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

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1answer
87 views

Yetter-Drinfeld modules as rigid category

I'm reading a proof of the following theorem If $H$ is a Hopf algebra with invertible antipode then Yetter-Drinfeld modules of finite dimension form a rigid category. In this proof we define ...
6
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1answer
190 views

When the restriction of a derived functor to a subcategory is the derived functor of the restriction

Let $\mathcal{D},\mathcal{E}$ be abelian categories and $\mathcal{C}$ be a Serre subcategory of $\mathcal{D}$. Let $D(\mathcal{C}), \, D(\mathcal{D})$ denote the derived categories of ...
4
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1answer
200 views

Properties of loop space functor from homotopy types to group objects in homotopy types

I am trying to understand some properties of categories enriched in homotopy types, and the following question has become important: When we take the loop-space of a (connected) homotopy type, we get ...
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2answers
150 views

Completion under weighted limits/colimits

Is there any further reference besides "Basic Concepts of Enriched Categories" (Kelly) for completion under T-(weighted) limits/colimits? (in which T is a set of weights) Thank you in advance
3
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0answers
167 views

Ring epimorphisms, and epimorphism in the category of small preadditive cats

This question is related to this other question I have asked some time ago. Let $R$ and $S$ be two rings and let $\phi:R\to S$ be a ring homomorphism. It is well-known that $\phi$ is an epimorphism ...
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211 views

Natural isomorphisms: what is the status now of “the Eilenberg/Mac Lane Thesis”?

I asked this some days ago over at math.se, and while the question got 10 upvotes, I didn't get too many answers. Although it is a "soft question", maybe the general issue is interesting enough to ...
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0answers
41 views

About cylinder and path functors

Let $\mathcal{K}$ be a locally presentable category. I recall that a cylinder $C:\mathcal{K}\to \mathcal{K}$ is by definition equipped with two natural maps $\gamma_X:X\sqcup X\to CX$ and ...
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2answers
471 views

Is there a “free abelian group of rank 1” in the category of affine group schemes?

Let's fix an algebraically closed field $k$. The group $\mathbb Z$, as a discrete group scheme, is not affine since it's not quasi-compact. Is there an affine algebraic scheme over $k$ whose ...
5
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1answer
142 views

Do models-and-homomorphisms always form an accessible category?

It's well-known that the category of models of any first-order theory $T$ form an accessible category if we take the elementary embeddings as morphisms. This is true in finitary first-order logic or ...
4
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1answer
270 views

Natural transformations induce homotopies - Is this true in the “fat” world?

Let $\mathcal{C}, \mathcal{D}$ be categories internal to topological spaces (or compactly generated Hausdorff spaces, if you like) $F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ be continuous functors ...
4
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0answers
149 views

Does the fat realization of simplicial spaces commute with finite limits up to homotopy?

I'm willing to work in the category of compactly generated Hausdorff spaces. The ordinary geometric realization of a simplicial space commutes with taking pullbacks and preserves the terminal object, ...
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1answer
169 views

Why do we need filtered categories to index ind-objects?

I edited the question in view of several helpful replies (thanks). When we define ind-objects in a category, we use in general filtered diagrams in a category, not just sequences $A_1 \rightarrow ...
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1answer
163 views

“Canonical” graph structure on $\text{Hom}(G, H)$

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq [V]^2 := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, ...
12
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1answer
441 views

Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) theme are there?

A linear operator $T:V\to V$ on a (say) vector space over a field $k$ is just a $k[T]$-module, and may be viewed as the sheaf $\mathscr F_T$ over $\mathbb A^1_k$, with fibre over $\lambda\in k$ equal ...
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0answers
65 views

Reduced products of (abelian and triangulated) categories: references?

For a filter $U$ on a set $X$ and for a family of categories $C_x$ indexed by $X$ I would like to consider the (corresponding categorical version of) reduced product of $C_x$ (for $x\in X$) with ...
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0answers
115 views

Higher algebra and terminology about 2-objects

It is well known that one way to build higher category theory is to use some induction process, where an $n$-category has as $0$-cells some $n-1$ categories, such that for two $0$-cells $\mathcal{A}$ ...
7
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2answers
484 views

What is descent data (of higher categories), conceptually?

First consider a scheme $X$ with an open cover $\mathcal{U}=\{U_i\}$. An object with descent data on $\mathcal{U}$ is a collection $(\mathcal{E}_i,\phi_{ij})$ where $\mathcal{E}_i$ is a ...
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1answer
132 views

Does the forgetful functor from presentable $\infty$-categories to $\infty$-categories preserve filtered colimits?

Marc's answer to my previous question gives a way to compute colimits in the category of presentable $\infty$-categories and continuous functors, using the (discontinuous) right adjoints to those ...
3
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1answer
117 views

What is the cokernel of a map of presentable stable $\infty$-categories?

Let $C$ and $D$ be presentable stable $\infty$-categories, and let $f:C \to D$ be a continuous functor between them. Let $0$ be the trivial stable $\infty$-category. What is the colimit of the ...
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2answers
311 views

Is the $\infty$-category of presentable $\infty$-categories presentable?

Let $\mathit{Pr}^L$ be the $\infty$-category of presentable $\infty$-categories and continuous functors in some universe. Is it presentable itself a larger universe?
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1answer
189 views

Rectifying the definition of a closed category

The definition of a closed category I'm using is here. Suppose $V$ is a closed category and that for each object $b\in V$, $[b,-]$ has a left adjoint $- \otimes b$. The result is nearly a monoidal ...
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0answers
305 views

Set as a (strict) infinite-category?

First, let me say that I have no idea if such a post has its place here. However, I believe that the ideas I'm going to present are important. The goal of this thread is three fold: 1) trying to ...
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0answers
57 views

Does it require Reedy fibrancy when we want the totalization to be weakly equivalent to the homotopy limit?

This question arises when I am reading the last two Chapter of Hirschhorn's "Model categories and their localizations" In Part (2) of Theorem 19.8.4 of that book it says If ...
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60 views

TTF triples are recollements

The notion of recollement $$ \mathcal{A}' \stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}} ...
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0answers
82 views

$n$-recollements and perverse t-structures

A recent preprint on arXiv brought my attention on the notion of $n$-recollement (def. 2) a generalization of the notion of recollement among three abelian or triangulated categories behaving like a ...
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0answers
158 views

Functors similar to $H^i(\cdot)$

Suppose $T$ is a contravariant functor from the category of pointed topological spaces to the category of abelian groups, then we have homomorphisms $\alpha\colon T(X)\times T(Y)\to T(X\times Y)$ and ...
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3answers
457 views

What are a couple of examples of finite sized but interesting categories?

I'm studying category theory and, given that I don't have a background in topology, I'm struggling to think of some finite categories that interesting. The main one I know of is finite preorders -- I ...
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0answers
35 views

Explicit computation of a limit of a cosimplicial object

Let $\Delta$ be the simplex category. Let $T_{n}$ be the standard topological $n$ simplex, i.e. it is the set of points of $\mathbb{R}^{n}$ such that $0\leq t_{1}\leq \dots \leq t_{n}\leq 1$. Its ...
13
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2answers
586 views

Does projective imply flat?

Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care ...
3
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0answers
123 views

When there exists some “cone” of a morphism of (ind-representable) cohomological functors?

I am interested in cohomological functors from a certain small triangulated category $C$ to abelian groups. The question is: given a tranformation $F\to G$ of two functors of this sort, is it ...
2
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2answers
171 views

A conservative, non faithful functor between triangulated categories

Suppose that we have: 1) triangulated categories $C,D$, each equipped with a $t$-structure. 2) triangulated functor $F: C \to D$ which is $t$-exact. 3) $F$ reflects isomorphisms, i.e. is ...
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0answers
99 views

Do smooth manifolds create colimits for complex manifolds?

Suppose we have a diagram $D$ in the category $\textrm{Diff}_\mathbb{C}$ of complex manifolds, and suppose this diagram has a colimit $L$ after inclusion in the category $\textrm{Diff}$ of smooth ...
1
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1answer
256 views

Van Kampen colimits

nLab uses the following definition of van Kampen colimits --- a colimit in a category $\mathbb{C}$ is called van Kampen iff it is preserved by the internal indexing functor $\mathbb{C}/(-) \colon ...
6
votes
1answer
464 views

Lurie's Endomorphism Space vs. Endomorphisms

In Jacob Lurie's book Higher Algebra, for an object $M$ of a monoidal $\infty$-category $\mathcal{C}$, he constructs a category $\mathcal{C}[M]$ which can be thought of as "maps in $\mathcal{C}$ of ...
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303 views

Is there a symmetric monoidal 2-category “SuperDuperVect”?

Recall that the category $\mathrm{SuperVect}$, as a category, consists of pairs of vector spaces, thought of as formal direct sums $V \oplus W\,\Pi$, where $\Pi$ is the "odd line". (Called "$\Pi$" ...
2
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0answers
104 views

Universality of the Simplex Category

This is not a research level question, but I have not been able to find a satisfactory proof of functoriality of a certain map, and have posted this on several sites, with no luck. I wonder if anyone ...
6
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1answer
218 views

Categorical definition of infinite symmetric product

Let $(C,\otimes,I)$ be a symmetric monoidal category with coequalizers and directed colimits. Fix some object $X$ and morphism $\tau\colon I\to X.$ Using $\tau$ one can construct a sequence of ...
3
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0answers
82 views

About Quillen equivalences between Bousfield localizations

Let $\mathcal{M}$ be a locally presentable category equipped with two left proper and left determined combinatorial model structures $\mathcal{M}_1$ and $\mathcal{M}_2$. There exist two sets $S_1$ and ...
4
votes
1answer
301 views

$\Omega X$-action on spectral $X$-bundles

I can generalize the notion of a space over $X$ (where $X$ is a based and connected space) to the notion of a spectrum over $X$ by considering functors of quasicategories $X\to Mod_\mathbb{S}$. In the ...
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3answers
2k views

Who needs Replacement anyway?

The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in ...
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0answers
676 views

Derived functors - homotopical vs homological approach

This question is a crosspost of the second part of this MSE question. In my first course in homological algebra, derived functors were defined in terms of universal $\delta$-functors. In the text ...
3
votes
1answer
243 views

Determining a scheme $X$ is affine from $Qcoh(X)$

My question is a subquestion of this question. And a repost from this MSE question. The settings is the following. Let $X$ be a scheme. Assume that the adjunction in TAG01BH of the stacks project is ...
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0answers
74 views

Monadicity of profinite algebras

We can show that the category of profinite algebras, cofiltered limits of finite algebras, is monadic over Stone spaces as follows. So, I wonder if there are any other examples. In case that I was ...
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2answers
373 views

A cosmos where coproduct injections are not monic

The injections (coprojections) of a coproduct in a category are very often monomorphisms. For instance, this happens in any extensive category (essentially by definition) and also in any category ...
0
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1answer
93 views

Trying to relate the fundamental groupoid to vector bundles

Fix a topological space $X$. Now consider a functor from the fundamental groupoid of $X$ to the category $Vect$. In other words, we assign a vector space to each point of $X$, we allow ourselves to ...
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0answers
99 views

Unique extendable functions… Is there a theory?

I also made this post in MSE, but I think it may fit here as well. Motivation: Take a continuous function $f$ from a topological space $X$ to a hausdorff space $Y$. If $g$ is a continuous function ...
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0answers
324 views

3 possible tensors in 2-categories?

let $\mathcal{A}$ be a 2-category, consider: $$ \mathcal{A}(W \otimes_i A, X) \simeq_i \mathcal{C}at(W, \mathcal{A}(A, X)) \;\;\; i = 1, \: 2, \: 3. $$ where $W$ is a category, and $A$, ...
1
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1answer
297 views

Construction of Highly Structured Quotient Groups in Quasicategories

Suppose we have a map of $E_n$-spaces $X\to Y$. Then there is a highly structured action of $X$ on $Y$, $X\wedge Y\to Y\wedge Y\to Y$, using the multiplication of $Y$. As such, I believe that there ...
8
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2answers
313 views

Reference for an unbiased definition of a symmetric monoidal category

In the references I know, a symmetric monoidal category is defined as a monoidal category equipped with some additional data, so in particular the monoidal product is a functor $$ \mathcal{C}\times ...
11
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1answer
222 views

How many Fréchet manifolds are there?

Clearly the title needs clarifying. Allow me to let "how many" to mean a set larger than a skeleton of the category of Fréchet manifolds and smooth maps, if this category is indeed essentially small. ...