# 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.

**3**

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23 views

### Induction principle from naturality

The Church encoding of the sum type $A + B$ goes like that:
$$\prod_{X:\mathsf{Set}_{\mathcal{U}}} (A\to X)\to (B\to X) \to X$$
But it lacks an induction principle.
According to this blog article ...

**8**

votes

**2**answers

251 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 ...

**3**

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**0**answers

77 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

208 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

203 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**

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**2**answers

220 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

270 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

116 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}^\...

**4**

votes

**0**answers

67 views

### Natural transformation of $A_\infty$-functors lifted from homology

Suppose you have two $A_\infty$-functors $\mathcal{F,G}:\mathcal{A}\longrightarrow \mathcal{B}$ which descend to $F,G:A \longrightarrow B$ in homology (here $A=H^0(\mathcal{A})$ and same for $\mathcal{...

**3**

votes

**1**answer

90 views

### Monoidalness of a model category can be checked on generators

If $C$ is a cofibrantly generated model category which is also monoidal biclosed, then to check that $C$ is a monoidal model category, it suffices to check that the Leibniz product of generating ...

**5**

votes

**1**answer

350 views

### Understanding the proof of Proposition 1.2.13.8 in Lurie's Higher Topos Theory

In the first chapter of Lurie's HTT, Proposition 1.2.13.8 shows that the functors out of over (infinity)-categories reflect infinity-colimits.
For the life of me I cannot follow the proof.
Can ...

**40**

votes

**2**answers

1k views

### Ultrafilters as a double dual

Given a set $X$, let $\beta X$ denote the set of ultrafilters. The following theorems are known:
$X$ canonically embeds into $\beta X$ (by taking principal ultrafilters);
If $X$ is finite, then there ...

**6**

votes

**2**answers

178 views

### When is a fold monomorphic/epimorphic

Given a functor $F : \mathcal C \to \mathcal C$ with initial algebra $\alpha : FA \to A$, and another algebra $\xi : FX \to X$, we obtain a unique morphism $\mathsf{fold}~\xi : A \to X$ such that $\...

**8**

votes

**2**answers

236 views

### Localization, model categories, right transfer

Suppose that we have a locally presentable category $M$ and $N$ is a locally presentable full subcategory of $M$. Both categories are complete and cocomplete. Lets suppose that we have an adjunction $...

**12**

votes

**2**answers

454 views

### Why is Kan's $Ex^\infty$ functor useful?

I've always heard that Kan's $Ex^\infty$ functor has important theoretical applications, but the only one I know is to show that the Kan-Quillen model structure is right proper. What else is it useful ...