elements and $k$ is a perfect field over $\textbf{F}_p$, not necessarily finite.
In this ...

**9**

votes

**1**answer

254 views

### Verdier localization for stable $\infty$-categories

Verdier localization is one of the more intuitive ways to localize a triangulated category, "killing" a suitable class of objects via a functor which is universal with respect to this property.
I ...

**4**

votes

**1**answer

357 views

### Proof without using Yoneda's lemma?

Let $\mathscr{T}$ be atriangulated category.
The third axiom for triangulated categories, namely,
if in the diagram
$$\begin{array} 0X ...

**2**

votes

**4**answers

684 views

### Are there natural examples of non-symmetric Frobenius algebras?

Symmetric Frobenius algebras arise everywhere, but the non-symmetric variety seem difficult to come by. Are there any natural examples/constructions that produce non-symmetric Frobenius algebras in ...

**0**

votes

**0**answers

95 views

### How to prove a Lefschetzesque principle for relative homology and absolute homology

Here is my conjecture:
If H is some homology theory from some abelian category A to some abelian category B, E is the class of all epimorphism in A, E' is some closed class of epimorphisms in A and ...

**2**

votes

**0**answers

117 views

### Which reflexive coequalizer diagrams are projectively cofibrant?

Consider the walking reflexive pair category W,
which consists of two objects 0 and 1 and three generating
morphisms f: 0→1, g: 0→1, and h: 1→0
satisfying the relation fh=gh=id₁.
Consider the ...

**14**

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 ...

**3**

votes

**1**answer

286 views

### How to define a generating subset for algebra in a category?

As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry here). What I would like to know is if the notion of ...

**3**

votes

**1**answer

375 views

### building a product of two categories [closed]

MacLane, as well as probably any other category book, does not hesitate to define a product of two categories as a category consisting of pairs of objects, etc.
Now my question is: what law of nature ...

**4**

votes

**1**answer

195 views

### How does associativity get twisted by elements of $H^3(G)$?

In Braided Monoidal Categories by Joyal and Street, §6 a monoidal category is $V =T(G,M,h)$ built using a recipe:
objects are are elements of $G$ ✓
$V_0(x,y) = M$ if $( x=y)$ or else ...

**6**

votes

**0**answers

141 views

### Picard-Brauer exact sequence for infinity categories

This question may be very naive, or the answer may be well-known. In any case, a good amount of googling did not bring up anything useful (maybe I'm using the wrong words?).
If $f:A\to B$ is a ...

**8**

votes

**3**answers

783 views

### Grothendieck's Homotopy Hypothesis - Applications and Generalizations

Grothendieck's homotopy hypothesis, is, as the $n$lab states:
Theorem: There is an equivalence of $(∞,1)$-categories $(\Pi⊣|−|): \mathbf{Top} \simeq \mathbf{\infty Grpd}$.
What are the ...

**3**

votes

**1**answer

118 views

### Khovanov $sl_2$ homology of a connected sum of some torus knots

Let $T_{p,q}$ be the (p,q) torus knot. Could anybody possibly compute either unreduced or reduced Khovanov $\mathfrak{sl}(2)$ homology of the connected sum $T_{2,3} \sharp T_{3,4}$ of the (2,3) and ...

**5**

votes

**0**answers

95 views

### Non-degenerate limits of topoi

Let $\mathcal{E}$ be an elementary topos with natural numbers object $\mathbb{N}$, and let $\mathbb{C}$ be a cofiltered internal category in $\mathcal{E}$. Suppose we have a diagram of shape ...

**13**

votes

**1**answer

325 views

### Unobstructedness of braided deformations of symmetric monoidal categories in higher category theory

Let $k$ be a field of characteristic zero, and $\mathcal{C}$ be a $k$-linear additive symmetric monoidal category. A braided deformation of $\mathcal{C}$ over a local artin ring $R$ with residue ...

**0**

votes

**0**answers

83 views

### Merging / combining categories

Given a category X (that will be used as an underlying category) and collection of categories C(i) for i in I with a faithful functor from each C(i) to X, a category C is called the a fibered product ...

**1**

vote

**0**answers

82 views

### A factorization system on ${\rm Ch}(R)$

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

**3**

votes

**1**answer

229 views

### Examples of functors $\mathbf{Set} \to \mathbf{Set}$ which are not analytic

Let $\mathbb{B}$ denote the groupoid of finite sets and bijections.
A functor $F : \mathbf{Set} \to \mathbf{Set}$ is analytic if it is the left Kan extension of some functor $G : \mathbb{B} \to ...

**3**

votes

**0**answers

75 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 ...

**1**

vote

**0**answers

63 views

### Adjoint action of semi-direct product

Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions
$\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...

**1**

vote

**0**answers

151 views

### Coarse moduli spaces and rational points [closed]

Let $K$ be a field (not necessarily algraically closed). Let $\mathcal{F}$ be a contravariant functor from the category of schemes over $K$ to sets and $M$ be a coase moduli space for the functor. So, ...

**90**

votes

**9**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

367 views

### An exact sequence which does not split

Let $X$ and $Y$ be indecomposable modules over a finite dimensional algebra and let $f \colon X \to Y$ be a non-zero morphism which is neither a monomorphism nor an epimorphism.
Suppose that it is ...

**3**

votes

**1**answer

131 views

### Pullbacks of $C^*$-algebras

I am reading the paper of Pedersen: "Pullback and Pushout Constructions in C^*-Algebra Theory". I try to work out the arguments from Proposition $3.1$ of his paper (you can find this article in the ...

**7**

votes

**1**answer

186 views

### Localizing 2-categories about a single morphism

This question is a straight-up reference request, but of course I will be grateful for an answer in the event that no references are readily available. Consider a strict $2$-category $\mathbf{C}$ and ...

**10**

votes

**0**answers

223 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 ...

**2**

votes

**0**answers

134 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

215 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

69 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 ...

**10**

votes

**1**answer

268 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

353 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

346 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

363 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

145 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

104 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

210 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 ...

**4**

votes

**0**answers

113 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

30 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

132 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

158 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

66 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

330 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 ...

**11**

votes

**1**answer

140 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 ...

**16**

votes

**4**answers

859 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

141 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

231 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

83 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

124 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

192 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 ...