# Questions tagged [locally-presentable-categories]

For questions about locally presentable categories.

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### Why injectivity class doesn't imply weakly reflective

In the snippet below (taken from Rosicky, Adamek: On injectivity in locally presentable categories) in the context of locally presentable categories it seems that injectivity class doesn't imply ...
43 views

### Grothendieck categories are locally presentable

The way I've read it, many fundamental facts about Grothendieck categories, such as completeness, are special cases of those about locally presentable categories (see, for example, this answer). ...
90 views

### Why do small objects have to preverse colimits from directed sets, rather than colimits from semilattices?

Let $\kappa$ be a regular cardinal. Let $\mathcal C$ be a category. Let $A : \mathcal C$ be an object of $\mathcal C$. As best I can tell, we say that $A$ is a small object of $\mathcal C$ if, for all ...
40 views

### $\omega$-pure subgraph in $\mathbf{Grph}$

Here on the page 7: I do not follow how $B$ looks like. It is written there: $B$ is a strong subgraph of $A$ over all nodes distinct from $x_k$ for $k\geq 1$". So $B$ is only $x_0$ as we have ...
50 views

### Essential smallness of finitely presentable subcategory

Consider the presheaf category $\left[C, \mathbf{Set}\right]$ where $C$ is small. I have read that this is a locally finitely presentable category. This makes sense to me except one detail: Why is ...
56 views

### Which limit sketches produce Grothendieck toposes?

A limit sketch $\mathcal S=(\mathcal A,L)$ consists of a small category $\mathcal{A}$, together with a set $L$ of cones in $\mathcal A$. A model (in the category of sets) of a limit sketch is a ...
34 views

### directed colimit in $\mathsf{Set}$

In my previous question here, how can I prove that $FC$ doesn't idenity too much from the fact that $F$ is bounded, i.e. by this fact: for every $X$ and $x∈FX$ there is a finite $Y$ and $i:Y→X$ ...
31 views

### Is the 2-Category of Groupoids Locally Presentable?

I am wondering if the 2-Category of groupoids is Locally Presentable? Locally presentable means the category is accessible and co-complete. Edit: It has been pointed out that the category of ...
50 views

### Reflections in locally presentable categories-unclear step in the proof

Here in the paper by Rosicky Adamek, Reflections in locally presentable categories on the page 90 in the proof theorem on the page 89, I do not follow the 3rd line there: ... and hence is reflective ...
43 views

### W-split coequalizers

The following snippet is from Adamek, Rosicky:Algebra and local presentability,how algebraic are. It is unclear to me the end of Example 5.1: Since $e$ is the coequalizer of $\bar{u}_1,\bar{u}_2$ in ...
70 views

### Is the category of chain complexes over an ring $R$ a locally presentable category?

I wonder if the category of chain complexes over an ring $R$ is a locally presentable category. I am trying proving that this category is combinatorial, I have seen some reference for the cofibrantly ...
64 views

### Reflections in locally presentable categories

In this paper, -4th line in the first paragraph on the (first) page 89, then each full subcategory of $\cal H$ closed under limits ... should or should not the word reflective be present: then each ...
49 views

What is the difference of $\text{CAT}^{\mathbb T}$ from $\text{VAR}$ in this paper sketched below?