homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
In abelian categories one talks of chain complexes; and in that context a composable pair $A \to B \to C$ is null iff $B \to C$ factors through the cokernel $B/(A)$:
and so forth. In a strict context, the factorization is unique.
In a pointed (∞,1)-category with (∞,1)-colimits of small 1-truncated diagrams, one may still consider factorizations through cofibers: $A \to B \to C \sim * : A \to C$ but now there is a choice to make, roughly parametrized by an action of $Map_* (\Sigma A, C)$. This leads to interesting structure, describing (with upper bounds!) how trivially a particular sequence of arrows may compose.
To begin, consider a sequence of maps $A_0 \to A_1 \to A_2 \to A_3$. If the composites $A_0 \to A_2$ and $A_1\to A_3$ are nulhomotopic, then one has a diagram
any choice of homotopies in the two squares gives a map $\Sigma A_0 \to A_3$.
Define $C$ and $D$ to be the cofibers of $A_0 \to A_1$ and $A_1\to A_2$, respectively. A choice of homotopy $A_0 \to A_2 \sim 0$ corresponds to a choice of factorization $A_1 \to C \to A_2$, which gives a diagram of pushout squares
It is to be noted that the map $\Sigma A_0 \to D$ and possibly the object $C'$ depend on the choice of factor $C \to A_2$, but that $A_2 \to D$ does not, in any meaningful sense, so depend: this is just the structure map of the cofiber of $A_1\to A_2$. Note that the cofiber $C'$ of $C\to A_2$ is thus equivalent to that of $\Sigma A_0 \to D$; but again the role of choices must be studied.
A sequence of maps $A_0 \to A_1 \to \cdots \to A_n$ will be called a bracket sequence (a novel phrase for the purposes of this entry) in either of two cases:
In all cases, a bracket sequence leads to a three-map sequence
in which consecutive maps compose trivially, and so there are induced choices of maps
The collection of all such maps, taking all compatible variations, is the Toda Bracket of the bracket sequence.
Among the bracket sequences, a particular family arises which here will be called null-bracket (again, a novel phrase). A sequence will be called null-bracket if
If the Toda bracket for a bracket sequence includes the trivial map $\Sigma^{m+1} A_0 \to A_{m+3}$ then the sequence is null-bracket.
By definition, if a sequence is a bracket sequence AND NOT a null-bracket sequence, it follows that all the relevant maps $\Sigma^{k} A_0 \to A_n$ are nontrivial. Things like these Toda brackets have been studied by many (FIXME: referrences later) and especially the length-three brackets used by H. Toda to describe most of $\pi_k \mathbb{S}^n$ for $k \lt 31$ or so.
In (Cohen, 1968) is given a criterion for stable maps of spheres to inhabit non-null Toda brackets; this turns out to be most of $\pi_* \mathbb{S}$, and furthermore the maps in the bracket sequences can be chosen from a very small set (_FIXME_: be more precise! degree maps $n \iota$, Hopf maps $\eta, \theta,\sigma$, and $\alpha_p$… )
Stanley Kochmann, section 5.7 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Hans-Joachim Baues, On the cohomology of categories, universal Toda brackets and homotopy pairs, K-Theory 11:3, April 1997, pp. 259-285 (27) springer
Boryana Dimitrova, Universal Toda brackets of commutative ring spectra, poster, Bonn 2010, pdf
C. Roitzheim, S. Whitehouse, Uniqueness of $A_\infty$-structures and Hochschild cohomology, arxiv/0909.3222
Steffen Sagave, Universal Toda brackets of ring spectra, Trans. Amer. Math. Soc., 360(5):2767-2808, 2008, math.KT/0611808