22,281 research outputs found
Free higher groups in homotopy type theory
Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the higher inductive type F (A)with constructors unit: F(A),cons : A → F(A) → F(A), and conditions saying that every cons(a)is an auto-equivalence on F(A). Equivalently, we can take the loop space of the suspension of A + 1. Assuming that A is a set (i.e. satisfies the principle of unique identity proofs), we are interested in the question whether F(A) is a set as well, which is very much related to an open problem in the HoTT book [20, Ex. 8.2]. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that ∥F(A)∥1 is a set
Rational homotopy theory and differential graded category
We propose a generalization of Sullivan's de Rham homotopy theory to
non-simply connected spaces. The formulation is such that the real homotopy
type of a manifold should be the closed tensor dg-category of flat bundles on
it much the same as the real homotopy type of a simply connected manifold is
the de Rham algebra in original Sullivan's theory. We prove the existence of a
model category structure on the category of small closed tensor dg-categories
and as a most simple case, confirm an equivalence between the homotopy category
of spaces whose fundamental groups are finite and whose higher homotopy groups
are finite dimensional rational vector spaces and the homotopy category of
small closed tensor dg-categories satisfying certain conditions.Comment: 28pages, revised version, title changed, to appear in JPA
Free higher groups in homotopy type theory
Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the higher inductive type F (A)with constructors unit: F(A),cons : A → F(A) → F(A), and conditions saying that every cons(a)is an auto-equivalence on F(A). Equivalently, we can take the loop space of the suspension of A + 1. Assuming that A is a set (i.e. satisfies the principle of unique identity proofs), we are interested in the question whether F(A) is a set as well, which is very much related to an open problem in the HoTT book [20, Ex. 8.2]. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that ∥F(A)∥1 is a set
A Rewriting Coherence Theorem with Applications in Homotopy Type Theory
Higher-dimensional rewriting systems are tools to analyse the structure of
formally reducing terms to normal forms, as well as comparing the different
reduction paths that lead to those normal forms. This higher structure can be
captured by finding a homotopy basis for the rewriting system. We show that the
basic notions of confluence and wellfoundedness are sufficient to recursively
build such a homotopy basis, with a construction reminiscent of an argument by
Craig C. Squier. We then go on to translate this construction to the setting of
homotopy type theory, where managing equalities between paths is important in
order to construct functions which are coherent with respect to higher
dimensions. Eventually, we apply the result to approximate a series of open
questions in homotopy type theory, such as the characterisation of the homotopy
groups of the free group on a set and the pushout of 1-types.
This paper expands on our previous conference contribution "Coherence via
Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the
language of higher-dimensional rewriting.Comment: 30 pages. arXiv admin note: text overlap with arXiv:2001.0765
A rewriting coherence theorem with applications in homotopy type theory
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution Coherence via Wellfoundedness by laying out the construction in the language of higher-dimensional rewriting
The relationship between the diagonal and the bar constructions on a bisimplicial set
AbstractThe aim of this paper is to prove that the homotopy type of any bisimplicial set X is modelled by the simplicial set W¯X, the bar construction on X. We stress the interest of this result by showing two relevant theorems which now become simple instances of it; namely, the Homotopy colimit theorem of Thomason, for diagrams of small categories, and the generalized Eilenberg–Zilber theorem of Dold–Puppe for bisimplicial Abelian groups. Among other applications, we give an algebraic model for the homotopy theory of (not necessarily path-connected) spaces whose homotopy groups vanish in degree 4 and higher
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