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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A question about the proof of Quillen's Theorem A

(I posted this question on Mathstack but I haven't received any answers or comments so I thought I might as well try my luck here. I apologize if it is not an appropriate question.) Theorem (Quillen) ...
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5answers
2k views

Origin of exact sequences

I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...
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116 views

Sheaffication using a $\lambda$-transfinite colimit

I asked this question on mathstack (long time ago), however I received no answers, so I'm trying it here. I don't know whether it's suitable for this site, anyway. I was reading this article ...
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1answer
225 views

Adjoining adjoints in a 2-category

For a given 2-category $C$, does there exist a faithful and locally faithful 2-functor $C \to C^*$, such that the image of every 1-morphism of $C$ has a right adjoint in $C^*$? Below are some of my ...
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2answers
387 views

Internal logic of the topos of simplicial sets

I am looking for a closed statement (i.e. not depending on any parameter objects) which is true in the internal logic of the topos of simplicial sets, but is not an intuitionistic tautology. Ideally, ...
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2answers
398 views

Microlocalizing Hochschild homology

A recent paper of Ben-Zvi and Nadler gives a general formalism for understanding "dimensions" in sheaf theories. Without getting too far into details, amongst other things, this formalism allows us ...
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165 views

Counting categories with at most $n$ morphisms

There are a number of results which count the number of possible algebraic structures on a set of $n$ elements. Notable previous MO questions are for example here and here. The analogous question for ...
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0answers
127 views

Equational theories determined by “identities without variables”

How to characterize equational theories $T$ which have the following property: for any two terms $t(x_1,...,x_n)$ and $t'(x_1,...,x_n)$ in the signature of $T$, if for any closed terms (i. e. terms ...
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1answer
153 views

In what generality does Eilenberg-Watts hold?

In homological algebra, the Eilenberg-Watts theorem states that if $F\colon\text{Mod}_R\to\text{Mod}_S$ is a right-exact coproduct preserving functor of modules, then $F\cong-\otimes_R F(R).$ The ...
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0answers
85 views

Intuitive idea: What is the motivation for relative homological algebra

Basically I was wondering what is the purpous of choosing a smaller class of epis in a category? Does it allow for the existence of "more" "projective-like" objects? What are some applications? For ...
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4answers
603 views

Functors and coverings

A category $C$ can be seen as a topological space via the geometric realization of the nerve of the category. Then a functor of categories gives a map of spaces. Is there a nice categorical ...
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0answers
86 views

Neeman's homotopy limits in stable $\infty$-categories

Related question (I hoped to turn this into an answer for the user, but in the end it became a question on its own right!): this In the book Neeman, Amnon. Triangulated categories. No. 148. ...
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1answer
264 views

Are left adjoints a left adjoint?

Let $\mathcal C$ be a strict, locally small 2-category. Consider a subcategory $\mathcal L$ of $\mathcal C$ such that $\mathcal L$ has the same objects as $\mathcal C$, and the arrows of $\mathcal L$ ...
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1answer
117 views

coends of stable infinity categories

Let $\mathcal{I}$ be a small ordinary category that I would like to think of as a diagram category. (If it helps: In my application $\mathcal{I}$ has only one object, i.e. comes from a monoid). Denote ...
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1answer
144 views

What do algebraic theories with strictly terminal trivial models look like?

By algebraic theory I mean one in the sense of Lawvere, i.e. a collection of finitary operations, including projections, together with a multi-composition satisfying the obvious axioms. (I believe ...
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78 views

How to show twisted complexes over a DG category is again a DG category?

In Bondal and Kapranov's paper enhanced triangulated categories, a twisted complex over a DG category $A$ is a set $\{(E_i)_{i\in \mathbb Z}, q_{ij}: E_i\to E_j\}$, where $E_i$ are objects in $A$, ...
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0answers
108 views

Which endomorphism algebras are not Morita-trivial?

Let $\mathcal C$ be a symmetric monoidal category — for example, $\mathcal C$ might be the category of $R$-modules for a commutative ring $R$. At a minimum, I request further that: $\mathcal C$ is ...
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2answers
123 views

Integral over integrals with different measures

I was trying to construct a category of measurable spaces and 'nondeterministic functions' (not the usual category of measurable spaces); specifically a map $(\Omega, \Sigma)\rightarrow (\Omega', ...
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0answers
98 views

Do coherent toposes descend along open surjection?

Let $f:\mathcal{T} \rightarrow \mathcal{S}$ be a geometric morphism between two toposes. Let $g:\mathcal{L}\rightarrow \mathcal{S}$ be an open surjection (of toposes) and assume that the map ...
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0answers
80 views

The classifying space of the groupoid $\pmb\Delta^n$

Consider the groupoid generated by the category $\{0\to 1\to\cdots\to n\}$; let's call this category $\pmb\Delta^n$ opposed to the category $\triangle^n$, which is "thinner". I'm trying to figure out ...
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1answer
359 views

Noncommutative geometry and category theory

The point in which one starts to talk about noncommutative geometry is the Gelfand Najmark theorem. It establishes an equivalence of the catgeories of commutative (non)unital $C^*$-algebras and the ...
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1answer
145 views

Cogroup objects are to groups what — are to $k$-modules

Let us place ourselves in a category $\mathcal C$ with finite coproducts $X\amalg Y$, even cocomplete if necessary. It is well known that the morphism set $\mathcal C(X,Y)$ carries an abelian group ...
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0answers
74 views

Categorizing epimorphisms in $\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)$

This is a follow up to this question of mine. The setup: Let $\mathcal{B}$ be an $\mathbb{F}_1$-linear category (Deitmar uses the term Belian); that is, $\mathcal{B}$ is pointed; balanced; contains ...
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1answer
228 views

Expressive power of first-order category theory

Given the signature $\lbrace \mathsf{dom}, \mathsf{cod}, \mathsf{id},\circ \rbrace$ and the axioms of category theory – which are expressible in the signature's first-order (FO) language – ...
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0answers
193 views

Sign problem for the shift functor on DG modules

Let $\mathcal{A}$ be a Differential graded $k$ category, where $k$ is a commutative ring. I want to extend the shift functor $[1]$ on chain complexes to dg modules over $\mathcal{A}$. Now a right dg ...
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1answer
266 views

Can the monoidal structure on manifolds be strictified?

I'm asking this question purely out of curiosity. Let $\{M_\alpha\}_{\alpha\in A}$ be a collection of closed smooth manifolds, with exactly one in every diffeomorphism class of closed smooth ...
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4answers
245 views

Graphs with dangling edges

In conventional Graph Theory the role of Nodes and Edges is skewed: nodes are perfectly ok being aloof, but poor edges are always drawn between existing nodes (that is, the two maps from EDGES to ...
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0answers
204 views

Why does this setting imply that a category is Grothendieck?

I came across the following Lemma in Mitsuyasu Hashimoto's Equivariant Twisted Inverses; it is Lemma 11.2 on page 107 of this pdf. Let $\mathcal{A}$ be an abelian category which satisfies the (AB3) ...
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0answers
104 views

Homotopy limits of weak diagrams

This question is somehow related to my question at http://math.stackexchange.com/questions/683915/derived-pseudo-functor and to the question here: A homotopy commutative diagram that cannot be ...
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1answer
483 views

On the foundations for large categories

There are some basic discussions on the motivations of large categories and small categories [here] (On the large cardinals foundations of categories) and [here] (Large cardinal axioms and ...
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3answers
408 views

Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
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5answers
1k views

(Short) Exact sequences with no commutative diagram between them

This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting. The problem is to produce an example of the following ...
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1answer
151 views

Homotopy limit of a cosimplicial category

Consider the usual model structure on Cat (category of small categories). Which are the fibrations of the injective model structure on the category of cosimplicial categories $Fun( \Delta ,Cat )$? ...
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1answer
160 views

n-limits and the Descent Category

Probably, this question could be at http://math.stackexchange.com/, but, for some reason, I can't access that site right now. So, I apologize in advance. At this article ...
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0answers
131 views

Is the collage of two spatial toposes a spatial topos?

Consider the collage operation along a profunctor, defined between two categories ${\bf C}, \bf D$. Suppose now that the two categories are toposes, say $Sh(X), Sh(Y)$ for two topological spaces ...
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1answer
282 views

Misunderstanding in the hypotheses of Schlessinger's criterion

Good day to everyone. In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion. Let's fix a representation $\bar{\rho}$ of a group ...
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162 views

Does there exist a terminal surjective discrete fibration out of $C$?

Let $DF$ denote the category whose objects are categories and whose morphisms $F\colon R\to S$ are the discrete fibrations. This category has applications to the real-world problem of structuring ...
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1answer
276 views

Not quite adjoint functors

What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the ...
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1answer
148 views

Expression for functor strengths?

It's well known that endofunctors on $Set$ have an unique strength. A strength for a functor $T : Set \to Set$ is a natural transformation $t_{A,B} : A \times T B \to T (A \times B)$ such that certain ...
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131 views

Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes. Equivalently (i) every interpretation of ...
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1answer
190 views

What is this name of this 2-category without very much structure?

I was wondering if there is a name for this 2-category which is like the 2-category of natural transformations, but does not actually require the 1-morphisms to be functors or the 2-morphisms to be ...
4
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1answer
172 views

Reference request: “unoriented composition” in generalized categories

I'm looking for a generalized notion of category (really of symmetric multicategory) which, roughly speaking, doesn't make a distinction between sources and targets. Each "morphism" in such a category ...
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0answers
50 views

A categorical analogue of Debreu's independent factors theorem

Background A major question in Decision Theory is that of the cardinal meaning of a utility function. That is, given a set $X$, a utility function $u:X\rightarrow \mathbb{R}$ represents the choices ...
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3answers
276 views

When does a monoidal functor between ribbon categories preserve cups and caps, but not necessarily braidings?

Imagine a monoidal functor between ribbon categories (i.e. monoidal, with a braiding, a twist and compatible left and right duals). An important example would be the restriction functor from the ...
3
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0answers
172 views

Invariants of groups that are invariant under passage to finite index subgroups

This question is mostly idle curiosity. Recall the following standard terminology: if $P$ is a property of groups, a group $G$ is said to be virtually $P$ if it has a subgroup of finite index which ...
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1answer
200 views

Cartesian closed category

Let $\bf{C}$ be a category with finite products. (1) An object $X$ of $\bf{C}$ is called cartesian if the functor $(-)\times X$ has a right adjoint. (2) A morphism ...
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1answer
346 views

Is $Lex(\mathcal B,\mathsf{Set}_*)$ an $\mathbb F_1$-linear category?

Following Anton Deitmar, let $\mathcal B$ be an "$\mathbb F_1$-linear category" (Deitmar uses the term "Belian"); i.e., $\mathcal B$ is balanced, pointed, contains finite products, kernels, and ...
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1answer
135 views

comparison between two monadic definitions for an operad

According to May, an operad $\mathcal{C}$ valued in sets is equivalent to a monad in Cat on the endofunctor $C\colon X\mapsto \coprod_i \mathcal{C}(i)\times X^i.$ According to Leinster, an operad is ...
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4answers
738 views

What other monoidal structures exist on the category of sets?

I know of the following monoidal structures over $\mathbf{Set}$ (taken from here): The Cartesian product $\otimes=\times$ (categorical product) The disjoint union $\otimes=+$ (categorial coproduct) ...
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1answer
175 views

A Criterion for a morphism to be a counit of an Adjunction

Suppose we have two functors $F:C\leftrightarrow D:G$ and a morphism $\varepsilon:FG\rightarrow\operatorname{Id}_D$. I am looking for a way to check whether $\varepsilon$ is the counit of an ...