# Tagged Questions

**3**

votes

**1**answer

301 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

**0**answers

109 views

### Cylinder in a topological space? [on hold]

There is a notion of path in a topological space $X$, namely a continuous function f with domain X and codomain the interval $[0, 1]$.
Given that, in a Quillen model category, the dual of a path ...

**0**

votes

**0**answers

106 views

### Understanding the homotopy category of chain complexes [on hold]

In the definition of the Homotopy category of chain complexes http://en.wikipedia.org/wiki/Homotopy_category_of_chain_complexes , One defines maps between chain complexes in a certain way that I ...

**3**

votes

**1**answer

196 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

**1**answer

39 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

**0**answers

52 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 ...

**-4**

votes

**0**answers

104 views

### show that L(X,Y)banach then Y banach

Let {Xα}α∈A be a collection of Banach spaces. It is easy to show that P={(xα):supα∥xα∥<∞} with ∥(xα)∥=supα∥xα∥ is a banach space.
If the indexing set A is finite, then it is easy to show that P ...

**6**

votes

**1**answer

330 views

+50

### 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

**1**answer

61 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 ...

**5**

votes

**1**answer

191 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

**1**answer

134 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

**1**answer

166 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

**0**answers

74 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 $\mathbb{R}^n$. $f,g: X \rightarrow [0,1]$ ...

**0**

votes

**0**answers

128 views

### On a paper by Yoneda [closed]

The reason why I asked this previous question was gathering some informations for the note on coend calculus I've just (almost) finished.
Unfortunately, I'm still unable to retrieve the original ...

**9**

votes

**0**answers

156 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. ...

**3**

votes

**2**answers

244 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 ...

**0**

votes

**0**answers

47 views

### How can I do this natural transformation? [closed]

I'm doing an exercise which asks me to give a natural transformation between functors $F,G:C \rightarrow C$. The functors are defined by, given fixed objects $X$ and $Y$, $F(A) = A^{X+Y}$ and $G(A) = ...

**3**

votes

**1**answer

124 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 ...

**-1**

votes

**0**answers

97 views

### What is a definition of functional morphism? [closed]

When reading Towards Higher Categories by John C Baez,J Peter May. I sometimes encounter such terms as functional and ...

**4**

votes

**0**answers

261 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 ...

**7**

votes

**1**answer

267 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

**1**answer

151 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

**1**answer

143 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

**2**answers

293 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

**2**answers

228 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 ...

**-3**

votes

**1**answer

112 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**

votes

**2**answers

218 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

**1**answer

210 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

**1**answer

131 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

**0**answers

151 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

**1**answer

96 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?

**10**

votes

**2**answers

611 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

**1**answer

235 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 ...

**5**

votes

**1**answer

85 views

### Site dependance of the Cech weak equivalences on simplicial sheaves

Let $\mathcal{T}= sh(C,J)$ a Grothendieck topos of sheaves over a (ordinary) site.
One endows the category of simplicial presheaves over $C$ with the "Rezk-Lurie" model structure: that is we start ...

**7**

votes

**2**answers

286 views

### Localizations or quotients of categories?

Motivation: In the classical construction of the derived category of an abelian category, one (roughly) starts with an abelian category $\mathcal{A}$, then considers the quotient category ...

**3**

votes

**1**answer

160 views

### Functoriality of the adjoint functor construction?

Say we have a category $\mathcal C$, and for every category $\mathcal A$, we have a category $\mathcal D_{\mathcal A}$ and a functor $F_{\mathcal A} : \mathcal D_{\mathcal A} \to \mathcal C$, and, ...

**8**

votes

**1**answer

214 views

### Non-abelian freeness of SU_2

The distribution of the trace of a random element of $SU_2$ is the Sato-Tate distribution.
The analogue of the Gaussian distribution in free probability theory is the Wigner semicircle distribution.
...

**6**

votes

**2**answers

184 views

### Exactness of an additive left Kan extension

Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair
$$\phi_!:R-Mod\rightleftarrows S-Mod:\phi^*,$$
where $\phi_!=(S\otimes_R -)$. The right adjoint $\phi^*$ is easily ...

**4**

votes

**1**answer

159 views

### When does $\mathbf{Top}/X$ embedd fully faithfully into $\mathbf{Top}$?

Under what conditions on the topological space $X$ is the overcategory $\mathbf{Top}/X$ of topological spaces over $X$ equivalent to a full subcategory of $\mathbf{Top}$? Surely if $X$ terminal i.e. a ...

**4**

votes

**0**answers

81 views

### Limits in Span(Vec)

Let Vec be the category of real vector spaces and linear maps. Let Span(Vec) be the bicategory of correspondences between real vector spaces. I am trying to understand lax limits in Span(Vec). What ...

**0**

votes

**0**answers

78 views

### About stable maps passing through fixed points

In "Notes On Stable Maps and Quantum Cohomology", Fulton and Pandharipande present some results, and their proofs, about the representability of the functor $\mathcal{M}_{g, n}(X, \beta)$, which maps ...

**3**

votes

**0**answers

61 views

### Cofree Lie Coalgebra

I have problems finding anything about the cofree Lie coalgebra functor
$\mathcal{L}ie^c$ out there.
Basically all I found was that it appears in Harrison cohomology and that,
given a ...

**5**

votes

**1**answer

268 views

### Framed version of braided monoidal category

The operad $\mathcal{D}_2$ of little $2$-disks is an operad whose $n$-th space is a $K(PB_n,1)$ where $PB_n$ denotes the pure braid group on $n$-strands. Algebras over $\mathcal{D}_2$ have a ...

**2**

votes

**0**answers

179 views

### What is the right categorical framework for diagonal approximations, cup and cap products and identities between them?

I wanted to know what is the right categorical analog for the diagonal approximation for the singular chain complex $\Delta: C\rightarrow C\otimes C$ as a morphism in the homotopy category of chain ...

**25**

votes

**6**answers

2k views

### Synthetic vs. classical differential geometry

To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of ...

**0**

votes

**0**answers

58 views

### Complex of “homotopy coherent dg-natural transformations” between dg-functors

Let $\mathcal B$ be a dg-category. I define the dg-category of morphisms $\underline{\mathrm{Mor}}(\mathcal B)$ as follows: objects are triples $(A,B,f)$, where $A,B \in \mathcal B$ and $f \in ...

**-1**

votes

**1**answer

209 views

### Kan extensions and special cases

Kan extensions specify the adjoint structures between $\mathbf{Sets^{C^{op}}}$ and $\mathbf{Sets^{D^{op}}}$, where there exists a functor $f:\mathbf{C} \to \mathbf{D}$ and $\mathbf{C}$ and ...

**0**

votes

**0**answers

75 views

### graded modules over graded algebra as graded objects

let $\cal{M}$ be the additive symmetric monoidal category of $\mathbb{Z}$-graded $R$-modules (for $R$ commutative ring), and $A$ a commutative algebra in $\cal{M}$. I still think about $Mod(A)$ as ...

**4**

votes

**1**answer

182 views

### Does “simplicial” commute with “Bousfield localization”?

Let $M$ be a model category and $S \subseteq \operatorname{Mor}(M)$ a set of arrows in (the underlying strict category of) $M$. Recall that the left Bousfield localization $L_SM$ of $M$ with respect ...

**2**

votes

**0**answers

121 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, ...