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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4
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2answers
295 views

Has this construction, which builds a symmetric multicategory from a commutative monoid, been described or studied anywhere, and if so, where?

Whenever $R$ is a commutative ring, write $R[x^{(n)}]$ for the set of all $p \in R[x]$ such that $p$ is a monic polynomial of degree $n$. Then $R[x^{(n)}]$ is not closed under sums, nor does it ...
3
votes
2answers
502 views

A naive question about SGA4

Can someone explain to me the meaning of remark 1.1.2 at the begining of SGA4.1? It says that if $C$ is a category that belongs to some universe $U$ (which I understand as "$\mathrm{Ob}(C)$ and ...
0
votes
0answers
116 views

Construction of Yoneda extension (repost)

This is a reposted question to pull a bit more attention. I unfortunately could not find a detailed construction of the Yoneda extension in literaure, namely, its action on morphisms. In "Category ...
2
votes
1answer
129 views

Terminology question for maps between posets

Let $P$ and $Q$ be two poset (partially ordered sets) and $\phi : P \to Q$ an order-preserving function. I would like to know whether there is a name and perhaps a different characterizations of such ...
4
votes
2answers
168 views

Colimit density and monads

Let $C$ be a cocomplete category, and suppose that it has an object that is colimit dense. Is $C$ automatically monadic over $Set$? And if not, is there an explicit counterexample?
1
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0answers
59 views

Getting a measure from a premeasure through an adjoint

Let's take the category of measure spaces with objects $(X,\mathcal{F},\mu)$ and avoid the morphisms for now (I'm not sure what they should be), where $X$ is a set, $\mathcal{F}$ is a ...
2
votes
1answer
155 views

Reference for “multi-monoidal categories”

I have attempted to find a definition of a monoidal category which incorporates $n$-fold tensor products instead of just binary tensor products. Definition. A "multi-monoidal category" consists of ...
11
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3answers
919 views

Non-abelian Grothendieck group

By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations ...
1
vote
2answers
128 views

The family of morphisms f such that Qf=identity for some functor Q

Localization of a category deals with the family of morphisms rendered invertible by a functor, i.e. the (saturated) family of weak equivalences, which I think resembles the kernel of a group or a ...
4
votes
2answers
473 views

2-category theory

I know that we can do a lot of 2-category theory, seeing 2-categories as Cat-enriched categories. Yet, I know that there are some limitations of this approach. I also know that there are many articles ...
6
votes
0answers
149 views

Correspondences as generalized morphism between $C^*$-algebras

While reading about Morita equivalence in the category of $C^*$-algebras I met also the following notion: a correspondence between two $C^*$-algebras $A,B$ is a pair $(X,\varphi)$ where $X$ is a ...
4
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0answers
212 views

Reference Request for TQFTs

I had originally asked this question on Math StackExchange but have not obtained any answers, so I decided to post this here. (I have flagged the MSE post to be moved to MathOverflow, for the ...
1
vote
0answers
99 views

The category of discontinuous Banach spaces

A banach space is discontinuous if it is isometric to $DC(X)$ for some Hausdorff topological space $X$. ($DC(X)$ is defined here. We denote by $DBan$, the category of all discontinuous ...
6
votes
0answers
234 views

A model category for descent?

Recall that an $(\infty,1)$-category $C$ is said to have descent if for any small diagram $X:I\to M$ with (homotopy) colimit $\overline{X}$, the adjunction between $C/\overline{X}$ and "equifibered" ...
9
votes
1answer
276 views

Example of a saturated class of morphisms which is not _obviously_ saturated?

By "saturated class of morphisms" in a category $\mathcal{C}$, I mean a subcategory $\mathcal{W} \subset \mathcal{C}$ such that the image of $\mathcal{W}$ in $\mathcal{C}[\mathcal{W}^{-1}]$ consists ...
3
votes
1answer
486 views

Opposite Symmetric Monoidal Structure on an Infinity Category

Given an $\infty$-category (in the sense of Lurie) $C$, and a symmetric monoidal structure on $C$ associated to a coCartesian fibration $p:C^\otimes\to N(Fin_\ast)$, Lurie says in Remark 2.4.2.7 of ...
8
votes
0answers
183 views

Retractions of Yoneda are retractors, i.e., left adjoints?

Background It is fairly well known that if a full subcategory $i: C \hookrightarrow D$ has a left adjoint $F: D \to C$, then the canonical counit $F i(c) \to c$ is an isomorphism. (A classical ...
6
votes
0answers
114 views

Preservation of universes in presheaves

In Lifting Grothendieck universes Hofmann and Streicher construct a universe in the category of presheaves over a small category given a Grothendieck universe in $\mathbf{Set}$. Suppose now I have ...
2
votes
0answers
96 views

Gluings and collages along profunctors

I'm dealing with this construction, available whenever you have two categories $\cal C,D$ and a profunctor $\varphi\colon {\cal C}\leadsto{\cal D}$ between them. Define a category ${\cal ...
0
votes
0answers
93 views

Distributive law between Kleisli triples

A distributive law of a monad $S$ over a monad $T$ is a natural transformation $l : T S \to S T$ such that: $l \circ T \eta^S = \eta^S T$ $l \circ \eta^T S = S \eta^T$ $\mu^S T \circ S l \circ l S = ...
2
votes
1answer
105 views

Kan extension pseudonatural transformations

Consider the 2-category $[S, H] $ of 2-functors $S\to H $ (in which, obviously, $S$ and $H$ are 2-categories). And consider a (possibly fully faithful) functor $T: S\to Z $ For simplicity, let's ...
3
votes
3answers
224 views

Tensor product over a monoid in a monoidal category

nLab article on tensor product says: "Given two objects in a monoidal category $(C,\otimes)$ with a right and left action, respectively, of some monoid $A$, their tensor product over $A$ is the ...
26
votes
2answers
698 views

Cantor's theorem for presheaves?

Some years back (before MathOverflow was born), Tom Leinster asked an interesting question at the $n$-Category Café which I don't recall ever seeing an answer for: Does there exist a ...
4
votes
2answers
663 views

What's so special about $1$-categories?

I have been pretty thoroughly convinced for some time now that, when thinking about mathematics, one really should be thinking 'categorically', that is, one should always be thinking of the morphisms ...
3
votes
1answer
282 views

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 ...
1
vote
1answer
45 views

Retractions and left-factoring morphisms

Let $\mathcal{C}$ be any category and let $A, B$ be objects. A retraction is a morphism $r: A\to B$ such that there is $s:B\to A$ such that $r\circ s:B\to B$ is the identity. A morphism $l: A\to B$ ...
1
vote
0answers
74 views

(Affine) Schemes and the point of view of morphisms with values in a field

Let $A$ be a commutative ring with unity. If $\varphi_1 : A \rightarrow K_1$ and $\varphi_2 : A \rightarrow K_2$ are ring morphisms from $A$ to fields, $\varphi_1$ and $\varphi_2$ are said to ...
6
votes
1answer
368 views

What is the universal property of quotienting a normaliser of the subgroup?

Let $G$ be a group, $H$ a subgroup and $X$ a $G$-set. By taking orbits $X/H = X \times_H 1$ or fixed points $X^H = \mathrm{Hom}_H(1,X)$ we obtain a set on which $H$ acts trivially, and we've destroyed ...
2
votes
1answer
68 views

Universal and left-factoring order-preserving maps

Trying to get a different angle for the question Fixed points and universal maps for posets, I want to compare universal maps to a different kind of functions. First recall that for posets $P,Q$ an ...
6
votes
1answer
324 views

Existence of internal toposes/inner models in a topos

It has been known for some time that one can define a topos as a model of a (finitary) essentially algebraic theory (or in other words, can be defined internal to any category with finite limits). In ...
2
votes
2answers
253 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 ...
4
votes
1answer
192 views

To what extent are homotopy colimits over a weakly contractible category “determined by local data”?

[I'd be very happy for a better question title, if anyone has any suggestions.] I have a category $C$, two functors $F,G : C \to \mbox{Cat}$, a natural transformation $\alpha : F \to G$, and a ...
2
votes
0answers
86 views

Pre-Order induced by continuous functions

I'm an newbie in category theory, but I want use it to solve a pre-order question I encountered in my research: Let $X$ be a convex&compact subset of $\mathbb{R}^n$. $f,g: X \rightarrow [0,1]$ ...
9
votes
0answers
198 views

Does the $(\mathbb Z/2)$-graded isomorphism $E_n \cong E_{n+2}$ have any nice properties?

This question assumes everything is dg. Let's decide to work over the "field" $\mathbb Q[\mu,\mu^{-1}]$ where $\mu$ has homological degree $+2$. Then chain complexes are just $\mathbb Z/2$-graded. ...
4
votes
2answers
287 views

Can one make a category concrete by “enlarging the universe”?

This is more or less a followup of this question. There, it was established that (it is well known that) the homotopy category of topological spaces is not concrete, in other words, there is no ...
3
votes
1answer
140 views

Model bicategories

From a conceptual point of view, the notion of a "model bicategory" is clear: a complete, cocomplete bicategory where there are two "very weak" factorization systems, where the commutativity of the ...
7
votes
1answer
512 views

The (un)reasonable (non-)ubiquity of the Grothendieck construction

Is there a way to export the Grothendieck construction to different contexts than $Cat$? in theory, if you build $\int F$ out of $F\colon \mathcal C\to Cat$, or $F\colon \mathcal C\to Sets$, as a ...
10
votes
1answer
459 views

Non-unique splittings of homotopy idempotents

By a homotopy idempotent I mean a map $f:X\to X$, where $X$ is a space, equipped with a homotopy $f\circ f \sim f$. In contrast to the situation in stable homotopy theory (where $X$ would be a ...
4
votes
1answer
176 views

“Universal maps” as a universal property

In the question Fixed points and universal maps for posets, we find the following definition: if $P, Q$ are partially ordered sets, an order-preserving map $u:P\to Q$ is said to be universal if for ...
5
votes
1answer
156 views

Cartesian product of small objects

Let's say we have a locally $\lambda$-presentable category and a pair of $\lambda$-presentable objects $A$ and $B$. Is it true that $A \times B$ is $\lambda$-presentable?
2
votes
2answers
329 views

Topological retraction vs categorical retraction

Let $(X,\tau)$ be a topological space. We say that $A\subseteq X$ is a topological retract if there is a continuous map $r:X\to A$ onto a subspace $A \subseteq X$ such that for all $a\in A$ we have ...
2
votes
2answers
252 views

Morphism on schemes induced by continuous morphism of sites

I am beginner in the theory of Grothendieck topologies and I have the following question. Let $X, Y$ be schemes over an algebraically closed field $k$. Denote by $X_{et}$ and $Y_{et}$ the Etale sites ...
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votes
1answer
121 views

Relations between ordinary functor categories and higher categories [closed]

Definitions of ordinary functor categories and higher categories are considered with very similar algebraic and geometric methods such as graph structures and simplicial sets. I know the differences ...
3
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2answers
261 views

Is the defining bijection for a pullback of topological spaces a homeomorphism?

I work in the category of CGWH spaces enriched over themselves. If a space $P$ is the pullback of $A \rightarrow B \leftarrow C$, then for every space $T$ the canonical map $$Top(T,P) \rightarrow Top ...
4
votes
1answer
240 views

Is the classifying space of a symmetric monoidal category an infinite loop space?

Wikipedia states: The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an infinite loop space. If my mind is correct, Segals delooping machine gives a ...
7
votes
1answer
152 views

Small objects vs Compact objects

Given a cocomplete category $C$, is there an example of an object which is small but not compact? I am working with the following definitions of small and compact: Given a cardinal $\kappa$ one ...
3
votes
0answers
168 views

Map of adjunctions

The following question must have been asked dozens of times, but I do not recall any non-trivial results. Consider an adjoint square where the arrows indicate directions of $F, G, H, K$. ...
1
vote
1answer
118 views

Does the category of posets have pushouts and pullbacks? [duplicate]

Let $\mathbf{Poset}$ be the category of partially ordered sets with order-preserving maps. Does $\mathbf{Poset}$ have both pushouts and pullbacks?
11
votes
2answers
671 views

History of integral notation for coends

I'm searching the wheres and whys about the integral notation for co/ends. Who was the first to adopt it? Can you give me a precise pointer or tell me the whole story about it? Was s/he motivated by ...
4
votes
1answer
250 views

$\omega$-nerve versus $\Theta$-nerve

To which extent the adjunction $F\dashv N_\omega$ generated by the $\omega$-nerve described at $n$Lab - oriental (obtained as a particular instance of the nerve-realization paradigm) is linked to the ...