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Jul
30
comment Why are Delta-generated spaces locally presentable?
It appears to me that indiscrete spaces of cardinality $\le \mathfrak{c}$, where $\mathfrak{c}$ is the cardinality of the continuum, can be obtained as quotients of $\Delta^1$. Thus we can construct an increasing sequence of ($\Delta$-generated) subspaces of cardinality $< \mathfrak{c}$ of a $\Delta$-generated space of cardinality $\mathfrak{c}$ whose union is the whole space. In particular, the presentability rank of a simplex is at least $\mathfrak{c}$.
Jul
28
comment Does every Lawvere theory arise in this way?
@goblin There's a problem with your question, though – the Lawvere theory of boolean algebras is not generated by $2$ as a boolean algebra but rather $2$ as a set (as you say). This is in contrast to vector spaces; after all, the opposite of the category of finite boolean algebras is the category of finite sets, whereas the opposite of the category of f.d. vector spaces is itself.
Jul
28
comment Does every Lawvere theory arise in this way?
What the two examples have in common is a good theory of dualisation. After all, the opposite of any Lawvere theory embeds in the category of algebras in a canonical way.
Jul
22
comment “Generalized theory of polynomials” for a given commutative Lawvere Theory
Yes, that is the procedure I am thinking of. It is a fact that the category of abelian group objects in the category of algebras for a Lawvere theory is itself equivalent to the category of algebras for a Lawvere theory (and, in particular, equivalent to the category of modules for a ring).
Jul
22
comment “Generalized theory of polynomials” for a given commutative Lawvere Theory
There is a general procedure that derives the theory of $R$-modules from the theory of commutative $R$-algebras, but I do not know of a way to do the reverse.
Jul
18
comment Exponential locales and a pointless version of the compact-open topology?
So you know, for instance, that there is a notion of locally compact locale, and that these are precisely the exponentiable locales?
Jul
18
comment Exponential locales and a pointless version of the compact-open topology?
Have you tried looking up exponentiable locales?
Jul
15
comment Stabilization of a generic pointed model category
Yes, Quillen's original definition only calls for finite limits and finite colimits. But I also have my own reasons for thinking about small model categories and universe enlargement...
Jul
14
comment Stabilization of a generic pointed model category
One can perfectly well have small model categories. There are no interesting small combinatorial model categories, however.
Jul
14
comment Natural isomorphisms: what is the status now of “the Eilenberg/Mac Lane Thesis”?
Sometimes, though, it turns out that naturality with respect to isomorphisms is trivial; for instance, the only isomorphisms in $\mathbf{\Delta}$ are the identities. I suppose this is no different to by-accident natural transformations...
Jul
13
answered Equivalent definition of a Kan fibration
Jul
12
comment $(\infty,1)$-categories and model categories
I would speculate that one might have to restrict to combinatorial model categories to get a fully faithful embedding.
Jul
6
answered What is descent data (of higher categories), conceptually?
Jul
6
comment What is descent data (of higher categories), conceptually?
I'm not sure I agree with your description of the category of descent data as a (homotopy) fibre product – you surely need to involve $\mathcal{F} (U \times_X U \times_X U)$.
Jul
5
comment a topos without an underlying boolean topos
What is an "underlying" boolean topos? Every topos has a boolean cover. Incidentally, $[\mathbf{Set}, \mathbf{FinSet}]$ is a topos.
Jul
4
comment Is the $\infty$-category of presentable $\infty$-categories presentable?
@YonatanHarpaz I would rather think of the $(\infty, 2)$-category of $(\infty, 1)$-categories as the archetypical $(\infty, 2)$-topos, just as the 1-category of 0-categories is the archetypical 1-topos.
Jul
3
comment Does it require Reedy fibrancy when we want the totalization to be weakly equivalent to the homotopy limit?
I expect that it is a typographical error.
Jul
1
comment Explicit computation of a limit of a cosimplicial object
Yes, you can make $F_*$ into a simplicial algebra that way, but do you still mean $\varprojlim_{\mathbf{\Delta}^\mathrm{op}} F_*$? There is a very easy formula for that, namely $F_0$.
Jul
1
comment Explicit computation of a limit of a cosimplicial object
Do you mean $(F_*, d, s)$ is a cosimplicial algebra? Do you mean $\varprojlim_{\mathbf{\Delta}} F_*$?
Jun
30
comment What is the most transparent, rigorous definition of the Univalence Axiom?
You really must understand type theory first – at minimum the notion of identity types. I think it would be enough to read Chapter 1 of the HoTT book, then skip to §2.10.