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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6
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1answer
216 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
votes
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
295 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 ...
18
votes
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 ...
13
votes
0answers
569 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 ...
0
votes
0answers
124 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 ...
1
vote
0answers
73 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 ...
9
votes
2answers
366 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
votes
1answer
88 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 ...
1
vote
0answers
98 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 ...
3
votes
0answers
319 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
vote
1answer
296 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
votes
2answers
304 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
votes
1answer
216 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. ...
5
votes
0answers
160 views

Is there a reasoned derivation of the coherence conditions for symmetric rig categories?

I know what the coherence conditions are, I can look them up in M. Laplaza, Coherence for distributivity, Lecture Notes in Mathematics 281, Springer Verlag, Berlin, 1972, pp. 29-72. In theory, ...
1
vote
2answers
159 views

Are monoids with zero and partial homomorphisms related?

Context: Let $\Sigma=\{U,C,A,G\}$ and $L\subset\Sigma^*$, i.e. $L$ is a language over the alphabet $\Sigma$. Let $\Sigma'=\{0,1\}$ and define a homomorphism $f:\Sigma^*\to\Sigma'^*$ by extending $U ...
2
votes
1answer
64 views

Morphisms $P \to M$ in the derived category of a dg-category, if $P$ is h-projective

Let $\mathbf A$ be a dg-category. Denote by $\mathsf{C}_{\mathrm{dg}}(\mathbf A)$ the dg-category of right $\mathbf A$-modules, and by $\mathsf{C}(\mathbf A) = Z^0(\mathsf{C}_{\mathrm{dg}}(\mathbf ...
1
vote
1answer
102 views

partial pullback-completion of a category

Let $\mathcal{C}$ be a (possibly enriched) category with all finite product, and $\mathbf{M}$ a class of morphisms. Can one construct completion of $\mathcal{C}$ w.r.t. all pullbacks along morphisms ...
2
votes
0answers
79 views

A generalization of the Spanier-Whitehead construction

What I call "Spanier-Whitehead stabilization" is a construction which extends a category $\bf C$ to a bigger one $\mathcal{SW}_\Omega({\bf C})$ where a given endofunctor $\Omega$ is invertible. The ...
0
votes
0answers
84 views

Representing topoi by topological groupoids

i was reading an article written by Butz and Moerdijk (https://www.math.uu.nl/publications/preprints/984.ps.gz) and i have a problem in understanding their proof of theorem $5.1$ (The one in which ...
3
votes
0answers
81 views

Problem with a proof in Wellfounded trees in categories

I'm reading the paper Wellfounded trees in categories by Moerdijk and Palmgren. I'm having trouble understanding the proof of Theorem 7.2 (page 216), i.e. that in $\mathbf{ML}_{< \omega} ...
4
votes
2answers
292 views

The classifying space of an infinite totally ordered set is contractible

I asked this question on math.stackexchange, but no one answered. Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This ...
6
votes
1answer
446 views

Vectorisation of a category

I have no experience with category theory at all, but I recently stumbled upon the following construction. Since it is extremely elementary and seems rather natural, it should be known, but I have not ...
5
votes
1answer
216 views

Can you “combine” Ord and Mon to get Cat?

Mon is the category of moniods, which can be seen as categories with one object. Ord is the category of preorders, which can be seen as categories with up to one morphism in each homset. Is there ...
6
votes
1answer
282 views

If two structures are elementarily equivalent, is there a zigzag of elementary embeddings between them?

Fix a first-order signature $\Sigma$. There is an equivalence relation $\sim$ on the class $\Sigma\mathrm{-Str}$ of all $\Sigma$-structures given by $M \sim N$ iff $M$ and $N$ are elementarily ...
4
votes
0answers
245 views

Generalize $\pi_0(B\mathcal{C})\cong\{\text{objects}\}/\{\text{morphisms}\}$ to categories internal to topological spaces

Warmup: Let $\mathcal{C}$ be an ordinary category. Denote by $$B\mathcal{C}=(\coprod_{i\in\mathbb{N_0}}N_{i}(\mathcal{C})\times\Delta^i)/\tilde{}$$ its classifying space, i.e. the geometric ...
1
vote
1answer
152 views

A delicate question about derived functors

Let $A\subseteq B \subseteq C$ be three triangulated categories, such that $A$ is a full triangulated sub-category of $B$, and $B$ is a full triangulated sub-category of $C = K(R)$. Let $F: C \to D$ ...
3
votes
2answers
263 views

How to show the following two definitions of homotopy monomorphism are equivalent?

Let $M$ be a model category. In Toen's The homotopy theory of dg-categories and derived Morita theory Page 11 it is written: a morphism $x \to y$ in a model category $M$ is called a homotopy ...
2
votes
1answer
98 views

A question about the morphisms in the homotopy category of dg-Cat

Let $dg-Cat$ denote the category of (small) dg-categories and $Ho(dg-Cat)$ denote the localization of $dg-Cat$ at quasi-equivalence. Using the model structure on $dg-Cat$ we can describe the morphisms ...
5
votes
1answer
132 views

Regular epimorphisms in the category of simple undirected graphs

Let $\textbf{Grph}$ be the category whose objects are graphs $G = (V,E)$ such that $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$. We sometimes write $E(G)$ for ...
13
votes
2answers
375 views

A model category of abelian categories?

Let $\mathcal{M}$ be the following category: The objects are small abelian categories with chosen zero object, biproducts, kernels, and cokernels. The morphisms are functors that preserve the ...
9
votes
2answers
175 views

Reflective Localizations vs. categories of local objects

Given a category $\mathcal{C}$ and a set (let's not bother with size issues here) $\mathcal{W} \subseteq \text{Mor}(\mathcal{C})$ we may form the category $\mathcal{C}[\mathcal{W}^{-1}]$ obtained by ...
3
votes
1answer
373 views

Formulating Kunen's inconsistency and Reinhardt cardinals in term of category theory

It is known that one can formulate certain large cardinal axioms (e.g. Vopenka's principle--see Mike Shulman's answer to Harry Gindi's mathoverflow question "Reasons to believe Vopenka's ...
14
votes
1answer
715 views

Do I know what “coherent sheaf” means if I know what it means on locally Noetherian schemes?

I've been trying to convince myself that "coherent sheaf" is a natural definition. One way I might be satisfied is the following: for modules over a Noetherian ring $A$, coherent and finitely ...
5
votes
1answer
178 views

What is the monoidal equivalent of a locally cartesian closed category?

If a closed monoidal category is the monoidal equivalent of a Cartesian closed category, is there an analogous equivalent for locally cartesian closed categories? Is there a standard terminology or ...
4
votes
2answers
272 views

Kan extensions in the $2$-category of monoidal categories

Kan extensions make sense in any $2$-category. But so far I have only really seen them in the case of the $2$-category of categories, functors, natural transformations and the $2$-category of ...
8
votes
2answers
298 views

Around Vopěnka: Accessible category with small full discrete subcategories of arbitrary size?

I believe the model-theoretic version of the question is: is there a theory in finitary first-order logic which has, for each cardinal $\lambda$, a set $C_\lambda$ of $\lambda$-many models, such that ...
27
votes
3answers
2k views

Is every abelian group a colimit of copies of Z?

More precisely, is every abelian group a colimit $\text{colim}_{j \in J} F(j)$ over a diagram $F : J \to \text{Ab}$ where each $F(j)$ is isomorphic to $\mathbb{Z}$? Note that this does not follow ...
2
votes
1answer
118 views

Regular and extremal monomorphisms in the category of graphs

Let $\textbf{Grph}$ be the category whose objects are graphs $G = (V,E)$ such that $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$. We sometimes write $E(G)$ for ...
6
votes
2answers
271 views

Strategies for proving a category is Noetherian?

Let $C$ be a small linear category over a commutative ring $R$. A representation of $C$ is an $R$-linear functor $C \to \mathrm{Mod}(R)$. For example, for each $c\in C$, there is a representation ...
3
votes
0answers
100 views

Homotopy (co)limit (co)cones

Let $\mathscr{M}$ be a model category and let $\mathscr{I}$ be a small category. Consider any homotopy colimit functor ...
3
votes
3answers
192 views

a (pseudo)adjunction for the functor sending a category C to PSh(C) the category of presheaves

I've read about free cocompletion of categories discussing on the adjunction between Cat and cocompleteCat (Cat: category of small categories, cocompleteCat: category of small cocomplete categories ...
1
vote
1answer
87 views

Induced topology on site + Reconstructing global sections of a scheme (Orlov)

Let $(C,T,O)$ be a ringed site. Let $X$ be a presheaf on C. We get an induced ringed site $(C/X,T_X,O_X)$. C/X is the over category wrt the presheaf X. The topology $T_X$ is the biggest topology ...
6
votes
2answers
165 views

Direct proof that the model category of cdgas is left proper

Let $k$ be a field of characteristic $0$. The projective model structure on the category $cdga$ of commutative differential graded $k$-algebras is proper. Since this model structure is transferred ...
12
votes
1answer
253 views

Is there something like “Noncommutative geometry internal to a category”?

I have heard that one can do algebraic geometry internal to symmetric monoidal categories. Topological quantum field theories also exist internal to symmetric monoidal categories, and the usual ...
2
votes
0answers
81 views

Do copairings provide dualities in derived categories?

Here is an elementary fact about vector spaces. Let $V,W$ be vector spaces over a field $\mathbb K$ and let $c : \mathbb K \to V \otimes W$ be an element of the tensor product. Then $c$ determines ...
2
votes
2answers
428 views

Exact sequences of pointed sets - two definitions

It seems to me that there are (at least) two notions of exact sequences in a category: 1) Let $\mathcal{C}$ be a pointed category with kernels and images. Then we call a complex (i.e. the composite ...
1
vote
1answer
188 views

Classifying space of a colimit of topological categories

Say I have a diagram $D:I\rightarrow\text{Cat}(\text{Top})$ of categories internal to compactly generated topological spaces. This induces a diagram $BD:I\rightarrow \text{Top}$ of classifying spaces. ...
21
votes
2answers
1k views

Why should have Peter May worked with CGWH instead of CGH in “The Geometry of Iterated Loop Space”?

This is a follow-up to Dan Ramras' answer of this question. The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here). The weak Hausdorff rather ...
1
vote
1answer
89 views

Is $\textbf{FHILB}$ locally regular?

Is the category, $\textbf{FHILB}$, of finite dimensional Hilbert spaces and linear maps locally regular, where `locally regular' is defined like this ...