# Questions tagged [ct.category-theory]

Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

**1**

vote

**0**answers

10 views

### Example of non accessible model categories

By curiosity, I would like to see an example of a model category with the underlying category locally presentable which is not accessible in this sense (and just in case: even by using Vopěnka's ...

**2**

votes

**0**answers

33 views

### Terminology: a certain semicategory with objects mor(C) (not the usual or twisted arrow category)

I’ve had recent cause to consider the following construction: given a category $\newcommand{\C}{\mathbf{C}}\C$, define a semicategory $M(\C)$, whose objects are arrows of $\C$, and where a map from $f ...

**2**

votes

**0**answers

47 views

### Does this elementary generalization (left and right sided products) of categorical product have an official name?

The question and definition at hand are related to the fact that in freshman calculus, to check pointwise continuity of $f:R \rightarrow R$, we ask if both the left and right hand limits exist, and if ...

**0**

votes

**0**answers

84 views

### Sketch of sketches, or sketch of presentations

In Sketches: Outline with References 4.3, Wells cites the result that sketches are sketchable by a finite limit sketch. I can't find the Burroni 1970a paper, and I am having a lot of trouble with Lair ...

**10**

votes

**1**answer

250 views

### Is there any references on the tensor product of presentable (1-)categories?

Is there any references on the tensor product of (locally) presentable categories ?
All I know about this is Lurie's book that deals with the $\infty$-categorical version, and a few references that ...

**1**

vote

**0**answers

54 views

### Reference request : Isomorphic stacks are given by Morita equivalent Lie groupoids

Let $\mathcal{G},\mathcal{H}$ be Lie groupoids. Let $B\mathcal{G}$ denote the stack of principal $\mathcal{G}$ bundles and $B\mathcal{H}$ denote the stack of principal $\mathcal{H}$ bundles. Then, we ...

**5**

votes

**1**answer

189 views

### Can groups be recovered as “monoids” in a bicategory?

Is there a bicategory $V$ and a definition of monoid in a bicategory so that $\text{Monoids}(V)$ is the category of groups and homomorphisms?
EDIT: For example, is there a bicategory $V$ so that ...

**7**

votes

**2**answers

368 views

### What are the advantages of simplicial model categories over non-simplicial ones?

Of course, there are general results allowing one to replace a model category with a simplicial one. But suppose I want to stay in my original non-simplicial model category (say for some reason I'm a ...

**9**

votes

**2**answers

395 views

### Examples of transfinite towers

I am looking for examples of constructions for transfinite towers $(X_{\alpha})_{\alpha}$ generated by structures $X$ where the problem of determining whether the tower $(X_{\alpha})_{\alpha}$ stops ...

**4**

votes

**0**answers

115 views

### Orthogonality and 2-filtered 2-categories

Let $C$ be a category. It is well known that $C$ is $\omega$-filtered if and only if it is weakly right orthogonal to every $A\to A^\rhd$, where $A^\rhd$ is the right cone of a finite category $A$.
...

**11**

votes

**1**answer

222 views

### Accessible functors not preserving lots of presentable objects

Let $F:\cal C\to D$ be an accessible functor between locally presentable categories. By Theorem 2.19 in Adamek-Rosicky Locally presentable and accessible categories, there exist arbitrarily large ...

**6**

votes

**1**answer

220 views

### Is the Thomason model structure the optimal realization of Grothendieck's vision?

In Pursuing Stacks, Grothendieck uses the category $Cat$ of small categories to model spaces. A recurring theme is the question of whether there is a Quillen model structure supporting this homotopy ...

**14**

votes

**2**answers

261 views

### $\mu$-presentable object as $\mu$-small colimit of $\lambda$-presentable objects

Remark 1.30 of Adámek and Rosicky, Locally Presentable and Accessible Categories claims that in any locally $\lambda$-presentable category, each $\mu$-presentable object (for $\mu\ge\lambda$) can be ...

**7**

votes

**2**answers

278 views

### For which categories of spectra is there an explicit description of the fibrant objects via lifting properties?

How explicit are the model structures for various categories of spectra?
Naive, symmetric and orthogonal spectra are obtained via left Bousfield localization of model structures with explicit ...

**8**

votes

**1**answer

122 views

### Weighted (co)limits as adjunctions

It's well known that a category $\mathcal{C}$ having (conical) limits/colimits of shape $\mathcal{D}$ is equivalent to the diagonal functor $\Delta^\mathcal{D}_\mathcal{C}:\mathcal{C}\to\mathcal{C}^\...