**3**

votes

**1**answer

82 views

### About a closed strucure on profunctors

Let $Prof$ the bicategory with profunctors (on small categories), arrows are like $D: \mathscr{A} \dashrightarrow \mathscr{B}$ and this means that $D: \mathscr{A}^{op} \times \mathscr{B}\to Set$.
...

**6**

votes

**0**answers

80 views

### Why do the model structures on dg-algebras and on dg-categories are not compatible?

First we talk about dg-algebras. According to this n-lab page, we write $dgAlg$ for the category of cochain dg-algebras in non-negative degree over a field $k$ of characteristic $0$. Write ...

**2**

votes

**1**answer

90 views

### Do the full subcategories have a simple structure in higher category theory?

Let $C \in Cat$ be an $(\infty,1)$-category.
Let $P$ be the partially ordered subset of full subcategories of $C$.
Is there a (canonical?) functor from the nerve of $P$ to $Cat$? I think the answer ...

**7**

votes

**1**answer

98 views

### Detecting positive endomaps of the formal reals

A locale is a sort of "formal topological space", which "may not have enough points to separate its open sets". For instance, there is a "locale of all real numbers that are both rational and ...

**0**

votes

**1**answer

88 views

### Adjointable Abelian Monoidal Functor

Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?

**2**

votes

**1**answer

92 views

### Does the following characterize local presentability?

Let $\mathcal C$ be a cocomplete category. Consider the following two conditions:
$\mathcal C$ is locally presentable.
The Yoneda embedding $$\mathcal C \hookrightarrow \{\text{continuous functors ...

**13**

votes

**1**answer

380 views

### The real numbers object in Sh(Top)

If $X$ is a sober topological space, the real numbers object in the topos $\mathrm{Sh}(X)$ is the sheaf of continuous real-valued functions on $X$. This is proven very explicitly in Theorem VI.8.2 of ...

**9**

votes

**1**answer

153 views

### Does every Lawvere theory arise in this way?

By a Lawvere theory, I mean a finite-product category $\mathsf{T}$ equipped with a distinguished object, such that every object of $\mathsf{T}$ can be expressed as a finite product of the ...

**6**

votes

**1**answer

146 views

### Rectifying the definition of a closed category

The definition of a closed category I'm using is here.
Suppose $V$ is a closed category and that for each object $b\in V$, $[b,-]$ has a left adjoint $- \otimes b$. The result is nearly a monoidal ...

**4**

votes

**1**answer

152 views

### An example for a construction on monads/operads?

Suppose that $C$ is a bicategory. (I only need a monoidal category, i.e. one object bicategory, but I will stick with bicategories, since theory of monads is more commonly stated in that setting). A ...

**54**

votes

**9**answers

3k views

### What are some examples of interesting uses of the theory of combinatorial species?

This is a question I've asked myself a couple of times before, but its appearance on MO is somewhat motivated by this thread, and sigfpe's comment to Pete Clark's answer.
I've often heard it claimed ...

**5**

votes

**0**answers

125 views

### “Generalized theory of polynomials” for a given commutative Lawvere Theory

I am trying to understand
Nikolai Durov's "New Approach to Arakelov Geometry" right now and it got me thinking about a particular thing.
Let $R$ be a commutative, associative ring with unit. We can ...

**2**

votes

**2**answers

152 views

### When is the adjoint to a monoidal functor monoidal?

Let $\mathcal C,\mathcal D$ be monoidal categories. Recall that a functor $F : \mathcal C \to \mathcal D$ is lax monoidal if it is equipped with maps $1_{\mathcal D} \to F(1_{\mathcal C})$ and $F(X) ...

**13**

votes

**0**answers

569 views

### Derived functors - homotopical vs homological approach

This question is a crosspost of the second part of this MSE question.
In my first course in homological algebra, derived functors were defined in terms of universal $\delta$-functors. In the text ...

**3**

votes

**0**answers

93 views

### What about “bilax” functors?

in [G] p.29, J.W Gray define the 2-comma category $[F, G]$ of two 2-functors $F: \mathcal{A}\to \mathcal{D},\ G: \mathcal{B}\to \mathcal{D}$. This definition work well also if we suppose $F$ a ...

**3**

votes

**0**answers

218 views

### Quasicategorical Construction of a Cosimplicial Map of Rognes

In John Rognes' Galois theory monograph he constructs something called the Hopf-cobar complex for a coalgebra object $H$ (in spectra) and a comodule algebra $X$. It is, intuitively, the object whose ...

**2**

votes

**1**answer

93 views

### Exponential locales and a pointless version of the compact-open topology?

TL;DR: compact-open topology for Homs of locales?
Let $\mathcal{L}$ be a full subcategory of the category $\mathcal{Loc}$ of locales.
For two locales, $A$ and $B$, is there a nice way to make an ...

**2**

votes

**1**answer

406 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} ...

**6**

votes

**1**answer

182 views

### When the restriction of a derived functor to a subcategory is the derived functor of the restriction

Let $\mathcal{D},\mathcal{E}$ be abelian categories and $\mathcal{C}$ be a Serre subcategory of $\mathcal{D}$.
Let $D(\mathcal{C}), \, D(\mathcal{D})$ denote the derived categories of ...

**3**

votes

**0**answers

157 views

### Ring epimorphisms, and epimorphism in the category of small preadditive cats

This question is related to this other question I have asked some time ago. Let $R$ and $S$ be two rings and let $\phi:R\to S$ be a ring homomorphism.
It is well-known that $\phi$ is an epimorphism ...

**0**

votes

**0**answers

49 views

### Yetter-Drinfeld modules as rigid category

I'm reading a proof of the following theorem
If $H$ is a Hopf algebra with invertible antipode then Yetter-Drinfeld modules of finite dimension form a rigid category.
In this proof we define ...

**1**

vote

**2**answers

146 views

### Completion under weighted limits/colimits

Is there any further reference besides "Basic Concepts of Enriched Categories" (Kelly) for completion under T-(weighted) limits/colimits?
(in which T is a set of weights)
Thank you in advance

**10**

votes

**2**answers

530 views

### Is there a notion of a “model category which admits left Bousfield localization?”

At a conference not too long ago I gave a talk on (left) Bousfield localization and was asked an interesting question afterwards. The question was whether I knew any examples of model categories which ...

**4**

votes

**2**answers

370 views

### How to characterize flasque sheaves in more functorial way?

The motivation to ask this question is some proposition of flasque sheaves.
Let's recall the definition of flasque sheaf:A sheaf $F$ on a topological space $X$ is flasque if for every inclusion ...

**4**

votes

**1**answer

187 views

### Properties of loop space functor from homotopy types to group objects in homotopy types

I am trying to understand some properties of categories enriched in homotopy types, and the following question has become important:
When we take the loop-space of a (connected) homotopy type, we get ...

**15**

votes

**4**answers

637 views

### Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...

**5**

votes

**2**answers

619 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

127 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 ...

**6**

votes

**0**answers

200 views

### Natural isomorphisms: what is the status now of “the Eilenberg/Mac Lane Thesis”?

I asked this some days ago over at math.se, and while the question got 10 upvotes, I didn't get too many answers. Although it is a "soft question", maybe the general issue is interesting enough to ...

**11**

votes

**2**answers

459 views

### Is there a “free abelian group of rank 1” in the category of affine group schemes?

Let's fix an algebraically closed field $k$.
The group $\mathbb Z$, as a discrete group scheme, is not affine since it's not quasi-compact. Is there an affine algebraic scheme over $k$ whose ...

**1**

vote

**0**answers

38 views

### About cylinder and path functors

Let $\mathcal{K}$ be a locally presentable category. I recall that a cylinder $C:\mathcal{K}\to \mathcal{K}$ is by definition equipped with two natural maps $\gamma_X:X\sqcup X\to CX$ and ...

**5**

votes

**1**answer

139 views

### Do models-and-homomorphisms always form an accessible category?

It's well-known that the category of models of any first-order theory $T$ form an accessible category if we take the elementary embeddings as morphisms. This is true in finitary first-order logic or ...

**4**

votes

**1**answer

260 views

### Natural transformations induce homotopies - Is this true in the “fat” world?

Let $\mathcal{C}, \mathcal{D}$ be categories internal to topological spaces (or compactly generated Hausdorff spaces, if you like) $F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ be continuous functors ...

**5**

votes

**2**answers

293 views

### Is the $\infty$-category of presentable $\infty$-categories presentable?

Let $\mathit{Pr}^L$ be the $\infty$-category of presentable $\infty$-categories and continuous functors in some universe. Is it presentable itself a larger universe?

**4**

votes

**0**answers

137 views

### Does the fat realization of simplicial spaces commute with finite limits up to homotopy?

I'm willing to work in the category of compactly generated Hausdorff spaces.
The ordinary geometric realization of a simplicial space commutes with taking pullbacks and preserves the terminal object, ...

**12**

votes

**1**answer

426 views

### Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) theme are there?

A linear operator $T:V\to V$ on a (say) vector space over a field $k$ is just a $k[T]$-module, and may be viewed as the sheaf $\mathscr F_T$ over $\mathbb A^1_k$, with fibre over $\lambda\in k$ equal ...

**10**

votes

**1**answer

582 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 ...

**2**

votes

**1**answer

168 views

### Why do we need filtered categories to index ind-objects?

I edited the question in view of several helpful replies (thanks).
When we define ind-objects in a category, we use in general filtered diagrams in a category, not just sequences $A_1 \rightarrow ...

**3**

votes

**1**answer

158 views

### “Canonical” graph structure on $\text{Hom}(G, H)$

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq [V]^2 := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, ...

**1**

vote

**1**answer

95 views

### When does a cogenerator determine a variety?

Two varieties of universal algebras are categorically equivalent iff their respective full subcategories of finitely generated free algebras are equivalent. Roughly speaking, this follows because they ...

**1**

vote

**0**answers

62 views

### Reduced products of (abelian and triangulated) categories: references?

For a filter $U$ on a set $X$ and for a family of categories $C_x$ indexed by $X$ I would like to consider the (corresponding categorical version of) reduced product of $C_x$ (for $x\in X$) with ...

**7**

votes

**2**answers

457 views

### What is descent data (of higher categories), conceptually?

First consider a scheme $X$ with an open cover $\mathcal{U}=\{U_i\}$. An object with descent data on $\mathcal{U}$ is a collection $(\mathcal{E}_i,\phi_{ij})$ where $\mathcal{E}_i$ is a ...

**0**

votes

**0**answers

111 views

### Higher algebra and terminology about 2-objects

It is well known that one way to build higher category theory is to use some induction process, where an $n$-category has as $0$-cells some $n-1$ categories, such that for two $0$-cells $\mathcal{A}$ ...

**2**

votes

**1**answer

64 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 ...

**12**

votes

**1**answer

262 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 ...

**3**

votes

**1**answer

139 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

119 views

### Does the forgetful functor from presentable $\infty$-categories to $\infty$-categories preserve filtered colimits?

Marc's answer to my previous question gives a way to compute colimits in the category of presentable $\infty$-categories and continuous functors, using the (discontinuous) right adjoints to those ...

**3**

votes

**1**answer

105 views

### What is the cokernel of a map of presentable stable $\infty$-categories?

Let $C$ and $D$ be presentable stable $\infty$-categories, and let $f:C \to D$ be a continuous functor between them. Let $0$ be the trivial stable $\infty$-category. What is the colimit of the ...

**0**

votes

**0**answers

296 views

### Set as a (strict) infinite-category?

First, let me say that I have no idea if such a post has its place here. However, I believe that the ideas I'm going to present are important. The goal of this thread is three fold:
1) trying to ...

**1**

vote

**0**answers

54 views

### Does it require Reedy fibrancy when we want the totalization to be weakly equivalent to the homotopy limit?

This question arises when I am reading the last two Chapter of Hirschhorn's "Model categories and their localizations"
In Part (2) of Theorem 19.8.4 of that book it says
If ...