# Questions tagged [accessible-categories]

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### Relation between Ind-completion and “additive”-ind-completion

Suppose that $\mathcal{C}$ is a skeletally small additive category. To enlarge $\mathcal{C}$ and produce a bigger category whose "small" objects can be identified with those in $\mathcal{C}$,...
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### Why are compactly generated $\infty$-categories closed under limits in $\operatorname{Cat}_{\infty}$?

In Proposition 5.5.7.6 of Lurie's Higher Topos Theory, Lurie states that the $\infty$-category $\operatorname{Pr}^R_{\omega}$ of compactly generated $\infty$-categories and filtered-colimit-preserving ...
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### In a locally presentable category, is every object (a retract of) the colimit of a chain of smaller objects?

Let $\mathcal C$ be an accessible category. For $C \in \mathcal C$, define the presentability rank $rk(C)$ of $C$ to be the minimal regular $\kappa$ such that $C$ is $\kappa$-presentable. Following ...
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### If $\mathcal C$ is a $\kappa$-accessible $\infty$-category, then is $Mor \mathcal C$ $\kappa$-accessible?

If $\mathcal C$ is a $\kappa$-accessible 1-category, then the category of morphisms $Mor \mathcal C$ is a $\kappa$-accessible 1-category, with the $\kappa$-presentable objects being those morphisms ...
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### Almost combinatorial accessible model categories

Theorem: Assume VP. Let $\mathcal{M}$ be an accessible model category such that there exists a set of generating cofibrations $I$ and such that all objects are fibrant. Then it is combinatorial. ...
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### Accessible 2-category of presheaves

Let $A$ be a locally small category and let $\mathbf{Cat}$ be the 2-category of small categories, functors and natural transformations and let $Ps(A)$ be the 2-category of presheaves (the objects are ...
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### How do you prove that the category of weak equivalences of sSet is accessible?

I am trying to prove that the category of simplicial sets is a combinatorial model category by using Proposition A.2.6.15 of Lurie's book, and this requires proving that the weak equivalences are ...
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### Characterizing the left / right classes of (weak) factorization systems in locally presentable categories

Let $\mathcal M \subseteq Mor(\mathcal C)$ be a class of morphisms in a locally presentable category. It's well-known that $\mathcal M$ is the left half of an accessible orthogonal factorization ...
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### presentability rank of categories of coalgebras

The following theorem is relatively classical: Theorem: Given an accessible endofunctor, (co)pointed endofunctor or (co)monad $T$ on a locally presentable category $C$, then the category of $T$-(co)...
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### Can the dual of a finitely-accessible category be accessible?

What is an example of an accessible category $\mathcal C$ which is not essentially small, such that $\mathcal C^{op}$ is finitely-accessible? More generally, what is an example of an accessible ...
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### Adjusting the definition of a well-powered category to category theory with universes: size issues

Wikipedia and Borceux (Handbook of Categorical Algebra, Part I) give the following definitions of subobjects and well-powered categories: A subobject of an object $X$ of a category $\mathsf{C}$ is an ...
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### Non-small objects in categories

An object $c$ in a category is called small, if there exists some regular cardinal $\kappa$ such that $Hom(c,-)$ preserves $\kappa$-filtered colimits. Is there an example of a (locally small) ...
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### Bousfield localization of a left proper accessible model category

What is known about the Bousfield localization of a left proper accessible model category by a set of maps ? (I mean not combinatorial which is already known)
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### 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 ...
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### What are the reflective subcategories of the category of presentable categories?

I am actually interested in the $\infty$-categorical case, but the same question is meaningful in the $1$-categorical situation as well. A nice property of presentable $\infty$-categories is that if ...
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### Can I check the accessibility of a functor on directed colimits of presentable objects?

Let $F: \mathcal{K} \to \mathcal{C}$ be a functor between $\lambda$-accessible categories, you can assume $\mathcal{C}$ to be Set if needed. Is it true that $F$ is $\lambda$-accessible if and only if ...
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### Saturated classes, generation by a set and pullbacks of categories

Assume that we have a pullback square $$\begin{array}{ccc} A & \rightarrow & B \\ \downarrow & & \downarrow \\ C & \rightarrow & D \\ \end{array}$$ with all functors ...
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### What was Burroni's sketch for topological spaces?

In a 1981 talk, René Guitart cites Albert Burroni as having given "A first interesting example of a mixed sketch...for the category of topological spaces" in 1970. This was apparently done in Burroni'...
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### About the Yoneda objects of a locally presentable category

This question is a follow-up of Extending functors defined on dense subcategories. Let $\mathcal{K}$ be a locally presentable category. An object $X$ of $\mathcal{K}$ is called a Yoneda object if ...
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### Intuition behind $\lambda$-pure subobjects

While reading about accessible categories in Locally Presentable and Accessible Categories I came accross the notion of $\lambda$-pure subobjects, which seem to be important while characterising ...
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### When are the categories of algebras over props (co)complete?

Suppose P is a (colored) prop in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable? It seems that one can relatively easy ...
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My question is ultimately about the model theory of $L_{\infty, \omega}$, but it is more convenient to phrase it in terms of category theory. Suppose I have finitely accessible categories $\mathcal{C},... 0answers 472 views ### Orthogonality relations and accessibility? Suppose I have a pair of locally presentable categories connected by an accessible functor. Then the preimage of an accessible subcategory of the codomain is an accessible subcategory of the domain. ... 1answer 325 views ### Equivalence of the two definitions of k-compactness/k-presentability We say that an object$X$of a category$C$is$\kappa$-compact (also$\kappa$-presentable and$\kappa$-accessible) for a cardinal$\kappa$if$h^X(\cdot):=Hom(X,\cdot)$commutes with all$\kappa$-... 1answer 620 views ### The poset of k-small downward-closed subposets of a poset P is k-filtered when k is a regular cardinal? Let$\kappa$be a cardinal, and let$P$be a poset. Let$\mathcal{P}_\kappa(P)$denote the poset of$\kappa$-small subposets of$P$and let$\mathcal{P}_\kappa^\downarrow(P)\subseteq\mathcal{P}_\...
Question: Let $D:A\to (X\downarrow C)$ be a $\kappa$-good $S$-tree rooted at $X$ for a collection of morphisms $S$ in $C$, where $\kappa$ is a fixed uncountable regular cardinal. Then according to ...