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
77 views

Left adjoint to Double Nerve?

The well known nerve functor from small categories to simplicial sets has a left adjoint, namely the fundamental category functor. Does the double nerve functor $N^2:2Cat\rightarrow sSSet$ from 2-...
3
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0answers
35 views

When is a functorial coverage a sheaf, and what universal property does it have?

In The Elephant (A.2.1.9), Johnstone defines the notion of a coverage on a category $\mathcal{C}$. Quoting verbatim, a coverage on $\mathcal{C}$ is a function assigning to each object $A$ of $\...
3
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0answers
154 views

Representability of the Weil restriction reference and proof

Proposition 2 of 7.6 of Néron Models [BLR] provides a sufficient condition for the representability of a Weil restriction $R_{S'/S}(X')$. The theorem is attributed to Grothendieck. Is there an ...
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1answer
76 views

A map between direct limits

Let $C$ be a category which has all small colimits. I have the following situation: $\{A_i\}_{i \in I}$ and $\{B_j\}_{j \in J}$ are two directed systems in $C$, with transition maps $\alpha_{i_1,i_2}...
3
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1answer
64 views

Saturated classes and cofibrantly generated model structures

There seem to be two definitions of what a saturated class should be: A class of morphisms closed under retracts, pushouts and transfinite composition. A class of monomorphisms containing all ...
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83 views

When do pushouts along epis preserve products?

A pushout diagram in a category $\mathcal{C}$ is a commutative square with a certain universal property; as usual, say that the pushout diagram is along an epi if at least one of the two arrows out of ...
2
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1answer
136 views

Does a binormal category always admit an additive structure?

Let $\mathsf{C}$ be a category. We call $\mathsf{C}$ binormal if it has a null object, has all equilizers and coequilizers, all monomorphisms are kernels and all epimorphisms are cokernels (whereby a ...
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3answers
950 views

Geometric models for classifying spaces of a group

For any given topological group $G$ we have Segal's construction/definition of $BG$. I'm recalling it in case the details turn out to be relevant. Form the disjoint union of $G^n\times\Delta_n$ ...
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149 views

Interaction of Grothendieck Construction with Coherent Nerve

There are a number of Grothendieck constructions: one for discrete categories, one for enriched categories (see Tamaki's paper here) and one for quasicategories (see the Unstraightening and ...
2
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0answers
149 views

Which locally ringed spaces are schemifiable?

(most of this question is re-asking Schemification (schematization?) of locally ringed spaces, which did not get answered) Given a locally ringed space $X$, say that a schemification of $X$ is a ...
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2answers
205 views

For a universal covering morphism $p:E\rightarrow B$, how to prove $E$ connected implies $B$ connected?

Definition. An arrow $\alpha:A\rightarrow B$ in $\mathsf C=\mathsf{Fam}(\mathsf A)$ is said to be a covering morphism if there exists an effective descent morphism $p:E\rightarrow B$ that splits it, i....
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4answers
425 views

Are there non-trivial infinite chains of adjoint functors?

There are self-adjoint functors $A \dashv A$. There are also functors $A$ that are both left- and right-adjoint to another functor $B$. $$A \dashv B \dashv A$$ There are also finite cyclic chains of ...
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2answers
419 views

Which spaces have enough curves

Let $\mathbf{Top}$ be the category of topological spaces, and let $I\in\mathbf{Top}$ be the unit interval $I=[0,1]\subset\mathbb{R}$. For any space $X$, let $|X|$ denote the underlying set of points; ...
5
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1answer
113 views

Necessity of shapes for coherence results in category theory

The classic coherence theorems of MacLane (Natural associativity and commutativity, Rice U. studies, 1963) talked about natural transformations between functors. By 1971 (Kelly-MacLane, Coherence in ...
13
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1answer
806 views

Can Vopenka's principle be violated definably?

One form of Vopenka's principle (a large cardinal axiom) states that no locally presentable category contains a full subcategory which is large (= a proper class) and discrete (= contains no ...
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0answers
183 views

Grothendieck Construction, Categories of Operators and Opposites

Given a symmetric monoidal category $C$, we can construct its endomorphism operad (or multicategory) $End(C)$ whose objects are the objects of $C$, and for which the multimorphisms from $\{c_1,\ldots,...
3
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1answer
145 views

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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292 views

Infinite Fubini rule for co/limits

It is a well known fact that given a functor $F\colon I\times J\to C$ then (when everybody exists) $$ \varinjlim_I \varinjlim_J F\cong \varinjlim_J \varinjlim_I F\cong \varinjlim_{I\times J}F $$ Now, ...
74
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11answers
6k views

Why do Groups and Abelian Groups feel so different?

Groups are naturally "the symmetries of an object". To me, the group axioms are just a way of codifying what the symmetries of an object can be so we can study it abstractly. However, this heuristic ...
1
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1answer
157 views

Product and coproduct for bipartite graphs

Consider the category $BiGraph$ of bipartite graphs (vertices are named "places" and "transitions" like in Petri nets) and continous maps. A subgraph is named open iff it is place-bordered, and a map ...
24
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2answers
1k views

Properties of functors and their adjoints

I am interested in collecting in this question a list of properties a functor $F$ may have and what those properties imply for left and right adjoints, $F^L$ and $F^R$, assuming they exist. There are ...
6
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1answer
270 views

Bar/Cobar Adjunction Between Modules and Comodules

There is a pretty well known, and widely written about, adjunction between augmented algebras and coaugmented coalgebras given by taking the bar construction on algebras and the cobar construction on ...
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0answers
157 views

$\mathbb{Z}_2$ as a colimit of $\mathbb{Z}^*$

Consider the multiplicative monoid $(\mathbb{Z}^*,\cdot)$ of nonzero integers. By multiplication, $\mathbb{Z}_2$ is a right (or left) $\mathbb{Z}^*$-set. According to the co-Yoneda lemma $\mathbb{Z}_2$...
4
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1answer
204 views

Categorification of covering morphisms

Given a category $\mathsf{A}$, let $\mathsf{Fam}(\mathsf{A})$ be its free coproduct cocompletion (which is always extensive). This means every object has a unique up to iso presentation as a coproduct ...
2
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1answer
299 views

Special objects in a category - terminology

For an object $A$ in a category $\mathfrak{C}$, consider the following property. ($*$) For every object $B$ in $\mathfrak{C}$, the set of morphisms $\text{Hom}(B,A)$ is either empty or consists ...
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101 views

Is the bar resolution of complexes dg-functorial?

Let $k$ be a commutative ring, and let $V$ be a complex of $k$-modules (more in general, we can take an $\mathcal A$-dg-module, where $\mathcal A$ is a dg-category. We can construct the bar resolution ...
9
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1answer
316 views

Coherence and rewriting

In category theory there are numerous coherence theorems (https://ncatlab.org/nlab/show/coherence+theorem). One example is the Mac Lane's coherence theorem for monoidal categories. This and probably ...
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0answers
222 views

The quotient of a scheme by a proper equivalence relation

Let $X$ be a scheme and $R$ be a proper equivalence relation on $X$. What can be said about the geometric structure of the quotient $X/R$? Is it representable by a stack, for example?
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0answers
78 views

Internal language type of power objects

It is a basic fact that in a category with finite limits the following are equivalent Each object $X$ has a (membership relation to a) power object $PX\times X\hookleftarrow\ni_X$ subject to the ...
4
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0answers
84 views

Can I combine the category of Drinfeld modules and the category of the base O_S

I am learning about Drinfeld modules,T-modules,...They are said to be analogues of elliptic curves, abelian varieties,... Let K be a finite extension of k = Frac(A), and $O_K$ the integral closure of ...
17
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3answers
2k views

Analysis from a categorical perspective

I have not studied category theory in extreme depth, so perhaps this question is a little naive, but I have always wondered if analysis could be taught naturally using categories. I ask this because ...
5
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1answer
176 views

Switching left and right adjoints in recollement situations

In Fascieaux pervers, Beilinson, Bernstein and Deligne define a recollement situation as a triple of triangulated categories $\mathcal{D}_U,\mathcal{D}_F$ and $\mathcal{D}$, together with functors $$ ...
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2answers
535 views

Generalized smooth spaces and infinite dimensional manifolds

There is a theorem due to Losik which shows that the category of Frechet manifolds embeds fully-faithfully into diffeological spaces. (Diffeological spaces are concrete sheaves on the site of (...
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0answers
94 views

“Sheaf” on the nerve of a category

So I have the nerve of a category and want to communicate information about the sets at each level in the simplicial set (nerve). Is there a way to regard the nerve as a category itself? Then one ...
10
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1answer
813 views

What is a field [Körper] really?

The notion of a field (a commutative ring $R$ with $0\neq 1$ and $R^\times=R-\{0\}$) seems to fit uncomfortably into modern algebra. To see what I mean, consider the following statements: The ...
21
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2answers
917 views

The homotopy category is not complete nor cocomplete

I understand that the homotopy category of (pointed) topological spaces and continuous maps is not complete. Nor is it cocomplete. In particular it neither has all pullbacks nor all pushouts. What ...
9
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1answer
695 views

Why do we denote (co)ends with integral notation (beyond Fubini's Theorem)?

I know that (co)ends (i.e. universal wedges) follow Fubini-like relation, i.e. $$ \int_{\langle c,d\rangle} F(c,d,c,d) \cong \int_c\int_d F(c,c,d,d) \cong \int_d\int_c F(c,c,d,d) $$ where we regard $F$...
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0answers
47 views

Dual equivalence for multioperators

This is a reference request question. But let's start with a few definitions. Let $L$ and $M$ be two bounded lattices. A multioperator $p$ for $L$ and $M$ is an application $$p : L \to Ft(M)^{op}$$ ...
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0answers
86 views

Mayer-Vietoris sequence for orbifolds

Is there a version of the Mayer-Vietoris long exact sequence for orbifolds? I am interested in orbifold homology as opposed to the homology of the underlying topological space.
28
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1answer
724 views

What is the meaning of this analogy between lattices and topological spaces?

Let me add one more edit to help explain why this is a serious question. Theorem 5 below is a sort of lattice version of Urysohn's lemma, and it has essentially the same proof. Theorem 6, the famous ...
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121 views

Categories where every Mono Splits

When every epi splits a category is said to satisfy the Axiom of Choice. When every idempotent splits a category is called Cauchy Complete or Idempotent complete. These look to be well-studied ...
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69 views

Reference for generalized ind-completions?

I am wondering whether any enriched versions of ind- and pro- completions have been studied? I can not find any literature on them, even though I believe people (most likely the Australian school) ...
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1answer
116 views

Equivalence of categeories-variants of definition [closed]

There is a notion of equivalence of categories which is the functor $F:\mathcal{C} \to \mathcal{D}$ such that there is a functor $G:\mathcal{D} \to \mathcal{C}$ such that $FG \cong id_{\mathcal{D}}$ ...
5
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3answers
131 views

Set of functions is not a bifunctor on Rel

Let Rel be the category whose objects are sets and whose morphisms are binary relations, with composition defined by $x (S \circ R) z \Leftrightarrow (\exists y : x R y \wedge y S z)$, and identity ...
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131 views

$\mathbb E$-descent maps in topological spaces in terms of different sites?

The paper Facets of Descent I by Janelidze and Tholen defines $\mathbb E$-descent maps as those for which $\Phi^p:\mathbb EB\longrightarrow \mathsf{Des}_\mathbb{E}(p)$ is an equivalence of categories. ...
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2answers
426 views

Topological characterization of injective metric spaces

Let $\ (X\ d)\ \,(Y\ \delta)\ $ be arbitrary metric spaces. A function $\ f:X\rightarrow Y\ $ is called a metric map (with respect to the given metrics $\ d\ \delta$) $\ \Leftarrow:\Rightarrow\ \...
6
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0answers
348 views

Competing notions of étaleness

I'm writing some notes to myself on algebraic geometry and I'm trying to get the most conceptual definitions. Having arrived at formally étale morphisms, I am pretty desperate. Here is a list of ...
2
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2answers
373 views

A Category-ish Structure with Morphism Domains containing Multiple Objects?

I am working on formalizing software design using category theory. However the most natural way for me to express what I want is with a Category where multiple morphisms can join into a single ...
7
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1answer
305 views

Étale morphisms in SDG

On page 70 here, Kock defines a formally étale morphism $f:M\rightarrow N$ as one for which the following square is a pullback for each $d:\mathbf 1\rightarrow D$ $$\require{AMScd} \begin{CD} M^D @>...