Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,395
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Ordered logic is the internal language of which class of categories?
Wikipedia says:
The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system.
"A Fibrational Framework for Substructural and Modal ...
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Inducing a model structure using a cosimplicial object
In a recent paper, Hiroshi Kihara induced a model structure on the category of diffeological spaces. He generates the classes of fibrations, cofibrations, and weak equivalences by constructing a ...
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Is the tensor product of symmetric pseudomonoids their coproduct?
The category of commutative monoid objects in a symmetric monoidal category is cocartesian, with their tensor product serving as their coproduct. This sort of result seems to date back to here:
...
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Prove that a Boolean two-valued topos in which supports split is well-pointed
In Lawvere and Rosebrugh's Sets for Mathematics, they write
It is a theorem [MM92] that a topos is well-pointed if and only if it is Boolean, two-valued, and supports split.
[MM92] is a reference to ...
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Lie monoids as monoids internal to the category of smooth manifolds?
This question can be thought as a complement to this one.
Lie groups can be defined as groups internal to the category of smooth manifolds. Lie monoids, however, as a particular case of Lie semigroups,...
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Locating the typed version of Hoàng Xuân Sính's thesis on Gr-categories
Hoàng Xuân Sính was a Vietnamese student of Grothendieck who defended her thesis on Gr-categories (now called weak 2-groups). The thesis, handwritten and in French, can be found at
https://pnp....
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Notions of Lie 2-groupoids
The term Lie $2$-groupoid is used in the literature in more than one context. Some examples are given below:
Ginot and Stiénon's paper $G$-gerbes, principal $2$-group bundles and characteristic ...
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Almost transferred model structures
Let $F : \mathcal{C} \leftrightarrows \mathcal{D} : U$ be a Quillen adjunction between cofibrantly generated model categories. The model structure on $\mathcal{D}$ is called transferred if $U$ ...
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The projective functor $\mathbb{P}^n: \operatorname{CRing} \to \operatorname{Set}$ is not representable: categorical argument
Using a "geometrical" argument of dimension, like the one here, one can show that the projective space is not affine.
I am interested in showing that, but using a categorical argument, i.e. I want ...
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Beck-Chevalley condition on pushouts
Let $C$ be a regular category with pushouts and $S(X)$ is the lattice of subjects of $X$. For every arrow $f\colon X\to A$, pulling back along $f$ gives a map $f^*\colon S(A)\to S(X)$ which has a left ...
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When a localization of a category is (non-)reflective?
Let $S$ be a class of maps of a category $\mathcal{C}$. The localization of $\mathcal{C}$ at $S$ is reflective when the localization functor $\mathcal{C} \to \mathcal{C}[S^{-1}]$ has a fully faithful ...
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Bicategory of bimodules over internal monoids
In a monoidal category $(C,\otimes)$ one can consider internal monoids and bimodules over these monoids (provided some additional requirements like existence of coequalizers hold). These monoids ...
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Tietze transformations for sites of toposes
I have read that people think of a site as a presentation of the corresponding sheaf topos. For instance, on page 7 of this text by Caramello: as Grothendieck observed himself, a site of definition ...
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Generalised Hodge Conjecture
Further to my question,
A Naive Question on Mixed Motives and Mixed Hodge Structures
that has received very good replies and suggestions, and I really appreciate it. I am going to ask a question on ...
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Is there an intrinsic definition of weak equivalence in Cat or RelCat?
It's known that Cat with the Thomason model structure serves as a model for $\infty\mathrm{Grpd}$, and that RelCat has a corresponding model structure that serves as a model for $\infty\mathrm{Cat}$. (...
user13113
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How many elementary embeddings can there be?
If $T$ is a complete first-order theory and $\kappa$ is a cardinal, let $\mathrm{Mod}_\kappa(T)$ be (a skeleton of) the category of $\kappa$-small models of $T$ (i.e. of cardinality $<\kappa$), ...
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Pushouts of injective monoid homomorphisms
Given a pushout square in the category of monoids
$$\begin{array}{ccc}A & \rightarrow & M \\ \downarrow && \downarrow \\ N & \rightarrow & P\end{array}$$such that $A \to M$ and ...
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Definition of dense functors
Definition. A functor $F:\mathsf C\rightarrow \mathsf D$ is dense if every $D\in \mathsf D$ is the vertex of the following colimit $$\varinjlim \left(F\downarrow D\rightarrow\mathsf C\rightarrow \...
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When do powers and ends in functor categories act pointwise?
$\newcommand{\C}{\mathcal C}\newcommand{\I}{\mathcal I}\newcommand{\D}{\mathcal D}\newcommand{\J}{\mathcal J}$Let $(\C, \otimes, I, \multimap)$ be a complete closed monoidal category and $\I$ a small ...
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How (if at all) does category theory deal with situations where the usual notion of isomorphism isn't right?
This question will potentially rub some people the wrong way; I can't do much about this, except state right here at the outset that this question is motivated by a genuine desire to understand, and ...
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For a simplicial set $X$, is the category of non-degenerate simplices of $X$ a full subcategory of the category of simplices of $X$?
I'm slightly confused, but I think you can help me. Let $X$ be a simplicial set. The category of simplices of $X$ and its subcategory of non-degenerate simplices are defined at http://ncatlab.org/nlab/...
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Kan extensions in concrete 2-categories
Kan extensions make sense in any 2-category. I am interested in Kan extensions in "concrete" 2-categories consisting of actual categories with some sort of structure (e.g., finite products, finite ...
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A More Advanced Version of Aluffi's Chapter 0
This is a crosspost of this question from MSE.
Paulo Aluffi's Book, Algebra, Chapter 0 aims to teach basic algebra from a categorical viewpoint. The first chapters of the book, however, introduce ...
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What do algebraic theories with strictly terminal trivial models look like?
By algebraic theory I mean one in the sense of Lawvere, i.e. a collection of finitary operations, including projections, together with a multi-composition satisfying the obvious axioms. (I believe ...
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Finitely presentable objects in functor categories
Given a locally finitely presentable category $\mathcal{C}$ it is well-known that every functor category $[\mathcal{A},\mathcal{C}]$ (where $\mathcal{A}$ is a small category) is also locally finitely ...
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Two questions about commutative theories
Let $\mathcal{T}$ be a commutative algebraic theory (for example sets, abelian groups, commutative monoids, but not groups etc.). References include the nlab and Borceux' Handbook of Categorical ...
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K-good trees and K-compactness of colimits over K-small downwards-closed subposets (500 point bounty if answered by Midnight EST))
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 ...
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For which rings does a projectivization of modules exist?
Let $R$ be a ring. Consider the inclusion functor from the category of finitely generated, projective $R$-modules to the category of all finitely generated $R$-modules. For which rings does it have a ...
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Can every weighted colimit in a $\mathbf{Pos}$-enriched category be rephrased as a conical colimit?
For ordinary category theory, we have the following fact.
A weighted colimit of a functor can always be equivalently expressed as a colimit of a different functor.
Specifically, the weighted colimit ...
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Comonoid homomorphisms in the bicategory of profunctors
Cartesian bicategories axiomatize the intuitively evident but mathematically elusive "cartesian" product on bicategories such as Rel, Span, and Prof. An important concept for cartesian ...
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The change-of-monoid adjunction between categories of modules induced by a morphism of monoids
Let $\mathcal{M}$ be a cocomplete closed symmetric monoidal category. Let $A, B$ be monoids in $\mathcal{M}$ and $f: A \rightarrow B$ be a morphism of monoids. The morphism $f$ induces the extension ...
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From the *usual* nerve of topological categories to $\infty$-categories
It is standard from work of Joyal and Lurie that there is a Quillen equivalence between the model category of simplicially enriched categories $Cat_\Delta$ and $\mathcal{S}\text{et}_\Delta$ with the ...
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Derived functor of functor tensor product
Suppose $\mathcal{A}$ is a Grothendieck abelian category with enough projectives, then $\mathcal{A}$ is tensored and cotensored over $\mathrm{Ab}$ with $\mathbb{Z}^{\oplus S}\otimes X\cong \bigoplus_S ...
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Category with few endofunctors?
Let $\mathcal C$ be a category whose skeleton has $\lambda$-many objects and $\kappa$-many morphisms. Then the skeleton of the endofunctor category $\mathcal C^{\mathcal C}$ has at most $\kappa^{3 \...
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Is there any Lie groupoid structure on $Hom(\mathcal{G}, \mathcal{H})$ where $\mathcal{G}$ and $\mathcal{H}$ are Lie groupoids?
We know that in general, there is no smooth manifold structure on $Hom(X,\, Y)$ where $X$ and $Y$ are smooth manifolds, but under certain nice conditions (see https://ncatlab.org/nlab/show/manifold+...
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Free extension of algebra for an operad
I fix $C$ a symmetric monoidal model category (with a cofibrant unit if it helps). I'm assuming that it is closed, or at least that the tensor product commutes to colimits in each variable.
If $X$ is ...
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What is the correct notion of morphism between statistical manifolds?
Given two statistical manifolds, is there a notion of "isomorphic"? What are morphisms?
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Any exact faithful functor is represented by a unique projective generator
In the book 'Tensor Categories' by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik on page 10 it says:
'Conversely, it is well known (and easy to show) that any exact faithful functor ...
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A semicartesian monoidal category with diagonals is cartesian: proof?
The nLab states that a semicartesian monoidal category equipped with natural transformations $\delta_x : x \to x \otimes x$ such that $\pi_1 \circ \delta_x = 1_x$ and $\pi_2 \circ \delta_x = 1_x$ (...
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Proposition in HTT on cofibrations of categories
Proposition A.3.3.9. in Higher Topos Theory is as follows:
Let $S$ be an excellent model category and let $f:C\rightarrow C'$ be a cofibration of small $S$-enriched categories. Then (1) for every ...
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If $\mathcal C$ has amalgamation, does $Ind(\mathcal C)$ have amalgamation?
Recall that a category $\mathcal C$ has amalgamation if every span admits a cocone. If $\mathcal C$ has amalgamation, then does $Ind(\mathcal C)$ have amalgamation?
The "obvious way to show this&...
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Are sifted (2,1)-colimits of fully faithful functors again fully faithful? (And a de-categorified variant)
1) Suppose that I have a sifted diagram of categories $\mathcal{C}_i$, another of the same shape $\mathcal{D}_i$, and that I have a system $F_i:\mathcal{C}_i\to\mathcal{D}_i$ commuting with the ...
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What is the name for a set endowed with a Lipschitz structure?
I am interested in the standard (or widely accepted) name for a mathematical structure, which is intermediate between the structures of a metric space and a topological space. I have in mind the ...
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Internal hom in $(\infty,2)$-categories
Let $X,Y$ be two $(\infty,2)$-categories, viewed as two fibrant objects in $\mathrm{Fun}(\Delta^{op},\mathrm{Set}_\Delta)$ with the complete Segal model structure (one uses the Joyal model structure ...
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Morphisms of principal bundles with different structure groups and associated bundles
Consider a pair of principal bundles $P \to M$ and $P' \to M'$ with groups $G$ and $G'$, respectively. A morphism from $P$ to $P'$ is a pair $(\Phi, \phi)$ where $\phi: G \to G'$ is a Lie group ...
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Equivalent definitions of Cartesian Fibrations between Quasi-Categories
In the paper by Verity and Riehl "Fibrations and Yoneda lemma in an ${\infty}$-cosmos" (https://arxiv.org/abs/1506.05500), they prove a Yoneda lemma that holds in any ${\infty}$-cosmos (see Corollary ...
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Do copowers commute with k-linear functors?
Fix a field $k$ and suppose $\mathcal{C}$ and $\mathcal{D}$ are $k$-linear additive categories and are enriched over the category $\mathcal{V}$ of finite-dimensional $k$-vector spaces. So we have ...
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The naive approach to deriving profunctors - What's wrong with it?
Let $(\mathcal{C,W})$ be relative category (equipped with a wide subcategory of weak equivalences satisfying 2 out of 3 property). Consider a profunctor $F: \mathcal{C \times D^{op}} \to \mathsf{Set}$ ...
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Lemma 2 from Beilinson's "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", intuition?
This is a followup to here.
Consider Lemma 2 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows.
Lemma 2. For any pair $i$, $j$ such that ...
user61522
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Generalizing indexed coproduct from $\mathrm{Set}$ to other monoidal categories
Consider the diagonal functor $\Delta_\mathcal{J} : \mathrm{Set} \to \mathrm{Set}^\mathcal{J}$, given by $\Delta_{\mathcal{J}}(X) = J \mapsto X$. This has left and right adjoints, which in the case ...
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